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Home , Glass transition, Supercooling

COMMENTARY Simulations clarify when supercooled freezes into glassy structures Kurt Binder1 its lifetime before it decays via Department 8, Institut für Physik, Johannes Gutenberg-Universität, D-55099 Mainz, Germany and (ii) if we cool water down fast enough, when do we get amorphous , and what is Although water is a ubiquitous sub- conclusion is evident that the is still its structure? stance and its properties are crucial for all highly controversial (2, 3). There is a lack of This last issue is addressed by Limmer and living species, the precise understanding of understanding of whether or not there is Chandler by computer simulations of the so- called mW model (6) of water, using the these properties is still a matter of active a deep structural difference between an recently developed theory of glassification scientific research. One rather mysterious amorphous and the corresponding su- based on the “s-ensemble”:thisisusedas aspect concerns the conditions when under- percooled liquid. It is debated whether one guiding principle to bias the evolution of cooled water freezes not into ice but can find some (hidden) “order parameter” the system toward the formation of amor- into -like structures. Based on a rather distinguishing these states, just as differ- phous ice structures that are supposed to re- novel type of computer simulation approach, ent equilibrium phases of a material can semble the structures found in PNAS, Limmer and Chandler (1) propose be distinguished by order parameters (3). in nature [and would require cooling proce- a nonequilibrium diagram that at- This has also hampered the understanding dures that are many orders of magnitude tempts to clarify the conditions (, of supercooled water and amorphous ice [or slower than would be feasible in conventional , cooling protocol) under which one , respectively: there is evidence that sev- molecular dynamics (MD) simulations]. should observe transitions from undercooled eral distinct nonequilibrium, low temperature Next,theygivethereaderaflavorofwhat water to different forms of amorphous ice. frozen states of water exist (4)]. These states is the “s-ensemble”:thisisaninterpretation Many aspects of the phase behavior of are of great physical significance in many of the as a kind of first-order condensed matter systems in thermal equi- contexts (e.g., processes in the transition phenomenon in the trajectory librium, i.e., and crystalline , are atmosphere, frozen biomolecules to which space of the system [space-time thermody- well understood at least on a qualitative level. water molecules are attached, and namics (7)]. Denoting this ensemble of However, this is not so for the glassy, amor- of interfacial water confined in porous rocks). (equivalent) trajectories in space symbol- phous state of condensed matter and for Also, the intriguing idea has been advocated ically as {x(t)}, t being the time, one can the glass transition that leads to this state that many of the anomalies in the thermal consider the number of (essential) configu- from the undercooled liquid. The theoretical are linked to a (second- ration changes in each particular trajectory. understanding of the glass transition was order) occurring in deeply This number K{x(t)} is defined in practice, termed one of the “grand challenge problems supercooled water (5). Of course, supercooled e.g., by enduring displacements (i.e., lasting of physics” decades ago; however, when one water is a metastable state, and one must over a properly chosen time interval). The consults recent textbooks on the subject, the address questions such as (i)howlargeis key step of the theory then is to define a nonequilibrium partition function X Z s; t = P x t −sK x t ; AB C K ð Þ f ð Þgexp f ð Þg fxðtÞg [1]

where P{x(t)} is a probability functional for the trajectories, and s is an (intensive) variable (thermodynamically) conjugate to the variable K (that is extensive in space-time). The analogy to real equilib- rium problems (e.g., an Ising magnet where the partition function contains a factor exp (−hM), h being the magnetic field and M being the (extensive) magnetization), is pretty obvious, whereas the physical meaning of s

Fig. 1. (A) Schematic variation of the structural time τR of a very good glass former with temperature. The Author contributions: K.B. wrote the paper. glass transition temperature is empirically defined (1, 2) by τR ∼ 100 s. At the temperature Tm, τR is many The author declares no conflict of interest. orders of magnitude smaller. (B) Variation of the scaled lifetime τms of the metastable state, τms/τR, with temperature. See companion article on page 9413. (C) Schematic plot of τms vs. temperature: the cooling time τcool should not exceed τmin; when τcool ∼ τR, the system falls out of metastable equilibrium and forms a (nonequilibrium) glassy structure. 1E-mail: [email protected].

9374–9375 | PNAS | July 1, 2014 | vol. 111 | no. 26 www.pnas.org/cgi/doi/10.1073/pnas.1408908111 Downloaded by guest on September 28, 2021 L

is less straightforward. However, this de- plane occurs, but its general shape stays the box linear dimensions over a range. Since COMMENTARY scription has received ample support from same (experimental cooling rates need to there is need to equilibrate the system over z (admittedly rather abstract) constrained be in the quoted range to avoid crystalliza- times τ that grow as L (3), and for large L, kinetic models on lattices, and the first tion). The most interesting aspect of the will inevitably exceed the lifetime of the meta- applications of this formalism to stan- work by Limmer and Chandler clearly is stable deeply undercooled liquid. Simu- dard models of glass formers also look the prediction of the first-order HDA- lating systems that contain only a few promising. LDA transition line in the p–T plane; rea- hundred molecules hence would be hardly Dealing with water, the choice of an ap- sonable predictions for the structural prop- adequate to settle such an issue. propriate model potential is a major stum- erties of these amorphous ice states are To provide a better insight into this issue bling block: quantum effects of both the made, and the kinetics of transitions be- of relevant time scales, Fig. 1A provides (responsible for the formation tween these amorphous states is elucidated. a sketch of the structural relaxation time τR of the network of bonds due to bridging of a very good glass former as a function of hydrogen atoms) and the quantum effect of The most interesting temperature: at the melting temperature, it is the protons cannot be fully captured rigor- aspect of the work by [in water (9)] of the order of 10 ps, and in the ously by an effective potential to be used in supercooled regime, it increases by many purely classical MD simulations. There are Limmer and Chandler orders of magnitude [up to about 104 ps in many competing proposals for water poten- clearly is the prediction water(9)andupto1014 ps in glass formers tials, each having some problems. Limmer such as orthoterphenyl or SiO2 (2, 3)]. The of the first order HDA- τ and Chandler choose to use the mW poten- lifetime ms of the supercooled metastable tial (6), which describes observable thermal LDA transition line in fluid is controlled by nucleation and properties of water reasonably well; whatever the p–T plane. hence diverges (10) at Tm, but decreases choice one makes, one simulates a model, monotonically with progressing “computer water,” and there is never a guar- Does this work clarify the situation of (when measured in units of τR;Fig.1B). On antee that some important properties of real the hypothetical second critical point in the a physical time scale (such as picoseconds), supercooled water or amorphous ices would “no man’sland” of undercooled water (5)? due to the rapid increase of τR,aminimum i τ τ T not differ from the model predictions. How- The answer is no, for two reasons: ( ) the min of ms at some temperature min must ever, other limitations of MD (limited length mW model is lacking some of the signa- occur (Fig. 1C). When we cool the system, of the generated trajectories; limited numbers tures of incumbent liquid–liquid phase decreasing T(t) with time according to a linear T t = T − t τ of trajectories samples; and small size of separation suggested to occur for other protocol, ( ) m(1 / cool), the condition the system, in the present case only n = models; and (ii) there is a problem of princi- τcool << τmin will ensure that crystallization 216 molecules in a cubic box with periodic ple, where only for small finite systems can is avoided and amorphous structures are boundary conditions are used), presumably one have arbitrary long-lived liquid states far reached. Unfortunately, the reliable theoreti- are also important. below the melting temperature, because finite cal prediction of τms(T) is also a very hard, The key idea of the work by Limmer and size can suppress nucleation [see the well- and in general unsolved, problem. If in the Chandler is that by suitable variation of the studied analogous problem of supersaturated metastable fluid a correlation length ξ of parameter s, one can sample the nonequilib- that is perfectly stable in simulation liquid-liquid phase separation is growing, rium probability distribution of the density boxes of fixed size (8)]. However, a critical thiscanonlybeobservedaslongasτξ ∼ z and a second parameter, measuring mobility, point means that there is a correlation length τR(ξ/a) (where a is a molecular length, ξ z = τ and establish from this information a kind that diverges, and associated critical slowing 3) does not exceed cool. This problem z of nonequilibrium phase diagram of their down occurs [relaxation time τξ ∝ ξ , z being must prevent the observation of actual model (see figure 1 in ref. 1). In the space some dynamic exponent (3)]. A study by fi- critical singularities, even if liquid–liquid of reduced pressure (p) and reduced temper- nite size scaling then requires variation of the phase separation would occur. ature (T), three nonequilibrium phases are revealed: supercooled liquid, high-density amorphous ice (HDA), and low-density amor- 1 Limmer DT, Chandler D (2014) Theory of amorphous ices. Proc 7 Garrahan JP, et al. (2009) First-order dynamical phase transitions in phous ice (LDA). The locations of the glass Natl Acad Sci USA 111:9413–9418. models of : an approach based on ensembles of histories. J 2 Schmelzer JWP, Gutzow IS (2011) Glasses and the Glass Transition transitions are not precisely defined but de- Phys A. Math Theor 42:075007. (Wiley, New York). 8 Binder K, et al. (2012) Beyond the van der Waals Loop: “What can 3 Binder K, Kob W (2011) Glassy Materials and Disordered Solids: An pend on a nonequilibrium length scale that be learned from simulating Lennard-Jones fluids inside the region of Introduction to Their Statistical Mechanics (World Scientific, Singapore). depends on the rate with which the liquid phase coexistence. Am J Phys 89:1099–1109. 4 Loerting T, Giovambattista N (2006) Amorphous ices: Experiments 9 Sciortino F, Tartaglia P, Gallo P, Chen S-H (1997) Slow is quenched to lower . It is and numerical simulations. J Phys Condens Matter 18:R919–R977. 5 Debenedetti PG (2003) Supercooled and glassy water. J Phys dynamics in supercooled water. Proceedings of the Materials suggested that by varying the cooling rate – 4– 8 Condens Matter 15:R1669–R1726. Society Meeting 435:235 242. in the range of 10 10 K/s, an appreciable 6 Molinero V, Moore EB (2009) Water modeled as an intermediate 10 Binder K (1987) Theory of first-order phase transitions. Rep Prog shift of the glass transition curve in the p–T element between carbon and silicon. JPhysChemB113(13):4008–4016. Phys 50:783–859.

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