摘要:
Properties of the double well potential and relaxation processes in a model glass coefficients of the fourth order polynomial describing the potential energy of the reaction coordinate, are found to be uncorrelated; (iv) using a factorized probability distribution function for these coefficients, the number of TLS per atom is indirectly estimated to be of the order of 10–5–10–6; (v) typically from 5 to 10 atoms participate in the motion in the DWP. In this paper we investigate by CS the topology of the potential energy hypersurface of a Lennard-Jones glass with the aim to check whether the TLS can be observed in this system, the characteristics of the DWPs, and the extent to which the harmonicity of the system is affected by the presence of DWPs. We will discuss the procedures used to search the minima of V({R}) and to identify the reaction coordinate (RC) and the least action path (LAP) joining minima of a pair.The main conclusions of the present work are the following. (a) No isolated double-wells are found in the system, i.e. in the investigated temperature range (T > 5 K) all the minima pertain to a network, and the thermally activated jumps among them are not controlled only by the energetic barriers, because, depending on temperature, a significant role is also played by the entropic term.16 (b) At variance with previous findings14,15 (see item (iii) above) and with the hypothesis that is the basis of the SPM12 the coefficients of the polynomial representation of the energy profile along the RC, are found to be highly correlated.(c) The degrees of freedom orthogonal to the LAP feel an almost harmonic potential, whose curvature is independent of the RC, i.e. of the position along the LAP itself. We investigated a sample of N = 864 atoms interacting via the 6-12 Lennard-Jones potential, and used e/KB = 125.2 K and s = 0.3405 nm, appropriate for argon. A microcanonical molecular dynamics simulation17 was carried out at the fixed density of 42 mol dm–3, in a cubic box with periodic boundary conditions. The glass was obtained as follows: a well-equilibrated fluid configuration at T » 500 K was gradually cooled down to a slightly supercooled liquid (T » 90 K). It must be noted that at this density the melting temperature T is » 100 K and the glass transition is expected18,19 to be at T » 50 K. The well equilibrated supercooled sample was then quenched down rapidly to T = 6 K.A new "equilibration" run (of the order of 1 ns) F. Demichelis,a G. Viliania and G. Ruoccob a Universitá di Trento and Istituto Nazionale di Fisica della Materia, I-38050, Povo, Trento, Italy. b Universitá di L'Aquila and Istituto Nazionale di Fisica della Materia, I-67100, L'Aquila, Italy Received 9th March 1999, Accepted 4th May 1999, Published 17th May 1999 We use computer simulation to investigate the topology of the potential energy V({R}) and to search for double well potentials (DWPs) in a model glass. By a sequence of newtonian and dissipative dynamics we find different minima of V({R}) and the energy profile along the least action paths joining them.At variance with previous suggestions, we find that the parameters describing the DWPs are correlated. Moreover, the trajectory of the system in the 3N-d configurational phase space follows a quasi-1-d manifold. The motion parallel to the path is characterized by jumps between minima, and is nearly uncorrelated from the orthogonal, harmonic dynamics. Among many unanswered questions concerning the dynamics of topologically disordered materials, two are related to the low energy excitations. The first one regards the excitations responsible for the low temperature thermal properties of glasses. Since the pioneering works of Phillips1 and Anderson et al.,2 who introduced the two level systems (TLS) as the origin of the anomalous specific heat behaviour below 1 K, many works have been devoted to identifying the degrees of freedom associated with the TLS in real glasses, but a definite answer has not yet been reached. The second question regards the excitations responsible for: (i) the thermal behaviour in the 1–10 K region, where a plateau in the thermal conductivity and a peak in cp/T 3 are observed;3 and (ii) the boson peak appearing in the inelastic neutron scattering (INS)4–6,11 and Raman spectra.7–11 This second question is also largely unanswered, and different hypotheses on the nature of these excitations have been proposed, ranging from an excess of acoustic-like modes to highly anharmonic and localized excitations.A link between the above questions is suggested by the so called soft potential model (SPM),12 which assumes the existence of few, highly anharmonic degrees of freedom with a potential energy function described by a fourth order polynomial. The coefficients of the polynomial are supposed to be uncorrelated, and to give rise to a large variety of curves, including double well potentials (DWPs). Among the DWPs, those that are isolated, have low asymmetry and have barrier heights resulting in a tunneling splitting around 1 K are candidates to be TLS. Computer simulations (CS) of model glasses might help to answer the previous questions, and, in particular, to verify the existence of TLS and to clarify the nature of DWPs in glasses.To our knowledge the direct CS inspection of the passage from minimum to minimum in glasses is limited to a few cases. Among them we mention the search for TLS performed by Stillinger and Weber13 and by Heuer and Silbey.14,15 The conclusions of these works can be summarized as follows: (i) low barrier bistable degrees of freedom exist in binary glasses with considerable variation in their physical parameters (barrier height, asymmetry, distance between minima, etc.); (ii) these parameters are strongly correlated, DWPs with large asymmetry typically have large barrier heights; (iii) in agreement with SPM, the PhysChemComm, 1999, 5i i was performed until the diffusion stopped, i.e.until when the mean square displacement at long times fluctuated around a constant value.The value of the constant <r2> was revealed to be » 10–2 nm2, in fair agreement with the thermal motions of the crystal at the same temperature. Starting from the "glass" at T = 6 K, we generated different initial configurations heating up the sample at T = 7.5, 10, 12 and 15 K, every time performing » 1 ns of annealing, and checking the value of the mean square displacement against that of the crystal at the corresponding temperature. At temperatures higher than 15 K the "glass" was no longer stable and, within the investigated run lasting 25 ns, the crystallization process rapidly took place. Each run (one for each temperature) consisted of a sequence of newtonian and dissipative dynamics.The newtonian trajectory was followed for 1 ps (50 integration time steps), subsequently a modified steepest descent method procedure (relaxationlike dynamics) was applied to quench the system and to find the inherent configuration {Xi}(i = 1...N), corresponding to a local minimum. The newtonian dynamics was then restarted from the same point in phase space where it was interrupted before the quenching, and the procedure was repeated up to 25,000 times for each temperature. The two minimum configurations obtained with successive quenching, {X a} and {X b}, are often the same. The structure factor was calculated at the beginning and at the end of each run in order to check for crystallization of the sample. No evidence of crystallization was found up to T = 15 K.a The adopted procedure ensures that different minima, when found, are close to each other in the 3N-d configurational space. About 130 different minima were found in the runs performed at the different investigated temperatures in the present work: minima of progressively high energy are found with increasing temperature, in the following we present the minima properties irrespective of the MD temperature at which they were found. The minima energies are spread over a range of about 1 J mol–1 centred at about 300 J mol–1 above the absolute minimum representing the crystal. All the minima belong to an interconnected network and the system is observed to perform temporarily closed loops in the configurational phase space.In order to study the topology of the potential energy, and to find the easiest way to jump among neighbouring minima, a fast and efficient algorithm has been developed to evaluate the LAP. The latter is determined as the path between {X } and {X b} that i i ba X minimizes the classical action integral, i ab X ò dsÖ(V[R(s)] – V0), with V0 = min{V({Xia}), V({Xib})}. We studied » 100 pairs of minima, evaluating the LAP joining them; in Fig. 1a we compare the potential energy profile evaluated along the LAP with that along the straight path. Although the LAP is not very far from being straight in the configurational space, as shown in Fig. 1b, the potential energy along the LAP is significantly lower, making the LAP itself highly preferred for the jump among minima.Moreover, the present analysis shows that the derivation of the DWP properties from the profile of the potential energy along the straight path between minima gives unreliable results. The potential energy profile along the LAP between the minima a and b was characterized by the energy difference between the two minima DVabm, their euclidean distance Dab, the displacement of the atom that moves most dab = maxi{|Xia – X b|}, the barrier height DV B (measured with respect to the higher minimum), and finally by the number of participation, defined as Pab = (Si|Xia – Xib|2)2/Si|Xia – Xib|4. It turns out that: (i) the participation is always around 20; (ii) among the 20 atoms involved in the jump, 1 or 2 account for 90% of the entire distance; (iii) for each pair of minima DVm is always higher than DVB.Fig. 1 (a) Comparison between the potential energy profile along the LAP (l) and along the straight path (.) joining two typical minima. Ex is the energy of the crystalline minima. (b) Orthogonal distance between the LAP and the straight path as a function of the reaction coordinate. Fig. 2 Example of correlation between the parameters describing the DWP. Each point represents the values found for the pair of parameters: (a) DVm, D; (b) w4, w3; (c) u3, u2; in the description of each DWP.Table 1 The upper right part of the three matrices reports the correlation coefficients r among all the various parameters used to describe the DWP in the physical representation {DVm, DVB, D, d, P}, and in the two different polynomial representations {w2, w3, w4} and {u0, u2, u3}.The lower left part of the correlation matrices indicates whether the two parameters are (l) or are not (.) correlated at the 5% level of confidence D d P DVm — DVB 0.45 — 0.86 0.72 ll l — l l l — 0.15 w2 — l DVm DVB DdP2 ww3 w4 . . l w w4 3 0.49 0.17 — 0.90 — l. Next we analyse the statistical correlation among the DWP parameters. In Fig. 2a we see that the different pairs of parameters that characterized the DWP are strongly correlated. To give a statistical significance to the correlation, for each pair of parameters we calculated the correlation coefficient r.20 The values of r, reported in the first part of Table 1, indicate statistical correlation between the parameters (with nDWP=141, at the significance level of 5%, the threshold for correlation is r = 0.165).Two other representations of the DWP are reported in the literature; these are the "u" representation, used in the soft potential model,12 E = eu0[u2(x/ s)2 – u3(x/ s)3 + (x/ s)4], and the "w" representation introduced in ref. 15, E = e[w2(x/ s)2 – w3(x/ s)3 + w4(x/ s)4]. The SPM assumes that there is no correlation among the parameters of the set {u0, u2, u3}, while the authors of ref. 15, noticing the correlation of the u set, made use of the lack of correlation among {w2, w3, w4} to determine the number of TLS in model glasses.As can be seen in Fig. 2b and 2c, and in Table 1, we found statistical correlation also in the case of the w set. It is beyond the scope of the present work to explain this inconsistency, which can depend on the particular glass examined or, more likely, on the different procedures used to determine the potential energy profile (LAP here, two straight paths, each one joining one minimum with the saddle point in ref. 15). In order to describe the low temperature anomalies of glasses, the most important quantity is the total splitting of the ground state associated with the TLS. To our knowledge, only one attempt has been made to estimate the energy splitting due to tunneling through TLS energy barriers in periodic systems.It (a) assumes the existence of isolated pairs of minima and (b) treats the problem as if it were 1-dimensional (1-d).14 From the present simulation, hypothesis (a) can be neither confirmed nor rejected because we did not directly identify any candidate TLS (i.e. pairs of minima with asymmetry < 1 K); recent results on argon clusters21 indicate that pairs of minima with low asymmetry are "isolated" in the sense that a third quasidegenerate, adjacent minimum has not been observed. As for item (b), the (3N – 3)-dimensional tunneling problem can be treated as a 1-d one if the Schrödinger equation can 0.78 0.72 0.85 0.13 0.02 0.25 — u u u 3 2 0 –0.23 — –0.22 — 0.87 ll l — be factorized into 3N – 3 independent equations.22 We shall assume that the reduction to 1-d can be made under the less restrictive condition that the relevant classical path (i.e.the LAP) is independent of all the others. To this end we have studied the curvatures of the potential energy surface along the LAP by calculating the dynamical matrix eigenvalues .u0 u2 u3 lj (n) [j = 1–(3N – 3)] and the corresponding eigenvectors in 42 (n = 1–42) equally spaced configurations along the LAP itself. In Fig. 3, we report, for each configuration along the LAP joining a typical pair of minima, the 20 (j = 1–20) lowest frequencies wj =¥| lj |, (we have assigned the minus sign to the frequencies associated with negative eigenvalues). As can be seen, in the present case only one eigenvalue becomes negative, indicating a first order saddle point in the 3N-dimensional space, and only the lowest frequencies change appreciably along the LAP.Fig. 3 The 20 lowest eigenfrequencies w=¥| l| of the system evaluated in 42 equispaced atomic configurations along the LAP are reported as a function of the reaction coordinate for a typical pair of minima. Negative eigenvalues l are reported as negative eigenfrequencies w. In Fig. 4 we report, for the six lowest eigenvalues, the projections of the eigenvectors on the local tangent to the LAP. Along the path, the main contribution comes from the lowest eigenvalue, while approaching the minima, an increasing contribution comes from other eigenvalues. The same results are found for all pairs of minima.They indicate that the dynamics parallel and orthogonal to the reaction coordinate are nearly independent: quasi-harmonic vibrations control the orthogonal dynamics, by a set ofeigenfrequencies that are independent of the position along the path. The reaction coordinate is associated with a single eigenvalue (often the lowest one), as shown by the large value of the projection of only one eigenvector in the path direction. Fig. 4 The projection on the LAP of the six lowest frequency eigenvectors of the modes of Fig. 3 are reported as functions of the reaction coordinate. DWP < 10–3 In conclusion, the emerging scenario is that of a network of connected minima, where each pair is joined together by a 1-dimensional path.The reaction coordinate along the path follows a LAP in the 3N-configurational space, described by DWPs with statistically correlated parameters (asymmetry, height, etc.). The potential energy experienced by the system along the LAP is significantly lower than that one along other paths, like, for example, the straight path. Orthogonal to the RC, the dynamics is harmonic, and the set of frequencies are nearly independent of the specific value of the RC. In the present simulation a single interconnected quasi-l-d path has been found, indicating that no more than one DWP is active at a time. Extrapolating this result, we can state that the number of DWPs coexisting with harmonic excitations is N atom–1. We are glad to acknowledge technical support by R.Iori (CISCA, Universitá di Trento), by V. Dimartino (CASPUR, Roma) and by S. Cozzini and M. Voli (CINECA, Bologna). References 1 W. A. Phillips, J. Low Temp. Phys., 1972, 3/4, 7. 2 P. W. Anderson, B. I. Halperin and C. M. Varma, Philos. Mag., 1972, 25, 1. 3 S. R. Elliott, Physics of Amorphous Materials, Longman, New York, 1990. 4 U. Buchenau, M. Prager, N. Nucker, A. J. Dianoux, N. Ahmad and W. A. 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Paper 9/01889A PhysChemComm © The Royal Society of Chemistry 1999
ISSN:1460-2733
DOI:10.1039/a901889a
出版商:RSC
年代:1999
数据来源: RSC