Statistical Mechanical Model of the Self-Organized Intermediate Phase in Glass-Forming Systems With Adaptable Network Topologies

ORIGINAL RESEARCH published: 14 February 2019 doi: 10.3389/fmats.2019.00011 Statistical Mechanical Model of the Self-Organized Intermediate Phase in Glass-Forming Systems With Adaptable Network Topologies Katelyn A. Kirchner and John C. Mauro* Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA, United States Edited by: Matthieu Micoulaut, Sorbonne Universités, France Reviewed by: Stefan Karlsson, RISE Research Institutes of Sweden, Sweden Guglielmo Macrelli, Independent Researcher, Este, Italy *Correspondence: John C. Mauro Specialty section: This article was submitted to Glass Science, a section of the journal Frontiers in Materials Received: 27 November 2018 Accepted: 25 January 2019 Published: 14 February 2019 Citation: Kirchner KA and Mauro JC (2019) Statistical Mechanical Model of the Self-Organized Intermediate Phase in Glass-Forming Systems With Adaptable Network Topologies. Front. Mater. 6:11. doi: 10.3389/fmats.2019.00011 Frontiers in Materials | www.frontiersin.org Non-equilibrium systems continuously evolve toward states with a lower free energy. For glass-forming systems, the most stable structures satisfy the condition of isostaticity, where the number of rigid constraints is exactly equal to the number of atomic degrees of freedom. The rigidity of a system is based on the topology of the glass network, which is affected by atomistic structural rearrangements. In some systems with adaptable network topologies, a perfect isostatic condition can be achieved over a range of compositions, i.e., over a range of different structures, giving rise to the intermediate phase of optimized glass formation. Here we develop a statistical mechanical model to quantify the width of the intermediate phase, accounting for the rearrangement of the atomic structure to relax localized stresses and to achieve an ideal, isostatic state. Keywords: glass, intermediate phase, topological constraint theory, statistical mechanics, modeling INTRODUCTION Within the field of topological constraint theory, there is growing interest in the ability of a glass network to adapt its topology to achieve isostaticity. A glass network is isostatic when the number of rigid constraints per atom, n, equals the number of translational degrees of freedom (Phillips, 1979). For a system in three-dimensional space, each atom has three degrees of freedom; hence, hni = 3 is the condition for achieving an isostatic network (Thorpe, 1983). If hni > 3 the system is overconstrained (stressed rigid), and if hni < 3 the system is underconstrained (floppy) (Thorpe, 1983). In the overconstrained region, additional rigidity, beyond hni = 3, creates localized stresses. Elimination of these stresses can be achieved through an imposition of crystalline order, which drives the network out of the glassy state (Thorpe, 1983). When topological constraint theory of glass was originally proposed by Phillips and Thorpe, the isostatic state was predicted to be achieved at a single composition, viz., the rigidity percolation threshold (Phillips and Thorpe, 1985). However, in 1999, Raman scattering and temperature-modulated differential scanning calorimetry (MDSC) experiments by Punit Boolchand et al. revealed a finite width of isostatic compositions in which the system can maintain stability, called the intermediate phase (IP) (Selvanathan et al., 1999; Boolchand et al., 2001b; Micoulaut, 2007; Moukarzel, 2013). Thorough investigations, particularly in chalcogenide systems, have revealed a difference between the onset of rigidity and the onset of stress, creating a finite width of compositions that enable the most stable, isostatic state (Selvanathan et al., 1999; Boolchand et al., 2001b). One of the most pronounced signatures of the intermediate phase was detected using MDSC measurements (Feng et al., 1997), which measures the non-reversible enthalpy of relaxation, 1H. The difference between the original 1 February 2019 | Volume 6 | Article 11 Kirchner and Mauro Self-Organized Intermediate Phase Phillips-Thorpe single percolation threshold result and Boolchand’s intermediate phase can be visualized in Figures 1A,B, respectively, where the blue circles indicate the lowest energy states of 1H and hence the isostatic composition(s). Although a consensus within the glass community has still not been reached regarding the existence of the intermediate phase, over the past 18 years understanding of the phenomenon has greatly advanced. Evidence of the IP has been found through numerical studies (Thorpe et al., 2000), analysis of finite size clusters (Micoulaut and Phillips, 2003), and thorough analyses using MDSC (Selvanathan et al., 1999, 2000; Boolchand et al., 2001b; Vaills et al., 2005; Novita et al., 2007) and Raman scattering (Selvanathan et al., 1999, 2000; Boolchand et al., 2001a; Wang et al., 2001; Novita et al., 2007). These studies all reveal two distinct thresholds marking the boundaries of the intermediate phase: the rigidity transition (the lower bound, below which there are floppy modes in the network) and the stress transition (the upper bound, above which the network is stressed-rigid). Between the two thresholds, fluctuations in the system can enable self-organization, as visualized in Figure 2. A challenge when studying the intermediate phase is the apparent irreproducibility of some of the experiments, causing the physical origins and very existence of the phase to be controversial. Careful sample preparation is necessary in order to detect the IP due to the experiment’s high sensitivity to impurities, inhomogeneities, and the thermal history of the glass (Bhosle et al., 2011, 2012). Some critics of the intermediate phase attribute the observed finite widths as possible experimental artifacts (Golovchak et al., 2008; Lucas et al., 2009; Shpotyuk and Golovchak, 2011). During MDSC experiments on Ge-Se glasses, the non-reversible enthalpy was shown to decrease in the IP domain, inferring a need for Ge-Se-Se isostatic structural fragments to account for the rigid but unstressed network (Micoulaut and Phillips, 2003; Massobrio et al., 2007; Sartbaeva et al., 2007). However, an extensive high-temperature nuclear magnetic resonance study revealed that these fragments were missing from the structure (Lucas et al., 2009). To account for this discrepancy, Lucas et al. disagreed with the existence of the intermediate phase and instead hypothesized that the previously observed phase could be an experimental artifact resulting from the use of a single modulation frequency in the MDSC experiments. However, subsequent modeling work showed that the frequency correction used in the analysis of the MDSC experiments provided non-reversing heat flows independent of the particular choice of modulation frequency (Guo et al., 2012). Another claim against the existence of the intermediate phase is the observation of physical aging in the intermediate phase glasses (Golov (...truncated)