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Melt-driven mechanochemical phase transformations in moderately exothermic powder mixtures.

Usually, mechanochemical reactions between solid phases are either gradual (by deformation-induced mixing), or self-propagating (by exothermic chemical reaction). Here, by means of a systematic kinetic analysis of the Bi-Te system reacting to Bi2Te3, we establish a third possibility: if one or more of the powder reactants has a low melting point and low thermal effusivity, it is possible that local melting can occur from deformation-induced heating. The presence of hot liquid then triggers chemical mixing locally. The molten events are constrained to individual particles, making them distinct from self-propagating reactions, and occur much faster than conventional gradual reactions. We show that the mechanism is applicable to a broad variety of materials systems, many of which have important functional properties. This mechanistic picture offers a new perspective as compared to conventional, gradual mechanochemical synthesis, where thermal effects are generally ignored.

Mechanical stress and strain can substantially affect chemical reactions, as in selective activation of covalent systems 1,2 , wear-induced chemical reactions 3,4 , or mechanochemicallyinduced unusual reactivity in solids [5][6][7][8] . Mechanochemical processing benefits from lower temperatures, reduced use of chemical solvents, and fewer processing steps than traditional synthesis routes 9 . Typically, mechanochemical processing involves powdered materials being milled, i.e., repeatedly forged, fractured, and re-welded under high-strain rates 5 , leading to nonequilibrium reactions to unique compounds and microstructures 10-12 . The nonequilibrium, or "driven", character of mechanical processing is generally assumed to be predominantly athermal, tending to homogenize the chemical composition rather than the chemical potential, as favoured by thermodynamic equilibrium [13][14][15] . The competition between these two tendencies forms the basis of driven systems theory 16 , and can explain why, e.g., immiscible alloy systems often form solid solutions during milling 13,14 ; if the rate of ballistic mixing exceeds the rate of chemical diffusion and phase separation then mixing will occur 16 .
Such driven reactions occur gradually over an extended period of time, e.g. several hours, as mechanical work is spread throughout the solid particles.
A simple diffusive-vs-ballistic mixing framework cannot explain all mechanochemical reactions however [17][18][19][20] ; in some cases, local temperature rise plays a role [21][22][23] . Mechanically induced self-propagating reactions (MSRs) are aided by the thermal energy from bond formation between dissimilar atoms; when two reactive species are mechanically ground together, the negative enthalpy of their mixing is released instantaneously, resulting in an autocatalytic propagation of the reaction 24,25 . During ball milling, MSRs are ignited spontaneously once the structure reaches a critical level of forced mixing, after which the reaction tends to consume the entire powder charge in an instantaneous manner.
Thus, two dominant types of driven reactions have been demonstrated: the gradual reactions promoted by mechanical energy input, and the abrupt MSRs promoted by the chemical energy of the exothermic reaction. In this paper we provide evidence for a third type of mechanochemical reaction, which exhibits some characteristics of the above two, but which is physically distinct and may have broad implications for low thermal effusivity materials of interest in energy and heat management applications. Specifically, we show that for at least some systems, the heating caused by mechanical work can lead to localized melting that in turn triggers local chemical reaction. The resulting new class of mechanochemical reactions provides some clarity on the controversial topic of contact melting in driven systems.

Impact characterization
The first step of our analysis involves quantitative microscopy of phase formation after a single mechanochemical unit process: a single impact of powder particles by tooling. Impact tests were performed on powders of elemental Bi and Te mixed in a molar ratio of 2:3, using a SPEX 8000 mixer mill (see methods). Fig. 1a shows a representative collision, imaged in a scanning electron microscope, in which perhaps a few hundred reactant powder particles of pure Bi and Te were flattened by the impact of a single 8 g steel ball. The resultant coldwelded compacts had an average measured impact radius of 680 μm and estimated thickness of 100 μm 26 . The reactant powder particles have reacted over a considerable scale, with a high volume fraction of product (intermetallic Bi2Te3) phase of about f = 0.25. The extent of phase formation is quantified in part (b), which shows the cumulative distribution of Bi2Te3 phase in 23 stereologically measured compacts. The average product volume fraction is f = 0.14.
The reaction morphology in Fig. 1 exhibits alloyed regions localized within bands that originate at the center of the compact and radiate outwards. We call such a localized event a critical loading condition (CLC). A CLC event is revealed in more detail in Fig. 1c, which shows the central region of the compact at higher magnification, where the components appear to have melted and flowed as a liquid (Fig. 1c). What is more, close inspection of locations where such bands reach the periphery of the compact reveals ejected material containing droplets (Fig. 1d) and rounded splash marks (Fig. 1e). Such observations strongly speak to the presence of liquid phase, and we deduce that the mechanical energy has induced melting through adiabatic heating. Based on this evidence we propose that molten Bi is first formed at the compact center once a critical amount of deformation is reached. Similar observations of deformation-induced melting have been made, e.g. during high-strain rate deformation of metallic glasses, thus melting a Sn surface coating 27 , or during high speed frictional loading on metallic surfaces, e.g. Pb, Sn, and Bi 28 . Indeed, contact melting during mechanical processing has been suggested theoretically 29,30 and observed in numerical simulations 31 -but not explicitly demonstrated experimentally before.

Kinetic analysis
To complement the above mechanistic observations, we now quantify the thermodynamic requirements for mechanochemical melting via systematic kinetic analysis. For this analysis, a different geometry of ball drop reactor, as schematized in Fig. 2a, was used to allow more precise control of the milling energy and temperature. However, the relevant parameters of the reactor such as the ball size, impact velocity, and ball-to-powder ratio are very similar, allowing direct comparison between experiments (see methods). Some typical X-ray diffraction (XRD) patterns of powder mixtures are shown in Fig. 2b, after about 1, 5, and 20 thousand collisions, respectively showing (i) almost exclusively reactants Bi + Te, (ii) a mixture of reactants Bi + Te and product Bi2Te3, and (iii) almost exclusively product Bi2Te3.
The patterns are best-fitted using the Rietveld method, from which the volume fraction of product (Bi2Te3) phase, f, can be calculated. In no pattern was any angular shift of the Bi and Te XRD peaks observed, suggesting negligible mutual dissolution of Bi and Te in one another. This conforms to the idea that Bi2Te3 forms directly via Te dissolution in molten Bi  The volume fraction of product Bi2Te3, f, as a function of the number of collisions, n, is plotted in Fig. 2c for a few representative datasets. In all cases, f undergoes a smooth monotonic increase, although the number of impacts required to complete the reaction varies considerably with processing conditions: between about 10 2 and 10 4 . The datasets are fitted with the following kinetic equation: (1) where k represents the apparent rate constant for the reaction. Equation (1) can adequately capture transformation kinetics, as shown by the good fit in Fig 2(c). The rate constant extracted from these fits, k, is related to the volume fraction of Bi2Te3 phase that would be formed during the first collision -i.e. in the absence of any pre-formed product phase. Fig. 2d shows that the rate constant k increases linearly with the collision energy E at all processing temperatures. However, each line of best-fit crosses the x-axis at a finite positive collision energy value, E0, which represents the smallest possible collision energy that gives rise to any reaction to the Bi2Te3 phase. We take this value of energy to be that required to initiate a CLC event in a given impact, such as those seen in Fig. 1.
As the processing temperature is increased, the rate constant increases, and E0 decreases, suggesting that less mechanical energy is needed to activate a CLC event at higher temperatures. Fig. 2e shows the monotonic decrease in E0 with processing temperature, T.
The data can be fitted with a straight line that crosses the x-axis at a temperature of T = 526 ± 7 K, which is close to the Bi melting point, Tm Bi = 544 K. The fact that the minimum collision energy necessary to activate mixing tends to vanish as the processing temperature T approaches Tm Bi supports the interpretation that Bi melting is the kinetically controlling process of the reaction. (The effect of pressure on the Bi melting point is analyzed in Supplementary Note 4, indicating that its effect can be neglected.) We note the value of E0 = 0.045 J at 300 K is about an order of magnitude larger than that typically required to induce plastic deformation during ball-milling 32 . Such values range between 0.004 J for soft FCC metals such as Cu and Ag and 0.008 J for harder BCC metals such as Fe and Ni. We can therefore assume that below E0 plastic deformation is still induced, but that -crucially -melting of Bi is not. Thus, we can identify the excess impact energy, E * = (E -E0), as being the energy above this threshold that is expended in melting Bi.
The transformation kinetics with respect to E * are shown in Fig. 3. The product volume fraction is plotted against the excess energy dose, defined as D * = n E * , at each processing temperature. Data points taken at different impact energies become superimposed for a given temperature, forming a single curve. This indicates that for mechanochemical activation, the excess energy dose is an invariant property 33   overlap when given as a function of the scaled dose, but are displaced with respect to the scaled dose, for a given processing temperature.

Reaction mechanism
In what follows, we seek to separate the mixing enthalpy and impact energy contributions to the reaction. We first consider the energy required to melt the quantity of Bi involved in each collision. The mass of powder converted in such an impact is simply m * = k m, where m is the total mass of the powder charge. The minimum thermal energy required to heat this quantity of powder from the ambient milling temperature T to the melting point of Bi, Tm Bi , and then melt it in the presence of a stoichiometric amount of Te, is: where MBi, ΔHm Bi , cp Bi and cp Te are the molar mass, enthalpy of fusion and specific heats, which are 209.0 g/mol, 11.3 kJ/mol, 25.7 J/mol-K and 25.4 J/mol-K respectively.
By comparing Emelt to the excess impact energy imparted, E * , we find the impact energy alone cannot explain all of the melting seen. I.e., under all experimental conditions E * < Emelt.
For example at 300 K and an impact energy of 0.06 J, E * = 0.015 J and Emelt = 0.026 J. We can therefore say that under all of our experimental conditions, mechanically-induced melting is not the only driving force for the reaction, and an additional source of energy is required to explain the amount of reacted product. One logical explanation is that the heat of the Bi + Te → Bi2Te3 reaction -which is measured by various authors to lie in the range 40 -80 kJ/mol 34-36 and is thus comparable to the fusion enthalpy of Bi -is supplying some excess thermal energy. Once triggered, the reaction energy could propagate the reaction further than the impact energy alone would promote.
We now consider a reaction mechanism where such reaction heat is taken into account. The amount of energy required to melt one mole of Bi2Te3 is The number of Bi moles that can undergo melting from the excess impact energy, E * , is Once the n0 moles of molten Bi have reacted with Te, they liberate the corresponding heat of reaction, n0 ΔHr/2. This heat could then be used to melt further Bi, and so on in a cycle of heat release and melting. The number of moles reacted, in the limit of no heat losses, in the ith cycle of such release, would be: In the limit of infinite steps, the total number of moles melting, i.e. ntot = n0 + n1 + …n∞, is a geometric series, which converges to: Equation (6) sets the upper limit for how much product can be formed using the impact energy and the heat of reaction. Fig. 4 compares this limit to the actual experimental values of product formation, on a plot of Emelt,mol vs. E * /ntot. The line corresponding to equation (6) intercepts the y-axis at ΔHr/2, while equation (2) -i.e. the case where the reaction enthalpy is neglected -intercepts at the origin. All experimental data points lie well above equation (2), confirming that mechanically induced melting is not the only driving force. Furthermore, the data lie below equation (6), indicating that not all of the enthalpy of reaction is transferred into extra product being formed. The discrepancy between values predicted by equation (6) and those measured suggests that a considerable amount of heat is lost to the surroundings before it can induce further reaction. Indeed, synthesis of Bi2Te3 by self-propagating high temperature synthesis was recently demonstrated, 37 which corroborates the view that during milling, full propagation of the reaction to consume the entire charge must be prevented by heat loss. product, E melt,mol is plotted as a function of energy supplied at impact, E * /n tot . All data lie well above equation (2), confirming that the impact energy alone cannot account for the amount of reaction, and thus the enthalpy of the Bi+Te→Bi2Te3 reaction (taken to be 79 kJ/mol 35 ) must account for the extra product formed. The fact that the data lies below equation (6) indicates that such propagation is only partial, thus a significant portion of the reaction heat must be dissipated. The dashed line is drawn as a guide to the eye.
It is worthwhile noting that the reaction rate in this system is much faster than seen in conventional mechanochemical processes. The chemical conversion kinetics of this system are compared in Fig. 5 to the kinetics of other gradually mixing systems, processed under identical conditions, extracted from previous studies, e.g. 33 Table 1).
The fact that product forms so rapidly in the Bi + Te system may explain why the reaction does not self-propagate; the reaction is mostly complete at energy doses where typical MSRs tend to ignite, indicating that the microstructure is not significantly refined before the reactants become consumed. Similarly, the reaction Ti + C → TiC will usually occur as MSR, but when milled at sufficiently high intensities, a more local reaction can occur, thus preventing self-propagation from happening 39 . transformation is about 1 kJ/g for Bi2Te3 but between 10 and 100 kJ/g for the formation of solid solutions or amorphization of intermetallics. However it is kinetically distinct from instantaneous MSR reactions.

Molten phase reactions in other systems
The above microscopy and kinetic analyses together corroborate the view that molten Bi From the criterion of equation (7), it can be inferred that if e (T -Tm) is below a critical value of φ set by the milling parameters, then the melting point should be reached during the collision process. behavior. The fact that it lies to the left of the SPEX mill line suggests melting is predicted to occur in Bi, along with other volatile elements such as Na, Li and K; these have incidentally all been seen to react rapidly in high energy ball mills 46 , which we may now suggest is associated with melting. Te is similarly positioned with Bi, suggesting it can melt also, but we suggest that Te's factor-of-three higher hardness 47 means that Bi deforms preferentially over Te, and thus governs the overall reactivity.
It is interesting to note that some other elements commonly processed by high energy ball

Outlook
With the present elaboration of a melt-driven mechanochemical reaction, it is hoped that this reaction pathway can be sought in many other systems. Thermal effects at the collision site are usually neglected in mechanochemical theories of mixing, and there is therefore scope for improved models. From an engineering viewpoint, this study offers potential guidance for rapid synthesis of functional compounds, many of which comprise one or more of the volatile elemental components that we predict are conducive to melt-driven mechanochemical reaction.

Kinetic analysis via ball-drop reactor
The reactor geometry was a cylinder roughly 20 cm in height, and 2 cm in diameter, with a rounded bottom base and stainless steel construction (Fig. 2a). Each milling experiment was prepared by adding to the reactor a single milling ball (also of stainless steel) of 20 g, 3 g of powders mixed according to Bi2Te3 stoichiometry, and sealed in Ar atmosphere. Milling was performed by attaching the reactor to a mechanical arm that subjected it to a harmonic oscillation of 5 s period along the vertical axis. The collision amplitude was varied to allow collision energies between about 0.06 and 0.24 J. The temperature of this reactor was actively controlled between 300 and 510 K using a metallic heating jacket. During each experiment the number of collisions, n, was recorded, after which the 3g of powders were removed for characterisation and the reactor refilled. The degree of phase transformation, i.e. 2Bi + 3Te → Bi2Te3, in the as-milled powders was characterized by XRD, using a Rigaku DMax/A diffractometer with a Cu Kα radiation source. All scans were collected over a scan range of 15 to 80 degrees 2θ, in discrete step mode with angular displacements of 0.05 degrees at a rate of 20 s per step (see Supplementary Note 3). Pattern intensity was fitted using the Rietveld method to extract relative phase fractions of Bi, Te and Bi2Te3. Raw data from these fits is provided in Supplementary Note 1.

Supplementary Note 1. Starting powders
To avoid oxidation, storage and handling of powders were performed in a glove box under high purity argon atmosphere, with oxygen and humidity levels below 2 ppm. Supplementary Figure   1 shows images of the starting powders taken in a Scanning Electron Microscope (SEM) under secondary electron mode. The average particle sizes of the powders were determined to be about 15, 3, 11 and 14 µm for Bi, Sn, Pb and Te respectively.

Supplementary Note 3. Treatment of X-ray diffraction patterns.
X-ray diffraction (XRD) analyses were carried out on disk-shaped specimens of powders compacted at about 1 bar. Occasionally, narrow 2θ ranges were studied in greater detail to clarify specific features of the peak profiles. A typical Rietveld fit of the experimental dataas used to determine the relative volume fractions of each phase -is shown in Supplementary   Figure 3. The high quality of fit indicates the lack of any amorphous phase -as would otherwise be present by a low angle 'hump'. To further confirm the lack of an amorphous phase, a similar analysis was performed on powders subjected to extended milling treatments -up to specific energy doses on the order of 10 5 J g -1 , i.e. about two orders of magnitude greater than the largest treatment reported in this study. Even in the most heavily deformed cases no amorphous hump could be detected.

Supplementary Note 4. Melting point depression in Bi.
In order to investigate the effect of pressure on the melting point of Bi, we estimate the maximum shear strain rate that could be experienced during a typical collision between a ball and the vial as: 54 ̇= ℎ ⁄ , where v is the impact velocity, h is the initial height of the trapped powder compact and M is the Taylor factor. Using the values of v = 5 m s -1 , h = 100 µm and M = 3, which are typical for SPEX mill, ̇≈ 2 x 10 4 −1 . At plastic strain rates of this magnitude, deformation becomes limited by phonon drag. 55 Here, we note that the following empirical law is generally obeyed for the shear stress of most metals: 55 where μ the shear modulus (12 GPa for Bi). These values predict a stress of τ = 50 MPa, which also corresponds to a hydrostatic stress of the same order. This hydrostratic stress is the driving force for any melting point depression, which is measured to be 38 56  3) Under SPEX high-energy milling, the chemical conversion kinetics are gradual (see Supplementary Table 1 and graphically illustrated in Supplementary Figure 6) and can be best-fitted with the same exponential kinetic law reported in the manuscript for