Chapter 2. Cluster Melting
|
|
- Ethan Page
- 5 years ago
- Views:
Transcription
1 Chapter 2 Cluster Melting A number of theoretical studies, largely by Berry and coworkers [12], showed in the 1980s, that finite clusters undergo an analogue of the solid-liquid melting phase transition. This change in the phase can be studied through a number of different indicators. All these attempt to compare properties of the system on either side of this finite-size first-order phase transition and we discuss the most commonly used among these in the next section. The Lindemann index which examines fluctuations in the interatomic distances is a measure that was originally [120] derived for bulk materials, but which also indexes the finite-size melting transition very well. The caloric curve [11, 97] and the specific heat are other similar indicators, adapted from analogous quantities used in the study of melting behaviour of bulk materials. These and other microscopic measures are discussed in the next section and their utility in the study of finite clusters is illustrated. The study of the thermal stability and the melting transition of clusters is an active field of research as increasingly finer experiments have become possible. It is possible to measure melting temperatures using the temperature dependence of fragmented mass spectra [172, 173]. A clear understanding of this phenomenon will help in the development of ideas concerning macroscopic phase transitions as well. 17
2 -}~ o :~ o j&;m~ o Iterations number Figure 2.1: The fluctuation parameter f as a function of time at different temperatures. The curves as obtained seems to indicate that the Ar7 cluster does not melt but fluctuates among its many isomers, even after the temperature where the Lindemann index shows appreciable bond length fluctuations. 2.1 Indicators of the Melting transition The caloric curve is a plot of the temperature T versus the total energy E. The temperature T is defined in terms of the average kinetic energy per freedom, 2 T = (3N _ 6) kb (E k (2.1) ) where kb is the Boltzmann constant. The Lindemann parameter is essentially the fluctuation 8 in the interatomic separations, and this determines whether the cluster is in a "solid" or "liquid" phase. This quantity is defined as (2.2) At the cluster melting transition, the caloric curve mayor may not show a prominent bend but usually the Lindemann index undergoes a sharp increase, exceeding the value of 0.1 which is taken as an indication of melting [31]. This is a criterion derived from bulk phenomena, and thus may not necessarily be 18
3 -~~ ~ M~h ~~~~!~~M~M~"i ~ o ~hm~~~ o ~' 0,06 ~ I 0.5 OL ~------~----~ o Iterations Figure 2.2: The fluctuation graph at various temperatures for a generic 7 atom transition metal cluster. The three different curves reflects that the cluster has already melted at the reduced temp true for cluster systems [10, 149, 185]. Hence it becomes necessary to monitor other quantities which may help to index melting behaviour more accurately. The specific heat capacity C in molecular dynamics (MD) simulations for an N atom cluster is defined as ~ = [N - N(l- 2 )(K)(K-1)t 1 NkB 3N - 6 (2.3) where (... ) denotes a time average. In Monte Carlo (MC) simulations the specific heat is evaluated as (2.4) with (... ) denoting configuration average. Another important indicator for understanding the melting behaviour is the mean-square displacement curve [185]. (2.5) 19
4 ':~ :~C:::=:J o :~E,~I,=-=;-] o Iterations number Figure 2.3: The fluctuation parameter f at various temperatures for the (C 6o h cluster. The cluster seems to be fluctuating vigorously around 400K and at 900K when the cluster has already melted as per the peak in the specific heat capacity, the cluster fluctuates more vigorously among the various higher energy minima. where nt is the number of different time origins toj selected. The mean square displacement gives a higher slope when the cluster has melted than when it is in a solid state, nt being the number of different time origins. The diffusion coefficient is directly related to the mean squared displacement 'curve and is.. given as (2.6) The mean square displacement can also be used to track the trajectories of independent atoms and thereby one can gain information on the cluster dynamics in greater detail. This may be particularly useful in understanding surface effects and surface melting behaviour of clusters in mixed species as well. Cheng et ai., [31] studied the dynamical properties of argon clusters "layer by layer" to establish the relation between surface melting, surface atoms and core atoms: well below the bulk melting temperature, the cluster surface becomes soft and exhibits well-defined diffusion constants while the core remains rigid.'
5 lrj C"<)... (), t J:: ~ o r- o K K K o~----~----~------~----~----~ o Iterations number Figure 2.4: The fluctuation parameter f for the Arl3 cluster, showing that the cluster actually melts. At temperatures around 34K the cluster switches back and forth among its many higher energy minima, liquid like states. The fluctuation index f measures how the atomic structure of the cluster changes over a period of time at different temperatures. f =!IIA - Aol12 (2.7) 2 A is the adjacency matrix corresponding to the structure present during the temporal evolution, A.. _ {I, for Iri - rjl < rn ZJ -. 0, ot h (2.8) erwise. The initial adjacency matrix Ao corresponds to the lowest energy configuration, while r n are the average interatomic distances in the ground state 21
6 ~J n : {; OOK 1 o ~----~ ~ ~ -7ooK 2... OL-~~----~~~--~~----~--~ o ~ ~ _ Figure 2.5: The fluctuation parameter f at various temperatures for the (C 6o h3 cluster. The cluster still seems to be fluctuating even at 900K when one or more molecules may have sublimed. configuration ofthe cluster. At low temperatures, the two adjacency matrices A and Ao usually have the same elements and hence f ~ o. At higher temperatures prior to melting, the cluster begins to fluctuate between its many higher energy minima and the ground state. After the cluster has melted, the nature of the curve changes appreciably and the cluster now moves among the many higher energy liquid-like minima [10, 11, 97]. The fluctuation parameter f behaves very differently in different cluster systems as can be seen in Figs where data is presented for 7 atom rare-gas, 13 atom rare-gas, metal and fullerene clusters respectively. The 7 atom rare gas cluster does not really melt, even though the Lindemann index crosses the value 0.1: as can be seen in Fig. 2.1, f keeps oscillating between o and 2 even at the highest temperature indicating that the cluster returns to its ground state geometry periodically. For the metal cluster, there is clear evidence of melting above the reduced temperature of '0.06, while for the 7 fullerene cluster, again there is melting, as also for the 13 atom Argon cluster. The 13 fullerene cluster probably sublimes. 22
7 Typically, three indicators are adequate to estimate the range of temperature over which the cluster can be said to melt [3]. Other indicators have also been studied. Lee et at., [114] examined the effect of melting on the potential energy distributions and they were consistently able to explain the non-monotonic variation of melting temperature with size. the dependence of melting, boiling and sublimation on the interatomic potentials. the existence of surface melted phase and the absence of a pre-melting peak in the heat capacity curves. They also identified a ne\\' type of premelting mechanism in the double icosahedral Pd 19 clusters where one of the two internal atoms' escape to the surface at the premelting temperature. In small clusters a remarkable phenomenon observed is phase co-existence, "'hereby the system fluctuates dynamically between its different phases. Coexistence requires that the timescales on which the fluctuation takes place to be smaller than the timescales in which the dynamics takes place on the potential energy surface. It appears that the change from solid phase to the liquid phase is mediated by intermediate phase changes, depending on whether the cluster has fully melted or not, characterized by increasing fluidity. In finite systems the different indicators may not coincide, the system being far away from the thermodynamic limit and also the various order parameters probe different aspects and regions of phase space. It is again widely acknowledged that the solid-liquid co-existence requires a gap between the quenched energy states. In mixed clusters such a co-existence can exist even in the absence of such gap. The timescales for fluctuating between the different phases being effectively reduced. While different phases can coexist at the melting temperature a necessary condition for dynamical coexistence is that each of the phases has a long persistence time, long enough to establish equilibrium properties characteristic of those phases and properties of which can be experimentally be determined [15]. There is nothing to prevent the co-existence of more than two phases, so ensembles of clusters may exist in two or more phase-like forms, the distinction between 'phase' and 'component' having been lost for clusters and the probability of finding the cluster in an intermediate state being small [15, 16]. There can also be instances when there is a negative specific heat capacity in the microcanonical molecular dynamics simulations. This was earlier theoretically predicted using simple models by considering the distribution of local minima which characterize the potential energy landscape [19]. In this study it was found that for high values of the parameter involving the 23
8 ratios of the vibrational energy of the global minimum isomer and the local minima, the caloric curve showed an S-bend, with a negative heat capacity in the vicinity of the melting point [19]. Recently experimental results showed a negative heat capacity for a 147 atom sodium cluster [79, 173]. The negative heat capacity was explained by assuming that the cluster avoids the melting states and that the excess kinetic energy gained is converted into potential energy [173]. However ab-initio methods based on first principles have not hinted at the existence of negative heat capacity. This may be due to the short simulation time [1, 147]. To find out whether the cluster has actually melted or evaporated, one should rather analyze the structure, take snapshots and study the interatomic distances [30]. This can be used to identify and discriminate between melting and evaporation or sublimation. Normally during evaporation or sublimation one or more atoms undergo large displacements. From a purely analytical point of view though, a complete picture of melting requires estimation of the free energies G(P, T, N) in the solid and liquid phase; at the transition these should match. On the other hand, from a purely dynamical point of "iew at the melting transition, one may anticipate that the dynamics becomes unstable [130, 151]. One method of detecting such a change is to examine the eigenvalues of the Hessian matrix: when these are negative, the associated mode is unstable [23, 184]. Such behaviour can be studied through the computation of the instantaneous normal modes: this is described in the following section. 2.2 Instantaneous Normal Modes The concept of instantaneous normal modes (INMs), namely that it is possible to define vibrational modes even for the liquid state, was initially developed by Keyes and co-workers [102]. Although normal modes of vibration can only be properly defined for the solid state [207] it is possible to compute the normal modes for each atomic configuration in the liquid state and consider the distribution of these INM frequencies [184]. These ideas have found interesting applications in studies on clusters, liquids and supercooled liquids. It can be argued that diffusion phenomena are related to negative eigenvalues in the INM spectrum [76, 112, 118] and correspond to the anharmonicities of the well. Simulation of supercooled liquids indicate that the '24
9 '8 8 8' 4 o ;-- E 8.!:! o T(oK) Figure 2.6: The adiabatic curves from MC and lvid simulations. They do not show any mode softening (crossing of the zero axis) over the temperature range where other indicators may suggest melting of the Ar7 cluster. frequency in diffusive directions tends to zero near the mode-coupling temperature [49, 148]. It has also been suggested [210] that barrier heights and hopping rates can be estimated from INM frequencies. Here we study the behaviour of INM spectra in a variety of cluster systems. These include the heterogeneous Ar9XelO cluster [93], the homogeneous Arlg, Ar, and Ar12 clusters as well as (C 60 )?, (C 60 h3 and a generic 7 atom transition metal cluster modeled by the Gupta potential [84]. To obtain INM frequencies, the potential is approximated at each instant in time as a quadratic function of coordinates, where Rtis the configuration at time t, expanded around the reference configuration, Ro. The instantaneous force matrix is F, while D is the Hessian matrix, namely the second derivative of the many body potential energy 25
10 4~------~----~~-- --~--~--, 3 ~-.. 2 '5 1 ~ -1-20L ~--~--~~--~--~~--~~~ 4~------~--~--~~~ ~,_ ~------~--~--~------~--~~ o T(~ Figure 2.7: The lowest frequency modes of the (C 6o h cluster softens as the cluster goes through the melting phase change. surface, V(R), a 2 v Djll,kll(Ro) = a a IRa' (2.10) rjll' rkll Clearly this approximation will be valid only for short times, but within this limitation, the (3N) x (3N) matrix of eigenvectors Ua,jll(Ro), of D(Ro) defines a unique set of "harmonic" modes. The eigenvalues of D give the squares of the normal mode frequencies, Aa = mu-';' = L Ua,jll(Ro)Djll,kll(Ro)UL,a(Ro) a == 1,...,3N. (2.11) j Il,kll INM spectra are obtained by accumulating the frequencies Wo. during a constant energy simulation, namely along a trajectory R t, and obtaining the 26
11 E 0 ' ' T(oK) Figure 2.8: The mode-softening for the (C 6o h3 duster as obtained from the Monte-Carlo and molecular dynamics results. The softening of many modes hints at sublimation even at low temperatures. distribution, P(w)dw _ Probability that an INM frequency Wa has a value between wand u.; + dw. (2.12) Quantities such as the Einstein frequency, the fraction of imaginary modes, etc. can be deduced from this distribution [184]. Note that since the squares of the frequencies are the eigenvalues of the Hessian, these can be negative. Such modes are unstable, having imaginary frequencies; traditionally, these are represented on the negative w axis. Three translational modes have frequency zero. These are usually not included in the analysis, as are the frequencies of the three purely rotational modes [184]. INMs are related to short time dynamics [108, 174, 183] and hence it is natural to consider these as being indicative of the melting process. The "universal" shape [33] for the density of states for all liquids (whether polar, nonpolar or atomic) is also typical of clusters. The \vide range in the melting temperature of different cluster types, for instance metallic versus nonmetallic, makes comparisons less easy and correspondence cannot be hence properly ascertained. On the other hand, the mean field theory [112] 27
12 's -1~~--~~--~--~~--~~--~~ o 0.Q1 0.Q 's 1-1~----~----~----~------~--~ o T('K) Figure 2.9: The adiabatic curves from Me and MD simulations for a generic 7 atom transition metal cluster. suggests that since mass differences may not play a major role, both the homogeneous and heterogeneous clusters may show similar INM spectra at least to a first approximation. Mass differences affect the dynamics in a different way in which the potential energy surface is explored and thereby alter the INM spectra. INM spectra [23, 184]' have been studied extensively in the past both within the classical dynamics [184] framework as well quantum mechanically [29]. In finite clusters the INM frequencies are well-separated and they do not mix owing to a consequence of a non-crossing rule, which applies to the Hessian matrix. The non-crossing rule due to [153] states that eigen-values 28
13 'e u ~ ~ ~ ~ ~~ o ~----~--~~--~----~----~----~--~ o T(K) Figure 2.10: MC and MD results for the adiabatic INM frequencies for the rare-gas cluster Ar12. corresponding to eigen-vectors of the same symmetry of the Hessian cannot cross[l11]. This allows us to define adiabatic INMs [40, 142] and study them at the cluster melting transition region. Specifically, during a MD (or MC) simulation, the eigenvalues of the Hessian matrix are evaluated at each step, and averages are computed over the different realizations of each eigenvalue. This is then converted into 29
14 an adiabatic normal mode frequency which can be studied as a function of temperature. 2.3 Applications Standard equilibrium molecular dynamics simulations of finite cluster systems are performed in a free volume. Interactions between the constituent atoms or molecules are the potentials discussed in Chapter 1; between raregas atoms, this is taken to be the usual Lennard-Jones form with parameters chosen appropriately, while for metal atoms we use the Gupta potential [84, 185]. For the Gupta potential the following values, p = giro, q = 3/ro and A = are found to be appropriate for the transition metals [185]. U is related to the bulk cohesive energy Ebulk = U which was kept at unity and ro was used as the unit of distance. For modeling (C 60 )N clusters the Girifalco potential [77] was used. For the mixed clusters which we study here in Section the Xe-Ar and Ar-Ar Lennard-Jones parameters were obtained by the Lorentz-Berthelot mixing rules [3], namely where a and (3 indicate the atomic identities. The values used are given in Table 2.1. Table 2.1: Parameters used in this study. II II Parameter t/k a/a Ar-Ar Xe-Ar Xe-Xe II The total potential is thus a sum of two-body terms, V(rij) = 4<a-P [ (a;:p) l2 _ (a;i~p) '], (2.13) the distance between atoms i (which is of type a) and j (of type (3) is rij =1 I4 - R j I, the R's being the coordinates of the various atoms. 30
15 The velocity Verlet algorithm was used to solve the equations of motion. For an N atom cluster, the internal temperature T is the (per mode) average kinetic energy E k. All quantities are measured in reduced units. The time step in molecular dynamics simulations was fixed at 0.01 in reduced units and in Monte-Carlo simulations the total number of sweeps was 10 6 iterations after equilibrating for 10 5 iterations. The Boltzmann constant in reduced units was fixed at unity. Melting of homogeneous or single component clusters differ from those which are mixed. A single impurity can cause the cluster to have different thermodynamic behaviours and ground state geometries. The 13 atom cluster, both homogeneous [41, 69, 70, 97, 161, 187] and heterogeneous [61, 80, 82, 122, 190] has been the subject of many studies. Investigations carried out by Frantz [69] for the binary rare-gas cluster Ar13-nKrn for o ~ n ~ 13 suggest that homogeneous clusters exhibit "magic number" effects for many of their properties due to their largely icosahedral ground state configurations. Because Ar is only about 11% smaller in radius than Kr, all 13-atom Ar-Kr clusters have icosahedral-like configurations. This results in categorization of various topological isomers which are similar to their homogeneous counterparts and permutational isomers which are based on various re-arrangements of the component atoms within a topological form. Ne and Ar have different sizes ( Ne atom is about 19% smaller than Ar), resulting in many non-icosahedral isomers with energies similar to the icosahedral like isomers. The size differences and the intermolecular potential differences (cne-ne/car-ar = and car-ar/ckr-kr = ) result in Ne-Ar cluster configurations dramatically different from their Ar-Kr counterparts [69]. Heat capacity curves for the 13-atom homogeneous clusters are characterized by a prominent peak in the solid-liquid "phase" transition [41, 69, 70, 123, 161, 187]. This owes to the fact that a large energy gap exists between the ground state icosahedral isomer and the higher energy non-icosahedral isomers [41, 69, 161, 187]. The heat capacity curves for mixed Ar-Kr clusters is very similar to the homogeneous counterpart except that a very small peak occurs at very low temperatures and some minor variations owing to whether the cluster is Ar centric or Kr centric [69]. This feature is reminiscent of order-disorder transition known to occur in some bulk alloy materials [123]. For Ne-Ar clusters the heat capacity curves are qualitath ely different. Another difference between the homogeneous and heterogeneous clusters is 31
16 that the latter generally suffer from inadequate mixing of the different components and hence show different characteristic thermodynamic behaviours. This also results in difficulty in optimization for finding the ground state configurations; see Chapters 3-5. At low temperatures the competition between interatomic forces and atomic sizes affects the dynamics causing the cluster's behaviour to change arising from different geometries available. Quenching carried out by Frantz [69] at low temperatures reveals that for the 13 atom Ar-Kr cluster, the low energy isomers were dominated by icosahedral like structures with the smaller Ar atom at the center. For the Ke-Ar, this was not the case especially in Ne12Ar and NellAr2 as both had :\fe core icosahedral like structures as their lowest energy isomers, but for all the other clusters the Ne core configurations were absent. ArnXe13-n clusters have been studied by Nayak and Ramaswamy [150] and earlier Robertson and Brown [165] had compared the phase transition behaviour of mixed cluster to the homogeneous case. They found that the melting temperatures exhibits plateaus in the case of equal mixture of (Ar- Xeh. Temporal correlations studied by Nayak and Ramaswamy [150] in the Ar-Xe mixture for n = 1 and n = 12 revealed that the spectrum of potential energy fluctuations for individual atoms have a marked 1/ f character, depending upon whether the cluster was solid or liquid and whether the atom chosen was Ar or Xe. Shown in Figs are the adiabatic INMs curves showing the modesoftening behaviour and consistent with other indicators. Both the MC and MD show similar behaviour. The Ar7 cluster does not melt and there is no mode softening as observed, whereas Ar12 does and the modes soften in this case. In contrast the other 7 atom cluster for the transition metal and the (C 6o h cluster both melt and show mode softening. (C 6o h3 shows sublimation effects as a number of modes soften at low temperature. Interesting features such as mentioned above prompts us to study the behaviour of the INM spectra in the case of the 19 atom cluster (both mixed and homogeneous) at the melting phase transition. This larger cluster is convenient for study since the ground state geometry of the homogeneous entity has the double icosahedral motif. The somewhat larger size of the cluster also allows for a better understanding of surface effects. 32
17 2.3.1 Melting of the heterogeneous cluster Ar9XelO Phase change phenomena in finite clusters mirror analogous bulk phase transitions. The INM spectra of mixed clusters have been studied in some detail previously [69, 112] and effects that devolve from the relative sizes' of the constituent atoms are germane to this study. For instance, mixed Ar-Kr clusters have properties that are similar to the pure cases, while Ne-Ar clusters have a wide range of behaviour which is very distinct from that of the pure clusters [69]; furthermore the properties vary drastically with composition. For instance: Ne4Ar9 and Ne3ArlO are very similar, but their complements Ne9Ar4 and NelOAr3 differ greatly. A number of studies have focussed on the behaviour of mixed cluster species [61, 80, 82, 106, 122, 123, 150, 167, 190] in order to examine the effect of the heterogeneity on the melting process induced in such clusters and other factors which influence the cluster solid -t liquid transition. Mixed clusters (especially those with comparable numbers of different species) are glassy in the sense that there are a large number of degenerate or nearly degenerate configurations even in the solid state. This leads to some interesting cases such as making it more difficult to locate the global minimum as the number of associated configurations and isomers grow exponentially with the size of the cluster. Equilibrium properties may be prone to errors if there is inadequate mixing of the different components and this may lead to large systematic errors at low temperatures. : At the melting transition, there may be co-exisiting phases with approxirriately no energy gaps and this can happen even with a single impurity [130]. The ground state geometry of the 19 atom homogeneous rare-gas cluster is that of two interpenetrating icosahedra. The point group for this arrangement is D Sh ; the high symmetry and the compact geometry endow this "magic" number cluster with a somewhat higher stability than the 18 or 20 atom cases. The mixed cluster, Ar9XelO, however, does not have the DSh symmetry, although the same double-ico arrangement is maintained in the ground state (see Fig. 2.11). This feature plays a significant role in the nature of the normal modes of the cluster, which melts around 60K. Shown in Fig is the caloric curve for this system, along with the Lindemann index as a function of the internal temperature. In comparison, the pure Ar19 cluster (see Fig. 2.13) melts around at 30K [11]. 33
18 Figure 2.11: Axial and transverse views of the ground state configurations for the Ar9XelO (left) and Ar19 (right) clusters. Xe atoms are coloured red while thear atoms are white. The atoms are linked by bonds in order to show the ~ymmetry; both clusters share the double-icosahedral motif, though only the homogeneous cluster belongs to the DSh symmetry group. The high level of symmetry for the pure Ar19 cluster, however, causes the normal modes (at OK) to have a high degree of degeneracy. Analysis of the vibrations of the 19 atom cluster show that the normal modes span the representations 5A~ + A~ + 6E~ + 5E~ + A~ +4A~ + 5 E~ +4 E~. From this it is clear that at low temperatures, in the cold solid phase, the 51 modes of vibration must be distributed in sets which are compatible with the above symmetry classes. (Accordingly we see that the levels at OK have significant degeneracies; see Fig. 2.16). The INM spectra at low temperatures takes the familiar form shown in (see Fig. 2.14). After melting, when there are significant imaginary normal mode. frequencies, the INM spectra becomes broadened (see Fig. 2.14). and picks up a large tail, both in the 34
19 100 1-~ ~~_ _r.~1 80 r 1 ~ 60 r j ~ :: r... B) j o~ I SO -70 ElF. 0.3 ~ I i <Q 0.2\ 0.1! ol... I j I :J TtK) Figure 2.12: (a) The caloric curve and (b) the variation of the Lindemann parameter with temperature, for Ar9XelO' 30 i2' 20 l ~ 10 I ~----- o L ~. -...,.,..... :... -::.:... -~ ElF. a) 0.3 r--. :.::... ~ I' o~ ----~~~-=--=-~~ ~_b_)~ o TtK) Figure 2.13: (a) The caloric curve and (b) the variation of the Lindemann parameter with temperature, for Ar19 high frequency regime, as well as in the range of negative w [184]. Ar9XelO behaves differently. Even at low temperatures, the large number of nearly degenerate levels for the mixed cluster gives rise to an inher- 35
20 Figure 2.14: INM spectra at (a) 10K, (b) 20K and (c) 32K for the homogeneous cluster Ar19. The cluster melts at around 30K. ently broader distribution (see Fig. 2.15). This is typical of glassy systems, where recently Parisi and coworkers have argued that random matrix theory should be applicable in order to determine the nature of the frequency distribution [38]. Melting still implies the broadening of the distribution, but this is mainly into the negative frequency domain rather than into the high-frequency region; see Fig Since for a cluster, the different modes are well-separated, it is possible to compute an average frequency for each mode as a function of the temperature. However, the noncrossing rule that applies to the Hessian matrix says that levels with the same symmetry do not cross. Here, since all the levels have no symmetry, they essentially have the same trivial symmetry and therefore do not mix [40, 142]. We define the adiabatic frequency of a mode as the time average of the mode frequency. The variation of these frequencies with temperature for Ar19 andargxelo is shown in Fig and Fig respectively. It is clear that while these vary smoothly for the most part, melting is heralded by the softening of the lowest few vibrational modes whose frequencies decrease to zero and eventually become imaginary. As discussed in the previous Section this behaviour is typical of all clusters including Ar19 shown in Fig It is clear, however, that the high level of symmetry of Ar19 gives a different 36
21 '. n N o ~_--'-'--~-..L~ a) 10k "._ , , j 0.03 ~ 3. ~ 0.02 L t( 0.01 ~ o,--i, /""l"~,"~... ", " 'x.. _~_~~~ -20 o ,, ~ r 0.02 r- l om ~ L Oi -20 o rc----,-.-- " '"~''''' b) 40k,-~-.;~ ' c) 74k 80,, 1 80 Figure 2.15: INM Spectra for Ar9XelO at (a) 10K, (b) 40K, (c) 14K. Contrast the spectra at 10K with Fig for Ar19' At such low temperatures the different glassy configurations of nearly the same energy gh-e rise to the jagged distribution. degeneracy structure to the adiabatic frequencies as compared to the case of Ar9XelO' For mixed as well as for homogeneous clusters, it is the 1m-vest few modes that appear to initiate the melting process. Eigenvector analysis of these can be illustrative ; in the pure cluster case, these have been seen to be highly symmetric modes which are like shear motions (Fig. 2.18). In the mixed case, the symmetry of the mode is more difficult to ascertain. The lowest mode for Ar9XelO is shown in Fig. 9, where it can be seen that the mobility rests largely with the heavier Xe atoms. Viewed in conjunction with Fig which gives the ground state structure of the cluster, it is clear that the' cluster melts first at the surface, after which the inner atoms begin to move independently. 37
22 70 IF=========:''====-::-:==.- :~='::~:::.::::-,:,~ ~r..,.", I I 'I: (,) 30 S TtK) 35 Figure 2.16: Variation of the adiabatic frequencies with temperature for Ar19' Figure 2.17: Adiabatic frequencies and their variation with temperature for Ar9XelO. Comparison with the Ar19 case (Fig. 2.16) shows that Ar9XelO is essentially a disordered glass-'-like system which lacks the degeneracy structure of Ar19' 2.4 Summary \Ve have reexamined the nature of melting in pure and mixed clusters from the point of view of instantaneolis normal modes. The main application has 38.
23 0>=10.91 em-i Figure 2.18: Lowest frequency adiabatic mode for the cluster Ar19. The displacements of the different atoms is indicated by the arrows (the length is proportional to the amplitude). Since the atoms are identical, the symmetric displacement of all atoms is clearly seen. been to the study of a glassy 19 atom heterogeneous cluster; the underlying potential energy surface of such systems has a large number of nearly degenerate minima, corresponding to configurations that are very close in energy. This has an effect on the nature of the INM frequencies which tend to have a wider distribution than the case of homogeneous clusters. In general we have shown here that the melting process can be understood as a mode-softening process when the vibrational modes are examined in detail. As a function of the internal temperature, the lowest normal modes soften-the frequency decreases to zero, and eventually become imaginary. This corresponds to a mode becoming unstable, and this causes the cluster to melt. Analysis of the corresponding eigenvectors for the mixed cluster 39
24 0>=5.61 cm- 1 Figure 2.19: As in (Fig. 2.18) for the mixed cluster Ar9XelO. The Xe atom displacements can be seen to be larger than the Ar atom displacements; thus this mode corresponds to a surface melting as discussed in the text. studied in detail show that the atomic displacements are consistent with a surface melting picture, where the heavier Xe atoms first become more mobile and capable of large scale intra cluster motions. The adiabatic mode softening is observed to be common to all the clusters studied. The study of such adiabatic modes thus throws considerable light on the detailed mechanics of the melting process. Similar analysis will be very useful in examining other phase change phenomena and may help in providing a novel and useful order-parameters for classifying phase transitions in finite systems. 40
Javier Junquera. Statistical mechanics
Javier Junquera Statistical mechanics From the microscopic to the macroscopic level: the realm of statistical mechanics Computer simulations Thermodynamic state Generates information at the microscopic
More informationAdvanced sampling. fluids of strongly orientation-dependent interactions (e.g., dipoles, hydrogen bonds)
Advanced sampling ChE210D Today's lecture: methods for facilitating equilibration and sampling in complex, frustrated, or slow-evolving systems Difficult-to-simulate systems Practically speaking, one is
More informationExploring the energy landscape
Exploring the energy landscape ChE210D Today's lecture: what are general features of the potential energy surface and how can we locate and characterize minima on it Derivatives of the potential energy
More informationMD Thermodynamics. Lecture 12 3/26/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky
MD Thermodynamics Lecture 1 3/6/18 1 Molecular dynamics The force depends on positions only (not velocities) Total energy is conserved (micro canonical evolution) Newton s equations of motion (second order
More informationPressure Dependent Study of the Solid-Solid Phase Change in 38-Atom Lennard-Jones Cluster
University of Rhode Island DigitalCommons@URI Chemistry Faculty Publications Chemistry 2005 Pressure Dependent Study of the Solid-Solid Phase Change in 38-Atom Lennard-Jones Cluster Dubravko Sabo University
More informationChapter 2. Dielectric Theories
Chapter Dielectric Theories . Dielectric Theories 1.1. Introduction Measurements of dielectric properties of materials is very important because it provide vital information regarding the material characteristics,
More informationIntroduction Statistical Thermodynamics. Monday, January 6, 14
Introduction Statistical Thermodynamics 1 Molecular Simulations Molecular dynamics: solve equations of motion Monte Carlo: importance sampling r 1 r 2 r n MD MC r 1 r 2 2 r n 2 3 3 4 4 Questions How can
More informationDynamics of melting argon clusters
PHYSICAL REVIEW E VOLUME 55, NUMBER 1 JANUARY 1997 Dynamics of melting argon clusters Gayle M. Tanner, 1,2 Aniket Bhattacharya, 1 Saroj K. Nayak, 3 and S. D. Mahanti 1 1 Department of Physics, Michigan
More informationChapter 10. Lesson Starter. Why did you not smell the odor of the vapor immediately? Explain this event in terms of the motion of molecules.
Preview Lesson Starter Objectives The Kinetic-Molecular Theory of Gases The Kinetic-Molecular Theory and the Nature of Gases Deviations of Real Gases from Ideal Behavior Section 1 The Kinetic-Molecular
More informationQuantum Disordering versus Melting in Lennard-Jones Clusters.
Quantum Disordering versus elting in ennard-jones Clusters. Jason Deckman and Vladimir A. andelshtam Chemistry Department, University of California at Irvine, Irvine, CA 92697, USA (Dated: December 7,
More informationMatSci 331 Homework 4 Molecular Dynamics and Monte Carlo: Stress, heat capacity, quantum nuclear effects, and simulated annealing
MatSci 331 Homework 4 Molecular Dynamics and Monte Carlo: Stress, heat capacity, quantum nuclear effects, and simulated annealing Due Thursday Feb. 21 at 5pm in Durand 110. Evan Reed In this homework,
More informationExample questions for Molecular modelling (Level 4) Dr. Adrian Mulholland
Example questions for Molecular modelling (Level 4) Dr. Adrian Mulholland 1) Question. Two methods which are widely used for the optimization of molecular geometies are the Steepest descents and Newton-Raphson
More information510 Subject Index. Hamiltonian 33, 86, 88, 89 Hamilton operator 34, 164, 166
Subject Index Ab-initio calculation 24, 122, 161. 165 Acentric factor 279, 338 Activity absolute 258, 295 coefficient 7 definition 7 Atom 23 Atomic units 93 Avogadro number 5, 92 Axilrod-Teller-forces
More informationChapter 10 Liquids, Solids, and Intermolecular Forces
Chapter 10 Liquids, Solids, and Intermolecular Forces The Three Phases of Matter (A Macroscopic Comparison) State of Matter Shape and volume Compressibility Ability to Flow Solid Retains its own shape
More informationMME 2010 METALLURGICAL THERMODYNAMICS II. Fundamentals of Thermodynamics for Systems of Constant Composition
MME 2010 METALLURGICAL THERMODYNAMICS II Fundamentals of Thermodynamics for Systems of Constant Composition Thermodynamics addresses two types of problems: 1- Computation of energy difference between two
More informationPre-yield non-affine fluctuations and a hidden critical point in strained crystals
Supplementary Information for: Pre-yield non-affine fluctuations and a hidden critical point in strained crystals Tamoghna Das, a,b Saswati Ganguly, b Surajit Sengupta c and Madan Rao d a Collective Interactions
More informationBasics of Statistical Mechanics
Basics of Statistical Mechanics Review of ensembles Microcanonical, canonical, Maxwell-Boltzmann Constant pressure, temperature, volume, Thermodynamic limit Ergodicity (see online notes also) Reading assignment:
More informationMonte Carlo. Lecture 15 4/9/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky
Monte Carlo Lecture 15 4/9/18 1 Sampling with dynamics In Molecular Dynamics we simulate evolution of a system over time according to Newton s equations, conserving energy Averages (thermodynamic properties)
More informationClassical Theory of Harmonic Crystals
Classical Theory of Harmonic Crystals HARMONIC APPROXIMATION The Hamiltonian of the crystal is expressed in terms of the kinetic energies of atoms and the potential energy. In calculating the potential
More informationFinite Ring Geometries and Role of Coupling in Molecular Dynamics and Chemistry
Finite Ring Geometries and Role of Coupling in Molecular Dynamics and Chemistry Petr Pracna J. Heyrovský Institute of Physical Chemistry Academy of Sciences of the Czech Republic, Prague ZiF Cooperation
More informationDynamics of Supercooled Liquids The Generic Phase Diagram for Glasses
Dynamics of Supercooled Liquids The Generic Phase Diagram for Glasses A normal liquid will crystallize at a melting temperature T m as it is cooled via a first-order phase transition (see figure above).
More informationThermal fluctuations, mechanical response, and hyperuniformity in jammed solids
Thermal fluctuations, mechanical response, and hyperuniformity in jammed solids Atsushi Ikeda Fukui Institute for Fundamental Chemistry, Kyoto University Atsushi Ikeda & Ludovic Berthier Phys. Rev. E 92,
More information12. Spectral diffusion
1. Spectral diffusion 1.1. Spectral diffusion, Two-Level Systems Until now, we have supposed that the optical transition frequency of each single molecule is a constant (except when we considered its variation
More informationIntensities and rates in the spectral domain without eigenvectors.
UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN: DEPARTMENT OF CHEMISTRY Intensities and rates in the spectral domain without eigenvectors. By: Dr. Martin Gruebele Authors: Brian Nguyen and Drishti Guin 12/10/2013
More informationRelevance of jamming to the mechanical properties of solids Sidney Nagel University of Chicago Capri; September 12, 2014
Relevance of jamming to the mechanical properties of solids Sidney Nagel University of Chicago Capri; September 1, 014 What is role of (dis)order for mechanical behavior? Andrea J. Liu Carl Goodrich Justin
More informationClassification of Solids, Fermi Level and Conductivity in Metals Dr. Anurag Srivastava
Classification of Solids, Fermi Level and Conductivity in Metals Dr. Anurag Srivastava Web address: http://tiiciiitm.com/profanurag Email: profanurag@gmail.com Visit me: Room-110, Block-E, IIITM Campus
More informationAP* Chapter 10. Liquids and Solids. Friday, November 22, 13
AP* Chapter 10 Liquids and Solids AP Learning Objectives LO 1.11 The student can analyze data, based on periodicity and the properties of binary compounds, to identify patterns and generate hypotheses
More informationChapter 10: Liquids, Solids, and Phase Changes
Chapter 10: Liquids, Solids, and Phase Changes In-chapter exercises: 10.1 10.6, 10.11; End-of-chapter Problems: 10.26, 10.31, 10.32, 10.33, 10.34, 10.35, 10.36, 10.39, 10.40, 10.42, 10.44, 10.45, 10.66,
More informationStructural probe of a glass-forming liquid: Generalized compressibility
Structural probe of a glass-forming liquid: Generalized compressibility Hervé M. Carruzzo* and Clare C. Yu Department of Physics and Astronomy, University of California Irvine, Irvine, California 92697
More informationA Study of the Thermal Properties of a One. Dimensional Lennard-Jones System
A Study of the Thermal Properties of a One Dimensional Lennard-Jones System Abstract In this study, the behavior of a one dimensional (1D) Lennard-Jones (LJ) system is simulated. As part of this research,
More informationStructural Signatures of Mobility in Jammed and Glassy Systems
Lisa Manning Sam Schoenholz Ekin Dogus Cubuk Brad Malone Tim Kaxiras Joerg Rottler Rob Riggleman Jennifer Rieser Doug Durian Daniel Sussman Carl Goodrich Sid Nagel Structural Signatures of Mobility in
More informationPhysics 53. Thermal Physics 1. Statistics are like a bikini. What they reveal is suggestive; what they conceal is vital.
Physics 53 Thermal Physics 1 Statistics are like a bikini. What they reveal is suggestive; what they conceal is vital. Arthur Koestler Overview In the following sections we will treat macroscopic systems
More informationKINETICE THEROY OF GASES
INTRODUCTION: Kinetic theory of gases relates the macroscopic properties of gases (like pressure, temperature, volume... etc) to the microscopic properties of the gas molecules (like speed, momentum, kinetic
More informationContents. 1 Introduction and guide for this text 1. 2 Equilibrium and entropy 6. 3 Energy and how the microscopic world works 21
Preface Reference tables Table A Counting and combinatorics formulae Table B Useful integrals, expansions, and approximations Table C Extensive thermodynamic potentials Table D Intensive per-particle thermodynamic
More informationOrder parameter by instantaneous normal mode analysis for melting behaviour of cluster Ag 17. Cu 2. Journal of Physics: Conference Series
Journal of Physics: Conference Series OPEN ACCESS Order parameter by instantaneous normal mode analysis for melting behaviour of cluster Ag 7 Cu To cite this article: Pin-Han Tang et al 03 J. Phys.: Conf.
More informationTB [103 marks] The damping of the system is now increased. Which describes the change in ƒ 0 and the change in A 0?
TB [103 marks] 1. A periodic driving force of frequency ƒ acts on a system which undergoes forced oscillations of amplitude A. The graph below shows the variation with ƒ of A. The maximum amplitude A 0
More informationComputer simulation methods (1) Dr. Vania Calandrini
Computer simulation methods (1) Dr. Vania Calandrini Why computational methods To understand and predict the properties of complex systems (many degrees of freedom): liquids, solids, adsorption of molecules
More informationTitle Theory of solutions in the energy r of the molecular flexibility Author(s) Matubayasi, N; Nakahara, M Citation JOURNAL OF CHEMICAL PHYSICS (2003), 9702 Issue Date 2003-11-08 URL http://hdl.handle.net/2433/50354
More informationLecture 20: Spinodals and Binodals; Continuous Phase Transitions; Introduction to Statistical Mechanics
Lecture 20: 11.28.05 Spinodals and Binodals; Continuous Phase Transitions; Introduction to Statistical Mechanics Today: LAST TIME: DEFINING METASTABLE AND UNSTABLE REGIONS ON PHASE DIAGRAMS...2 Conditions
More informationPhase transition phenomena of statistical mechanical models of the integer factorization problem (submitted to JPSJ, now in review process)
Phase transition phenomena of statistical mechanical models of the integer factorization problem (submitted to JPSJ, now in review process) Chihiro Nakajima WPI-AIMR, Tohoku University Masayuki Ohzeki
More informationCan imaginary instantaneous normal mode frequencies predict barriers to self-diffusion?
Can imaginary instantaneous normal mode frequencies predict barriers to self-diffusion? J. Daniel Gezelter, Eran Rabani, and B. J. Berne Department of Chemistry and Center for Biomolecular Simulation,
More information3. General properties of phase transitions and the Landau theory
3. General properties of phase transitions and the Landau theory In this Section we review the general properties and the terminology used to characterise phase transitions, which you will have already
More informationThe Harmonic Oscillator: Zero Point Energy and Tunneling
The Harmonic Oscillator: Zero Point Energy and Tunneling Lecture Objectives: 1. To introduce simple harmonic oscillator model using elementary classical mechanics.. To write down the Schrodinger equation
More informationMichael W. Mahoney Department of Physics, Yale University, New Haven, Connecticut 06520
JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 23 15 DECEMBER 2001 Quantum, intramolecular flexibility, and polarizability effects on the reproduction of the density anomaly of liquid water by simple potential
More informationCHAPTER V. Brownian motion. V.1 Langevin dynamics
CHAPTER V Brownian motion In this chapter, we study the very general paradigm provided by Brownian motion. Originally, this motion is that a heavy particle, called Brownian particle, immersed in a fluid
More informationPhysics of Materials: Classification of Solids On The basis of Geometry and Bonding (Intermolecular forces)
Physics of Materials: Classification of Solids On The basis of Geometry and Bonding (Intermolecular forces) Dr. Anurag Srivastava Atal Bihari Vajpayee Indian Institute of Information Technology and Manegement,
More informationChapter 19 Entropy Pearson Education, Inc. Slide 20-1
Chapter 19 Entropy Slide 20-1 Ch 19 & 20 material What to focus on? Just put out some practice problems Ideal gas how to find P/V/T changes. E.g., gas scaling, intro to the ideal gas law, pressure cooker,
More informationQuantum Annealing in spin glasses and quantum computing Anders W Sandvik, Boston University
PY502, Computational Physics, December 12, 2017 Quantum Annealing in spin glasses and quantum computing Anders W Sandvik, Boston University Advancing Research in Basic Science and Mathematics Example:
More informationCase study: molecular dynamics of solvent diffusion in polymers
Course MP3 Lecture 11 29/11/2006 Case study: molecular dynamics of solvent diffusion in polymers A real-life research example to illustrate the use of molecular dynamics Dr James Elliott 11.1 Research
More informationStructural transition in (C 60 ) n clusters
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Axel Enders Publications Research Papers in Physics and Astronomy 2002 Structural transition in (C 60 ) n clusters W. Branz
More informationSIMULATIONAL ANALYSIS OF GLASSES PREPARED VIA DIFFERENT INTERATOMIC POTENTIALS
Journal of Optoelectronics and Advanced Materials Vol. 7, No. 4, August 2005, p. 1915-1922 SIMULATIONAL ANALYSIS OF GLASSES PREPARED VIA DIFFERENT INTERATOMIC POTENTIALS Y. Sano, J. K ga, F. Yonezawa Department
More informationComputer simulation methods (2) Dr. Vania Calandrini
Computer simulation methods (2) Dr. Vania Calandrini in the previous lecture: time average versus ensemble average MC versus MD simulations equipartition theorem (=> computing T) virial theorem (=> computing
More informationChanges of State. Substances in equilibrium change back and forth between states at equal speeds. Main Idea
Section 4 s Substances in equilibrium change back and forth between states at equal speeds. A liquid boils when it has absorbed enough energy to evaporate. Freezing occurs when a substance loses enough
More informationCorrelated 2D Electron Aspects of the Quantum Hall Effect
Correlated 2D Electron Aspects of the Quantum Hall Effect Outline: I. Introduction: materials, transport, Hall effects II. III. IV. Composite particles FQHE, statistical transformations Quasiparticle charge
More informationComputational study of the structures and thermodynamic properties of ammonium chloride clusters using a parallel jump-walking approach
Computational study of the structures and thermodynamic properties of ammonium chloride clusters using a parallel jump-walking approach Alexander Matro and David L. Freeman Department of Chemistry, University
More informationTemperature and Heat. Prof. Yury Kolomensky Apr 20, 2007
Temperature and Heat Prof. Yury Kolomensky Apr 20, 2007 From Mechanics to Applications Mechanics: behavior of systems of few bodies Kinematics: motion vs time Translational and rotational Dynamics: Newton
More informationarxiv: v1 [cond-mat.stat-mech] 23 Jan 2019
arxiv:1901.07797v1 [cond-mat.stat-mech] 23 Jan 2019 Accelerated relaxation and suppressed dynamic heterogeneity in a kinetically constrained (East) model with swaps 1. Introduction Ricardo Gutiérrez Complex
More informationComputer simulations as concrete models for student reasoning
Computer simulations as concrete models for student reasoning Jan Tobochnik Department of Physics Kalamazoo College Kalamazoo MI 49006 In many thermal physics courses, students become preoccupied with
More informationA MOLECULAR DYNAMICS SIMULATION OF A BUBBLE NUCLEATION ON SOLID SURFACE
A MOLECULAR DYNAMICS SIMULATION OF A BUBBLE NUCLEATION ON SOLID SURFACE Shigeo Maruyama and Tatsuto Kimura Department of Mechanical Engineering The University of Tokyo 7-- Hongo, Bunkyo-ku, Tokyo -866,
More informationChemistry A: States of Matter Packet Name: Hour: Page 1. Chemistry A States of Matter Packet
Chemistry A: States of Matter Packet Name: Hour: Page 1 Chemistry A States of Matter Packet Chemistry A: States of Matter Packet Name: Hour: Page 2 Worksheet #1: States of Matter In this packet we will
More information3.091 Introduction to Solid State Chemistry. Lecture Notes No. 5a ELASTIC BEHAVIOR OF SOLIDS
3.091 Introduction to Solid State Chemistry Lecture Notes No. 5a ELASTIC BEHAVIOR OF SOLIDS 1. INTRODUCTION Crystals are held together by interatomic or intermolecular bonds. The bonds can be covalent,
More informationPotts And XY, Together At Last
Potts And XY, Together At Last Daniel Kolodrubetz Massachusetts Institute of Technology, Center for Theoretical Physics (Dated: May 16, 212) We investigate the behavior of an XY model coupled multiplicatively
More informationSome notes on sigma and pi bonds:
Some notes on sigma and pi bonds: SIGMA bonds are formed when orbitals overlap along the axis between two atoms. These bonds have good overlap between the bonding orbitals, meaning that they are strong.
More informationChapter 4. The Physical transformations of pure substances Fall Semester Physical Chemistry 1 (CHM2201)
Chapter 4. The Physical transformations of pure substances 2011 Fall Semester Physical Chemistry 1 (CHM2201) Contents Phase Diagrams 4.1 The stabilities of phases 4.2 Phase boundaries 4.3 Three representative
More informationUsing Molecular Dynamics to Compute Properties CHEM 430
Using Molecular Dynamics to Compute Properties CHEM 43 Heat Capacity and Energy Fluctuations Running an MD Simulation Equilibration Phase Before data-collection and results can be analyzed the system
More informationCO 2 molecule. Morse Potential One of the potentials used to simulate chemical bond is a Morse potential of the following form: O C O
CO 2 molecule The aim of this project is a numerical analysis of adsorption spectra of CO2 molecule simulated by a double Morse potential function. In the project you should achieve following tasks: 1.
More informationPhase Transitions. µ a (P c (T ), T ) µ b (P c (T ), T ), (3) µ a (P, T c (P )) µ b (P, T c (P )). (4)
Phase Transitions A homogeneous equilibrium state of matter is the most natural one, given the fact that the interparticle interactions are translationally invariant. Nevertheless there is no contradiction
More informationChapter 2 Ensemble Theory in Statistical Physics: Free Energy Potential
Chapter Ensemble Theory in Statistical Physics: Free Energy Potential Abstract In this chapter, we discuss the basic formalism of statistical physics Also, we consider in detail the concept of the free
More informationCollective behavior, from particles to fields
978-0-51-87341-3 - Statistical Physics of Fields 1 Collective behavior, from particles to fields 1.1 Introduction One of the most successful aspects of physics in the twentieth century was revealing the
More informationSOLIDS AND LIQUIDS - Here's a brief review of the atomic picture or gases, liquids, and solids GASES
30 SOLIDS AND LIQUIDS - Here's a brief review of the atomic picture or gases, liquids, and solids GASES * Gas molecules are small compared to the space between them. * Gas molecules move in straight lines
More informationName: Class: Date: SHORT ANSWER Answer the following questions in the space provided.
CHAPTER 10 REVIEW States of Matter SECTION 1 SHORT ANSWER Answer the following questions in the space provided. 1. Identify whether the descriptions below describe an ideal gas or a real gas. a. The gas
More informationAnalysis of the simulation
Analysis of the simulation Marcus Elstner and Tomáš Kubař January 7, 2014 Thermodynamic properties time averages of thermodynamic quantites correspond to ensemble averages (ergodic theorem) some quantities
More informationBig Idea #5: The laws of thermodynamics describe the essential role of energy and explain and predict the direction of changes in matter.
KUDs for Unit 6: Chemical Bonding Textbook Reading: Chapters 8 & 9 Big Idea #2: Chemical and physical properties of materials can be explained by the structure and the arrangement of atoms, ion, or molecules
More informationTHEORY OF MOLECULE. A molecule consists of two or more atoms with certain distances between them
THEORY OF MOLECULE A molecule consists of two or more atoms with certain distances between them through interaction of outer electrons. Distances are determined by sum of all forces between the atoms.
More informationLECTURE 1: Disordered solids: Structure properties
LECTURE 1: Disordered solids: Structure properties What is an disordered solid: examples amorphous solids or glasses inorganic compounds (e.g. SiO 2 /silicates, B 2 O 3 /borates, GeO 2 /germanates, P 2
More informationPhysics 541: Condensed Matter Physics
Physics 541: Condensed Matter Physics In-class Midterm Exam Wednesday, October 26, 2011 / 14:00 15:20 / CCIS 4-285 Student s Name: Instructions There are 23 questions. You should attempt all of them. Mark
More informationdynamics computer code, whiccan simulate the motion of atoms at a surface and We coembininewton's equation for the nuclei,
DUI FILE COpy T DTIC ANNUAL LETTER REPORT BY 0. F. ELEC33 20KE AD-A224 069 P.1. FOR N00014-85-K-0442 U Theory of electronic states and formations energies of.defect comp exes, interstitial defects, and
More information1. Introduction to Clusters
1. Introduction to Clusters 1.1 The Field of Clusters Atomic clusters are aggregates of atoms containing from few to a few thousand atoms. Due to their small size, the properties of the clusters are, in
More information2. As gas P increases and/or T is lowered, intermolecular forces become significant, and deviations from ideal gas laws occur (van der Waal equation).
A. Introduction. (Section 11.1) CHAPTER 11: STATES OF MATTER, LIQUIDS AND SOLIDS 1. Gases are easily treated mathematically because molecules behave independently. 2. As gas P increases and/or T is lowered,
More informationPhonons and lattice dynamics
Chapter Phonons and lattice dynamics. Vibration modes of a cluster Consider a cluster or a molecule formed of an assembly of atoms bound due to a specific potential. First, the structure must be relaxed
More informationBasics of Statistical Mechanics
Basics of Statistical Mechanics Review of ensembles Microcanonical, canonical, Maxwell-Boltzmann Constant pressure, temperature, volume, Thermodynamic limit Ergodicity (see online notes also) Reading assignment:
More informationChapter 11. Intermolecular Forces and Liquids & Solids
Chapter 11 Intermolecular Forces and Liquids & Solids The Kinetic Molecular Theory of Liquids & Solids Gases vs. Liquids & Solids difference is distance between molecules Liquids Molecules close together;
More information- intermolecular forces forces that exist between molecules
Chapter 11: Intermolecular Forces, Liquids, and Solids - intermolecular forces forces that exist between molecules 11.1 A Molecular Comparison of Liquids and Solids - gases - average kinetic energy of
More informationADVANCED CHEMISTRY CURRICULUM. Unit 1: Mathematical Representation in Chemistry
Chariho Regional School District - Science Curriculum September, 2016 ADVANCED CHEMISTRY CURRICULUM Unit 1: Mathematical Representation in Chemistry OVERVIEW Summary Measurements are fundamental to the
More informationLecture PowerPoints. Chapter 13 Physics: Principles with Applications, 7 th edition Giancoli
Lecture PowerPoints Chapter 13 Physics: Principles with Applications, 7 th edition Giancoli This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching
More informationCHAPTER FIVE FUNDAMENTAL CONCEPTS OF STATISTICAL PHYSICS "
CHAPTE FIVE FUNDAMENTAL CONCEPTS OF STATISTICAL PHYSICS " INTODUCTION In the previous chapters we have discussed classical thermodynamic principles which can be used to predict relationships among the
More informationThis is a statistical treatment of the large ensemble of molecules that make up a gas. We had expressed the ideal gas law as: pv = nrt (1)
1. Kinetic Theory of Gases This is a statistical treatment of the large ensemble of molecules that make up a gas. We had expressed the ideal gas law as: pv = nrt (1) where n is the number of moles. We
More informationMatter changes phase when energy is added or removed
Section 12.4 Phase Changes Explain how the addition and removal of energy can cause a phase change. Interpret a phase diagram. Matter changes phase when energy is added or removed Energy Changes Accompanying
More informationUnit 6. Unit Vocabulary: Distinguish between the three phases of matter by identifying their different
*STUDENT* Unit Objectives: Absolute Zero Avogadro s Law Normal Boiling Point Compound Cooling Curve Deposition Energy Element Evaporation Heat Heat of Fusion Heat of Vaporization Unit 6 Unit Vocabulary:
More informationMonte Carlo (MC) Simulation Methods. Elisa Fadda
Monte Carlo (MC) Simulation Methods Elisa Fadda 1011-CH328, Molecular Modelling & Drug Design 2011 Experimental Observables A system observable is a property of the system state. The system state i is
More informationChapter 11. Freedom of Motion. Comparisons of the States of Matter. Liquids, Solids, and Intermolecular Forces
Liquids, Solids, and Intermolecular Forces Chapter 11 Comparisons of the States of Matter The solid and liquid states have a much higher density than the gas state The solid and liquid states have similar
More informationIII.H Zeroth Order Hydrodynamics
III.H Zeroth Order Hydrodynamics As a first approximation, we shall assume that in local equilibrium, the density f 1 at each point in space can be represented as in eq.(iii.56), i.e. [ ] p m q, t)) f
More informationName Date Class THE NATURE OF GASES
13.1 THE NATURE OF GASES Section Review Objectives Describe the assumptions of the kinetic theory as it applies to gases Interpret gas pressure in terms of kinetic theory Define the relationship between
More informationUnit 4: Gas Laws. Matter and Phase Changes
Unit 4: Gas Laws Matter and Phase Changes ENERGY and matter What is 에너지 A fundamental property of the universe that cannot be easily defined. Energy No one knows what energy is, only what it does or has
More informationCARBON 2004 Providence, Rhode Island. Adsorption of Flexible n-butane and n-hexane on Graphitized Thermal Carbon Black and in Slit Pores
CARBON Providence, Rhode Island Adsorption of Flexible n-butane and n-hexane on Graphitized Thermal Carbon Black and in Slit Pores D. D. Do* and H. D. Do, University of Queensland, St. Lucia, Qld 7, Australia
More informationStates of matter Part 2
Physical Pharmacy Lecture 2 States of matter Part 2 Assistant Lecturer in Pharmaceutics Overview The Liquid State General properties Liquefaction of gases Vapor pressure of liquids Boiling point The Solid
More informationStatistical Mechanics
Franz Schwabl Statistical Mechanics Translated by William Brewer Second Edition With 202 Figures, 26 Tables, and 195 Problems 4u Springer Table of Contents 1. Basic Principles 1 1.1 Introduction 1 1.2
More informationSOLIDS AND LIQUIDS - Here's a brief review of the atomic picture or gases, liquids, and solids GASES
30 SOLIDS AND LIQUIDS - Here's a brief review of the atomic picture or gases, liquids, and solids GASES * Gas molecules are small compared to the space between them. * Gas molecules move in straight lines
More informationChapter 1 and Sections
Chapter 1 and Sections 3.1-3.3 Major Goals of Chapter 1: 1. Define the term chemistry. 2. Identify substances (matter) as chemicals. 3. Describe some physical and chemical properties of matter. 4. Describe
More information3.320 Lecture 23 (5/3/05)
3.320 Lecture 23 (5/3/05) Faster, faster,faster Bigger, Bigger, Bigger Accelerated Molecular Dynamics Kinetic Monte Carlo Inhomogeneous Spatial Coarse Graining 5/3/05 3.320 Atomistic Modeling of Materials
More information