Glassy clusters: Relations between their dynamics and characteristic features of their energy landscape
Sandip De, Bastian Schaefer, Ali Sadeghi, Michael Sicher, D. G. Kanhere, Stefan Goedecker
aa r X i v : . [ phy s i c s . a t m - c l u s ] N ov Glassy clusters: Relations between their dynamics and characteristic features of theirenergy landscape
Sandip De, ∗ Bastian Schaefer, Ali Sadeghi, Michael Sicher, D. G. Kanhere, and Stefan Goedecker Department of Physics, Universit¨at Basel, Klingelbergstr. 82, 4056 Basel, Switzerland Centre for Simulations and Modeling, University of Pune, 411007 India (Dated: October 8, 2018)Based on a recently introduced metric for measuring distances between configurations, we in-troduce distance-energy (DE) plots to characterize the potential energy surface (PES) of clusters.Producing such plots is computationally feasible on the density functional (DFT) level since it re-quires only a set of a few hundred stable low energy configurations including the global minimum. Bycomparison with standard criteria based on disconnectivity graphs and on the dynamics of Lennard-Jones clusters we show that the DE plots convey the necessary information about the character ofthe potential energy surface and allow to distinguish between glassy and non-glassy systems. Wethen apply this analysis to real systems on the DFT level and show that both glassy and non-glassyclusters can be found in simulations. It however turns out that among our investigated clusters onlythose can be synthesized experimentally which exhibit a non-glassy landscape.
The features of the potential energy surface (PES) [1]and the resulting consequences for the physical propertiesof a system are subject to intensive research. Because ofthe technological importance of glassy bulk materials, ex-tended glassy systems have been studied extensively [2–4]. During the last decades or so a number of advanceshave been made in understanding the nature of the glasstransition using powerful simulation and analytical meth-ods [5–7]. However a number of issues such as non expo-nential relaxation processes, rapid growth of relaxationtimes with decreasing temperatures, the role of potentialenergy surface (PES) and configurational entropy andspatial heterogeneity continue to be debated [8, 9]. Thequalitative understanding is based on the nature of theenergy landscape [1]. It was shown [10] that glassy sys-tems have a large number of local minima of similar en-ergy which are separated by barriers of various heights.Turning to finite systems, the electronic structure,equilibrium geometries and many properties of atomiccluster have also been studied extensively at various lev-els of theory. The PES and related properties of theLennard Jones (LJ) clusters with up to 1000 atoms arewell understood [1]. Atomic clusters are known to dis-play size sensitive properties. For example, some clusterssuch as the LJ (Lennard Jones cluster of 55 atoms ofsame type) are structure seekers that exhibit a strong ten-dency to fall into their unique ground state [11], whereasothers such as LJ have a multi-funnel character whichmakes it much harder to fall into the ground state [11].By ground state we denote in this article the geometri-cal configuration corresponding to the global minimum ofthe PES. For gold clusters the basic structural motif ofthe ground state can for instance change by the additionof a single atom [12]. Ground state geometries frequentlyexhibit amorphous structures. [13–15]. This can lead toa flat heat capacity in gallium and aluminum clusters,whereas highly symmetric clusters of the same materialgive a peaked heat capacity [16, 17]. Though it is believed that a glassy landscape wouldalso lead to glassy dynamics in clusters, the reportedwork has been rather sporadic and evidence in terms ofdynamical behavior at low temperature is missing [18–20]. One of the early attempts to seek glassy behav-ior in clusters was by Rose and Berry in their study of( KCl ) clusters [18], and by Nayak, Jena and Berry [19].In a more recent work, Banerjee and Dasgupta have in-vestigated the dynamics of glass forming liquids using amaster equation approach within a network model [20].Unfortunately their cluster was a structure seeker with awell defined ordered structure. Nevertheless they did ob-tain clear indications of glassy behavior by removing thelow energy part of the spectrum. The standard approachto probe the glassy nature is via very long molecular dy-namics (MD) runs at various temperatures. Althoughfeasible for LJ clusters, this is prohibitively expensivefor a realistic treatment using Density functional theory(DFT). An alternative is to characterize the PES usingthe associated disconnectivity graphs [21] which showsthe relation between the energy differences of the localminima and the barrier heights. However determinationof a large number of saddle points is computationally alsovery expensive at the DFT level. For this reason studieson glassy clusters based on a realistic description, suchas DFT, are virtually nonexistent.The present work has two main objectives. First we in-troduce a novel approach based on distance-energy (DE)analysis [22], and show that a DE plot represents the es-sential characteristics of a potential energy landscape. Toestablish this, we carry out long time MD and computerelevant dynamical susceptibility for two model LJ clus-ters. Second we demonstrate the utility of the approachby applying it to four clusters on the DFT level and showthat one cluster has a glassy character whereas the otherones are structure seekers.The basic idea is illustrated for a one-dimensionalmodel in Fig 1, where a glassy landscape is transformedinto the landscape of a structure seeker by lowering theenergy region around the global minimum with respectto the regions further away. During this transformationthe energy differences between the global minimum andthe low energy local minima are obviously increased andsome barriers disappear which in turn causes some lo-cal minima to disappear as well. This can be explainedmathematically by the Tomlinson model [23]. The DEplots for the PES at the four stages of the transforma-tion are given by the locations of the local minima andshown by discs in the same color as the correspondingPES. Obviously the distance of a disk along the x axis isthe distance of this local minimum from the global mini-mum in configurational space whereas the distance alongthe y axis is the energy of the local minimum with re-spect to the global minimum. For the structure seeker(red PES) the energy increases more rapidly with dis-tance and has fewer points which are close according tothe configurational distance compared to the case of theglassy landscape (black discs).For realistic PESs which are very high-dimensional asuitable generalization of the distance is needed. A globalfingerprint describing a cluster can be obtained from theeigenvalues of an overlap matrix of atom centered gaus-sians whose width is given by the covalent radius of theatom on which it is centered. The root mean square of thedifference vector between two fingerprint vectors is thena distance measure which fulfills all the properties of ametric [22]. As we shall see it is this distance between theground state and all metastable states along with theirenergies which reveals the character of a PES. Since forthe LJ model systems the bond-length can not be approx-imated by the sum of the covalent radii, we use the follow-ing slightly modified matrix C for the calculation of theeigenvalues of the LJ systems: C i,j = exp( − r i,j / (2 σ ij )),where r i,j is the distance between atom i and j and σ ij the parameter of the LJ potential (specified in the sup-plementary material) which takes on 3 different valuesdepending on whether the atoms i, j are of A or B type.Since all the matrix elements used for the calculation ofthe configurational distance are scaled with respect to theequilibrium bond-lengths, the configurational distance isindependent of the bond-length and systems with dif-ferent bond-lengths can be compared. Our results arerather insensitive to the exact functional form chosen forthe calculation of the matrix elements C i,j and it is tobe expected that even distances based on other descrip-tors of the chemical environments [24] will lead to similarresults.The high dimensional character of the true PES leadsto an important modification of the simple picture shownin Fig. 1. Because local minima can be found in somany directions around the global minimum the num-ber of minima within a certain configurational distancewill be much larger than in our one-dimensional modeland the density of points in the plot will be much higher. E n e r g y ( a r b . un i t ) Configurational Coordinate E n e r g y R e l a t i v e t o G l o b a l Config. Dist. From Global Minimum
FIG. 1: Simple one-dimensional model for the transformationof a glassy into a non-glassy energy landscape. The movementof the local minima, indicated by the discs, show the evolutionof the DE plots during this transformation.
An even larger increase occurs for the saddle points whichlead to neighboring minima. The global minimum of the LJ cluster for instance is surrounded by 535 local min-ima which are connected to the global minimum by 911saddle points [25].Furthermore, there are 911 structurally distinct tran-sition states connecting 535We will next show that DE plots convey all the nec-essary information to judge whether a system has glassycharacter or not. To do so we study two binary LJ sys-tems (BLJ) having 45 (13 of type A and 32 of type B)and 55 (13 of type A and 42 of type B) atoms. The LJpotential parameters used for these two cluster are givenin the supplementary material. Then we establish theglassy nature of the 55 atom cluster using standard toolssuch as disconnectivity graphs and dynamical suscepti-bilities obtained from molecular dynamics. The sameexamination of the 45 atom cluster on the other handshows that it is a structure seeker. We next computeand examine the DE plots and will see that they giveinformation which is in agreement with the informationobtained by the previous methods.In order to compute long time dynamical proper-ties we have performed constant temperature MD us-ing DLPOLY [26] at five temperatures in the range T ∈ [0 . , . Q ( t ), Q ( t ) = Z d~rρ ( ~r, t ) ρ ( ~r, t + t ) ∼ N X i =1 w ( | ~r i ( t ) − ~r i ( t + t ) | ) , where ρ ( ~r, t ) are space-time dependent particle densi-ties. w ( r ) = 1, if r ≤ a and zero otherwise. The av-eraging over the initial time t is implied. The windowfunction w of width [ a = 0 .
30] treats particle positionsseparated by an amplitude smaller than .3 as identical.The dynamical susceptibility is defined as the fluctuationin Q ( t ), χ ( t ) = N [ h Q ( t ) i − h Q ( t ) i ]. It is well estab-lished that for glassy systems, χ ( t ) has a non-monotonictime dependence, and peaks at a time τ that is propor-tional to the structural relaxation time. The time de-pendence of χ ( t ) is shown in Fig. 2 for BLJ . Thenote worthy feature is the increase in τ m by two ordersof magnitude as the temperature decreases. The behav-ior is very similar to the behavior of a glassy extendedsystem [8] and quantitatively establishes in the glassycharacter of the BLJ cluster. On the other hand ap-plying the same analysis to the BLJ cluster does notgive such a temperature dependence (see supplementarymaterial). -2 -1 χ ( t ) t (LJ unit) T=0.29T=0.28T=0.24T=0.20 FIG. 2: Dependence of the dynamic susceptibility χ on timefor different temperatures in the case of BLJ We also looked at an experimentally measurable fin-gerprint of glassy systems namely the heat capacity. Asshown in Fig. 3, it is rather flat for
BLJ and showsno well defined peak, indicating the absence of a first or-der like transition. For comparison Fig. 3 also showsthe specific heat for another cluster of the same size,namely LJ , which is known to be a strong structureseeker. These calculated heat capacities are quite simi-lar to the experimentally observed specific heats of Ga and Ga cations [16], which were termed as meltersand non melters. The phenomenon has been explainedon the basis of their respective geometries, the magicmelters being relatively more ordered and non melters be-ing disordered [17]. The same explanation applies in thiscase. The non-glassy LJ cluster is a icosahedral struc-ture, whereas the glassy system is disordered.Fig. 4 shows the disconnectivity trees for our two modelbinary LJ clusters. The differences are obvious. Thestructure seeker has a ground state which is consider-ably lower in energy than the next metastable configu-rations. In the case of the glassy system there are manymetastable configurations which are close in energy tothe ground state and the barriers that have to be sur-mounted to get from one metastable configuration intoanother one are of variable height and frequently much C v ( L J un i t) T (LJunit) FIG. 3: The heat capacity C V as a function of temperaturefor the glassy cluster BLJ (black curve) and the non-glassy LJ one (red curve). larger than the energy difference between the local min-ima. . . Δ E ( L J un i t s ) Δ E ( L J un i t s ) FIG. 4: Disconnectivity graphs for the glassy BLJ55 (top)and non-glassy BLJ45 (bottom) binary Lennard Jones clus-ters. The graphs were produced with the disconnectionDPSsoftware [27].
Now we present and discuss the DE plots (Fig. 5) forboth systems. The differences are striking and by com-parison with our model PES of Fig 1, it is clear thatthe
BLJ has a glassy PES whereas BLJ does nothave a glassy character. As has already be seen from thedisconnectivity plot the global minimum is much lowerin energy than any other metastable state for the struc-ture seeker. In addition the first metastable structurehas also a rather large configurational distance from theglobal minimum. For the glassy system, on the otherhand, there exists a large number of local minima closeto the global minimum which implies that the density ofstructures in the configurational space is higher for theglassy system. Indirectly the large number of close-byminima also indicates that low saddle points exist aroundthe global minimum. This is related to the Bell-Evans-Polanyi principle [28] which states that barriers are lowif the educt and product of a chemical reaction are simi-lar. Hence the system has a distribution of low and highbarriers characteristic of glassy systems. BLJ BLJ ∆ E L J - un i t / a t o m Fingerprint Distance
BLJ BLJ FIG. 5: DE plots for the glassy (black dots) and non-glassy(red dots) binary investigated Lennard Jones systems to-gether with their ground state structures. Only the lowest5000 configurations are considered.The two solid lines showleast-square fits to the two data sets. Their slope is a mea-sure for the average driving force towards the ground state.