Possible charge ordering and anomalous transport in graphene/graphene quantum dot heterostructure.

Observations of superconductivity and charge density waves (CDW) in graphene have been elusive thus far due to weak electron-phonon coupling (EPC) interactions. Here, we report a unique observation of anomalous transport and multiple charge ordering phases at high temperatures (T1∼213K,T2∼325K) in a 0D-2D van der Waals (vdW) heterostructure comprising of single layer graphene (SLG) and functionalized (amine) graphene quantum dots (GQD). The presence of functionalized GQD contributed to charge transfer with shifting of the Dirac point ∼ 0.05 eV above the Fermi level (ab initio simulations) and carrier densityn∼-0.3×1012 cm-2confirming p-doping in SLG and two-fold increase in EPC interaction was achieved. Moreover, we elucidate the interplay between electron-electron and electron-phonon interactions to substantiate high temperature EPC driven charge ordering in the heterostructure through analyses of magnetotransport and weak anti-localization (WAL) framework. Our results provide impetus to investigate strongly correlated phenomena such as CDW and superconducting phase transitions in novel graphene based heterostructures.

W18O49 Nanowhiskers Decorating SiO2 Nanofibers: Lessons from In Situ SEM/TEM Growth to Large Scale Synthesis and Fundamental Structural Understanding.

Tungsten suboxide W18O49 nanowhiskers are a material of great interest due to their potential high-end applications in electronics, near-infrared light shielding, catalysis, and gas sensing. The present study introduces three main approaches for the fundamental understanding of W18O49 nanowhisker growth and structure. First, W18O49 nanowhiskers were grown from γ-WO3/a-SiO2 nanofibers in situ in a scanning electron microscope (SEM) utilizing a specially designed microreactor (µReactor). It was found that irradiation by the electron beam slows the growth kinetics of the W18O49 nanowhisker, markedly. Following this, an in situ TEM study led to some new fundamental understanding of the growth mode of the crystal shear planes in the W18O49 nanowhisker and the formation of a domain (bundle) structure. High-resolution scanning transmission electron microscopy analysis of a cross-sectioned W18O49 nanowhisker revealed the well-documented pentagonal Magnéli columns and hexagonal channel characteristics for this phase. Furthermore, a highly crystalline and oriented domain structure and previously unreported mixed structural arrangement of tungsten oxide polyhedrons were analyzed. The tungsten oxide phases found in the cross section of the W18O49 nanowhisker were analyzed by nanodiffraction and electron energy loss spectroscopy (EELS), which were discussed and compared in light of theoretical calculations based on the density functional theory method. Finally, the knowledge gained from the in situ SEM and TEM experiments was valorized in developing a multigram synthesis of W18O49/a-SiO2 urchin-like nanofibers in a flow reactor.

Impact of d-states on transition metal impurity diffusion in TiN.

In this work, we studied the energetics of diffusion-related quantities of transition-metal impurities in TiN, a prototype ceramic protective coating. We use ab-initio calculations to construct a database of impurity formation energies, vacancy-impurity binding energies, migration, and activation energies of 3d and selected 4d and 5d elements for the vacancy-mediated diffusion process. The obtained trends suggest that the trends in migration and activation energies are not fully anti-correlated with the size of the migration atom. We argue that this is caused by a strong impact of chemistry in terms of binding. We quantified this effect for selected cases using the density of electronic states, Crystal Orbital Hamiltonian Population analysis, and charge density analysis. Our results show that the bonding of impurities in the initial state of a diffusion jump (equilibrium lattice position), as well as the charge directionality at the transition state (energy maximum along the diffusion jump pathway), significantly impact the activation energies.

Long-Range Magnetic Order in Nickel Hydroxide-Functionalized Graphene Quantum Dots.

In this work, we demonstrate the prospect of chemically synthesizing transition metal (Ni) doped magnetic graphene quantum dots (GQDs) with the sole aim of shedding light on their magnetic properties. Our results show that adsorption of nickel hydroxide on predominantly paramagnetic GQDs reveals antiferromagnetic ordering in the M-T profile around 10 K with change of the spin exchange coupling deviating from J = 1/2 to J = 1, mainly arising from the d-p mixing hybridization between the p orbital of carbon from the GQD and the d orbital of Ni. Furthermore, our results are well complemented by ab initio simulations showing asymmetry of the up and down spins around the Fermi level for nickel hydroxide-doped GQDs with long-range spin polarization. Furthermore, the magnitude of the net magnetic moment generated for doped GQDs on the carbon atoms is found to be site-dependent (surface or edge).

Probing the charge transfer and electron-hole asymmetry in graphene-graphene quantum dot heterostructure.

In recent years, graphene-based van der Waals (vdW) heterostructures have come into prominence showcasing interesting charge transfer dynamics which is significant for optoelectronic applications. These novel structures are highly tunable depending on several factors such as the combination of the two-dimensional materials, the number of layers and band alignment exhibiting interfacial charge transfer dynamics. Here, we report on a novel graphene based 0D-2D vdW heterostructure between graphene and amine-functionalized graphene quantum dots (GQD) to investigate the interfacial charge transfer and doping possibilities. Using a combination ofab initiosimulations and Kelvin probe force microscopy (KPFM) measurements, we confirm that the incorporation of functional GQDs leads to a charge transfer induced p-type doping in graphene. A shift of the Dirac point by 0.05 eV with respect to the Fermi level (EF) in the graphene from the heterostructure was deduced from the calculated density of states. KPFM measurements revealed an increment in the surface potential of the GQD in the 0D-2D heterostructure by 29 mV with respect to graphene. Furthermore, we conducted power dependent Raman spectroscopy for both graphene and the heterostructure samples. An optical doping-induced gating effect resulted in a stiffening of theGband for electrons and holes in both samples (graphene and the heterostructure), suggesting a breakdown of the adiabatic Born-Oppenheimer approximation. Moreover, charge imbalance and renormalization of the electron-hole dispersion under the additional influence of the doped functional GQDs is pointing to an asymmetry in conduction and carrier mobility.

Surface Energy of Au Nanoparticles Depending on Their Size and Shape.

Motivated by often contradictory literature reports on the dependence of the surface energy of gold nanoparticles on the variety of its size and shape, we performed an atomistic study combining molecular mechanics and ab initio calculations. We show that, in the case of Au nanocubes, their surface energy converges to a value for ( 0 0 1 ) facets of bulk crystals. A fast convergence to a single valued surface energy is predicted also for nanospheres. However, the value of the surface energy is larger in this case than that of any low-index surface facet of bulk Au crystal. This fact can be explained by the complex structure of the surface with an extensive number of broken bonds due to edge and corner atoms. A similar trend was obtained also for the case of cuboctahedrons. Since the exact surface area of the nanoparticles is an ill-defined quantity, we have introduced the surface-induced excess energy and discuss this quantity as a function of (i) number of atoms forming the nano-object or (ii) characteristic size of the nano-object. In case (i), a universal power-law behaviour was obtained independent of the nanoparticle shape. Importantly, we show that the size-dependence of the surface is hugely reduced if the surface area correction is considered due to its expansion by the electronic cloud, a phenomenon specifically important for small nanoparticles.

An Ab Initio Study of Magnetism in Disordered Fe-Al Alloys with Thermal Antiphase Boundaries.

We have performed a quantum-mechanical study of a B2 phase of Fe 70 Al 30 alloy with and without antiphase boundaries (APBs) with the {001} crystallographic orientation of APB interfaces. We used a supercell approach with the atoms distributed according to the special quasi-random structure (SQS) concept. Our study was motivated by experimental findings by Murakami et al. (Nature Comm. 5 (2014) 4133) who reported significantly higher magnetic flux density from A2-phase interlayers at the thermally-induced APBs in Fe 70 Al 30 and suggested that the ferromagnetism is stabilized by the disorder in the A2 phase. Our computational study of sharp APBs (without any A2-phase interlayer) indicates that they have moderate APB energies (≈0.1 J/m 2 ) and cannot explain the experimentally detected increase in the ferromagnetism because they often induce a ferro-to-ferrimagnetic transition. When studying thermal APBs, we introduce a few atomic layers of A2 phase of Fe 70 Al 30 into the interface of sharp APBs. The averaged computed magnetic moment of Fe atoms in the whole B2/A2 nanocomposite is then increased by 11.5% w.r.t. the B2 phase. The A2 phase itself (treated separately as a bulk) has the total magnetic moment even higher, by 17.5%, and this increase also applies if the A2 phase at APBs is sufficiently thick (the experimental value is 2-3 nm). We link the changes in the magnetism to the facts that (i) the Al atoms in the first nearest neighbor (1NN) shell of Fe atoms nonlinearly reduce their magnetic moments and (ii) there are on average less Al atoms in the 1NN shell of Fe atoms in the A2 phase. These effects synergically combine with the influence of APBs which provide local atomic configurations not existing in an APB-free bulk. The identified mechanism of increasing the magnetic properties by introducing APBs with disordered phases can be used as a designing principle when developing new magnetic materials.

An Ab Initio Study of Vacancies in Disordered Magnetic Systems: A Case Study of Fe-Rich Fe-Al Phases.

We have performed quantum-mechanical calculations to examine the impact of disorder on thermodynamic, structural and electronic (magnetic) properties of Fe-Al systems with vacancies. A series of supercells was used and their properties were computed employing density-functional theory (DFT) as implemented in the VASP package. Our case study is primarily aimed at a disordered solid solution Fe 81.25 Al 18.75 but we have compared our results also with those obtained for the ordered Fe 3 Al intermetallic compound for which experimental data exist in literature. Both phases are found in Fe-Al-based superalloys. The Fe-18.75at.%Al solid solution was simulated using special quasirandom structures (SQS) in three different disordered states with a different distribution of Al atoms. In particular, we have considered a general disordered case (an A2-like variant), the case without the first nearest neighbor Al-Al pairs (a B2-like distribution of atoms) and also the case without both the first and second nearest neighbor Al-Al pairs (the D0 3 -like variant, in fact, an Fe-rich Fe 3 Al phase). The vacancy formation energies as well as the volumes of (fully relaxed) supercells with vacancies showed a large scatter for the disordered systems. The vacancy formation energies decrease with increasing concentration of Al atoms in the first coordination shell around the vacancy (an anti-correlation) for all disordered cases studied. The computed volumes of vacancies were found significantly lower (by 25-60%) when compared with the equilibrium volume of the missing atoms in their elemental states. Lastly, we have analyzed interactions between the vacancies and the Fe atoms and evaluated vacancy-induced changes in local magnetic moments of Fe atoms.

Thermal Expansion and Other Thermodynamic Properties of α2-Ti3Al and γ-TiAl Intermetallic Phases from First Principles Methods.

Anisotropic thermal expansion coefficients of tetragonal γ -TiAl and hexagonal α 2 -Ti3Al phases were calculated using first principles methods. Two approaches with different computational costs and degrees of freedom were proposed. The predicted values were compared with available experimental data showing that for γ -TiAl, the more computational demanding method with decoupled impact of volume and temperature effects on the cell shape leads to significantly better results than that with only ground-state optimised unit cell geometry. In the case of the α 2 -Ti3Al phase, both approaches yielded comparable results. Additionally, heat capacity and bulk modulus were evaluated as functions of temperature for both phases, and were fitted to provide an analytical formula which can be further used.

Experimental Chemistry and Structural Stability of AlNb3 Enabled by Antisite Defects Formation.

First-principles evolutionary algorithms are employed to shed light on the phase stability of Al⁻Nb intermetallics. While the tetragonal Al3Nb and AlNb2 structures are correctly identified as stable, the experimentally reported Laves phase of AlNb3 yields soft phonon modes implying its dynamical instability at 0 K. The soft phonon modes do not disappear even upon elevating the temperature in the simulation up to 1500 K. X-Ray diffraction patterns recorded for our powder-metallurgically produced arc cathodes, however, clearly show that the AlNb3 phase exists. We propose that AlNb3 is dynamically stabilised by ordered antisite defects at the Al sublattice, leading also to a shift of the Nb content from 75 to ∼81 at.%. Unlike the defect-free AlNb3, the antisite-stabilised variant hence falls into the compositional range consistent with our CALPHAD-based phase diagram as well as with the previous reports.

An Ab Initio Study of Pressure-Induced Reversal of Elastically Stiff and Soft Directions in YN and ScN and Its Effect in Nanocomposites Containing These Nitrides.

Using quantum-mechanical calculations of second- and third-order elastic constants for YN and ScN with the rock-salt (B1) structure, we predict that these materials change the fundamental type of their elastic anisotropy by rather moderate hydrostatic pressures of a few GPa. In particular, YN with its zero-pressure elastic anisotropy characterized by the Zener anisotropy ratio A Z = 2 C 44 / ( C 11 - C 12 ) = 1.046 becomes elastically isotropic at the hydrostatic pressure of 1.2 GPa. The lowest values of the Young's modulus (so-called soft directions) change from 〈100〉 (in the zero-pressure state) to the 〈111〉 directions (for pressures above 1.2 GPa). It means that the crystallographic orientations of stiffest (also called hard) elastic response and those of the softest one are reversed when comparing the zero-pressure state with that for pressures above the critical level. Qualitatively, the same type of reversal is predicted for ScN with the zero-pressure value of the Zener anisotropy factor A Z = 1.117 and the critical pressure of about 6.5 GPa. Our predictions are based on both second-order and third-order elastic constants determined for the zero-pressure state but the anisotropy change is then verified by explicit calculations of the second-order elastic constants for compressed states. Both materials are semiconductors in the whole range of studied pressures. Our phonon calculations further reveal that the change in the type of the elastic anisotropy has only a minor impact on the vibrational properties. Our simulations of biaxially strained states of YN demonstrate that a similar change in the elastic anisotropy can be achieved also under stress conditions appearing, for example, in coherently co-existing nanocomposites such as superlattices. Finally, after selecting ScN and PdN (both in B1 rock-salt structure) as a pair of suitable candidate materials for such a superlattice (due to the similarity of their lattice parameters), our calculations of such a coherent nanocomposite results again in a reversed elastic anisotropy (compared with the zero-pressure state of ScN).

Quantum-Mechanical Study of Nanocomposites with Low and Ultra-Low Interface Energies.

We applied first-principles electronic structure calculations to study structural, thermodynamic and elastic properties of nanocomposites exhibiting nearly perfect match of constituting phases. In particular, two combinations of transition-metal disilicides and one pair of magnetic phases containing the Fe and Al atoms with different atomic ordering were considered. Regarding the disilicides, nanocomposites MoSi 2 /WSi 2 with constituents crystallizing in the tetragonal C11 b structure and TaSi 2 /NbSi 2 with individual phases crystallizing in the hexagonal C40 structure were simulated. Constituents within each pair of materials exhibit very similar structural and elastic properties and for their nanocomposites we obtained ultra-low (nearly zero) interface energy (within the error bar of our calculations, i.e., about 0.005 J/m 2 ). The interface energy was found to be nearly independent on the width of individual constituents within the nanocomposites and/or crystallographic orientation of the interfaces. As far as the nanocomposites containing Fe and Al were concerned, we simulated coherent superlattices formed by an ordered Fe 3 Al intermetallic compound and a disordered Fe-Al phase with 18.75 at.% Al, the α -phase. Both phases were structurally and elastically quite similar but the disordered α -phase lacked a long-range periodicity. To determine the interface energy in these nanocomposites, we simulated seven different distributions of atoms in the α -phase interfacing the Fe 3 Al intermetallic compound. The resulting interface energies ranged from ultra low to low values, i.e., from 0.005 to 0.139 J/m 2 . The impact of atomic distribution on the elastic properties was found insignificant but local magnetic moments of the iron atoms depend sensitively on the type and distribution of surrounding atoms.

Impact of Nano-Scale Distribution of Atoms on Electronic and Magnetic Properties of Phases in Fe-Al Nanocomposites: An Ab Initio Study.

Quantum-mechanical calculations are applied to examine magnetic and electronic properties of phases appearing in binary Fe-Al-based nanocomposites. The calculations are carried out using the Vienna Ab-initio Simulation Package which implements density functional theory and generalized gradient approximation. The focus is on a disordered solid solution with 18.75 at. % Al in body-centered-cubic ferromagnetic iron, so-called α -phase, and an ordered intermetallic compound Fe 3 Al with the D0 3 structure. In order to reveal the impact of the actual atomic distribution in the disordered Fe-Al α -phase three different special quasi-random structures with or without the 1st and/or 2nd nearest-neighbor Al-Al pairs are used. According to our calculations, energy decreases when eliminating the 1st and 2nd nearest neighbor Al-Al pairs. On the other hand, the local magnetic moments of the Fe atoms decrease with Al concentration in the 1st coordination sphere and increase if the concentration of Al atoms increases in the 2nd one. Furthermore, when simulating Fe-Al/Fe 3 Al nanocomposites (superlattices), changes of local magnetic moments of the Fe atoms up to 0.5 µ B are predicted. These changes very sensitively depend on both the distribution of atoms and the crystallographic orientation of the interfaces.

An Ab Initio Study of Connections between Tensorial Elastic Properties and Chemical Bonds in Σ5(210) Grain Boundaries in Ni3Si.

Using quantum-mechanical methods we calculate and analyze (tensorial) anisotropic elastic properties of the ground-state configurations of interface states associated with Σ 5(210) grain boundaries (GBs) in cubic L1 2 -structure Ni 3 Si. We assess the mechanical stability of interface states with two different chemical compositions at the studied GB by checking rigorous elasticity-based Born stability criteria. In particular, we show that a GB variant containing both Ni and Si atoms at the interface is unstable with respect to shear deformation (one of the elastic constants, C 55 , is negative). This instability is found for a rectangular-parallelepiped supercell obtained when applying standard coincidence-lattice construction. Our elastic-constant analysis allowed us to identify a shear-deformation mode reducing the energy and, eventually, to obtain mechanically stable ground-state characterized by a shear-deformed parallelepiped supercell. Alternatively, we tested a stabilization of this GB interface state by Al substituents replacing Si atoms at the GB. We further discuss an atomistic origin of this instability in terms of the crystal orbital Hamilton population (COHP) and phonon dispersion calculations. We find that the unstable GB variant shows a very strong interaction between the Si atoms in the GB plane and Ni atoms in the 3rd plane off the GB interface. However, such bond reinforcement results in weakening of interaction between the Ni atoms in the 3rd plane and the Si atoms in the 5th plane making this GB variant mechanically unstable.

Strength and Brittleness of Interfaces in Fe-Al Superalloy Nanocomposites under Multiaxial Loading: An ab initio and Atomistic Study.

We present an ab initio and atomistic study of the stress-strain response and elastic stability of the ordered Fe 3 Al compound with the D0 3 structure and a disordered Fe-Al solid solution with 18.75 at.% Al as well as of a nanocomposite consisting of an equal molar amount of both phases under uniaxial loading along the [001] direction. The tensile tests were performed under complex conditions including the effect of the lateral stress on the tensile strength and temperature effect. By comparing the behavior of individual phases with that of the nanocomposite we find that the disordered Fe-Al phase represents the weakest point of the studied nanocomposite in terms of tensile loading. The cleavage plane of the whole nanocomposite is identical to that identified when loading is applied solely to the disordered Fe-Al phase. It also turns out that the mechanical stability is strongly affected by softening of elastic constants C ' and/or C 66 and by corresponding elastic instabilities. Interestingly, we found that uniaxial straining of the ordered Fe 3 Al with the D0 3 structure leads almost to hydrostatic loading. Furthermore, increasing lateral stress linearly increases the tensile strength. This was also confirmed by molecular dynamics simulations employing Embedded Atom Method (EAM) potential. The molecular dynamics simulations also revealed that the thermal vibrations significantly decrease the tensile strength.

Origin of the Low Magnetic Moment in Fe2AlTi: An Ab Initio Study.

The intermetallic compound Fe 2 AlTi (alternatively Fe 2 TiAl) is an important phase in the ternary Fe-Al-Ti phase diagram. Previous theoretical studies showed a large discrepancy of approximately an order of magnitude between the ab initio computed magnetic moments and the experimentally measured ones. To unravel the source of this discrepancy, we analyze how various mechanisms present in realistic materials such as residual strain effects or deviations from stoichiometry affect magnetism. Since in spin-unconstrained calculations the system always evolves to the spin configuration which represents a local or global minimum in the total energy surface, finite temperature spin effects are not well described. We therefore turn the investigation around and use constrained spin calculations, fixing the global magnetic moment. This approach provides direct insight into local and global energy minima (reflecting metastable and stable spin phases) as well as the curvature of the energy surface, which correlates with the magnetic entropy and thus the magnetic configuration space accessible at finite temperatures. Based on this approach, we show that deviations from stoichiometry have a huge impact on the local magnetic moment and can explain the experimentally observed low magnetic moments.

Surface energy of nanoparticles - influence of particle size and structure.

The surface energy, particularly for nanoparticles, is one of the most important quantities in understanding the thermodynamics of particles. Therefore, it is astonishing that there is still great uncertainty about its value. The uncertainty increases if one questions its dependence on particle size. Different approaches, such as classical thermodynamics calculations, molecular dynamics simulations, and ab initio calculations, exist to predict this quantity. Generally, considerations based on classical thermodynamics lead to the prediction of decreasing values of the surface energy with decreasing particle size. This phenomenon is caused by the reduced number of next neighbors of surface atoms with decreasing particle size, a phenomenon that is partly compensated by the reduction of the binding energy between the atoms with decreasing particle size. Furthermore, this compensating effect may be expected by the formation of a disordered or quasi-liquid layer at the surface. The atomistic approach, based either on molecular dynamics simulations or ab initio calculations, generally leads to values with an opposite tendency. However, it is shown that this result is based on an insufficient definition of the particle size. A more realistic definition of the particle size is possible only by a detailed analysis of the electronic structure obtained from initio calculations. Except for minor variations caused by changes in the structure, only a minor dependence of the surface energy on the particle size is found. The main conclusion of this work is that surface energy values for the equivalent bulk materials should be used if detailed data for nanoparticles are not available.

An Ab Initio Study of Thermodynamic and Mechanical Stability of Heusler-Based Fe2AlCo Polymorphs.

We use quantum-mechanical calculations to test a hypothesis of Glover et al. (J. Mag. Mag. Mater. 15 (1980) 699) that Co atoms in the Fe 2 AlCo compound have on average 3 Fe and 3 Co atoms in their second nearest neighbor shell. We have simulated four structural configurations of Fe 2 AlCo including the full Heusler structure, inverse Heusler polymorph and two other phases matching this idea. The highest thermodynamic stability at T = 0 K is indeed predicted for one of the phases with the distribution of atoms according to Glover and et al. However, small energy differences among three of the studied polymorphs lead to a disordered CsCl-structure-like (B2-like) phase at elevated temperatures. The fourth variant, the full Heusler phase, is predicted to be mechanically unstable. The global magnetic states are predicted to be ferromagnetic but local magnetic moments of Fe and Co atoms sensitively depend on the composition of the first and second coordination shells.

Ab initio inspired design of ternary boride thin films.

The demand to discover new materials is scientifically as well as industrially a continuously present topic, covering all different fields of application. The recent scientific work on thin film materials has shown, that especially for nitride-based protective coatings, computationally-driven understanding and modelling serves as a reliable trend-giver and can be used for target-oriented experiments. In this study, semi-automated density functional theory (DFT) calculations were used, to sweep across transition metal diborides in order to characterize their structure, phase stability and mechanical properties. We show that early transition metal diborides (TiB2, VB2, etc.) tend to be chemically more stable in the AlB2 structure type, whereas late transition metal diborides (WB2, ReB2, etc.) are preferably stabilized in the W2B5-x structure type. Closely related, we could prove that point defects such as vacancies significantly influence the phase stability and even can reverse the preference for the AlB2 or W2B5-x structure. Furthermore, investigations on the brittle-ductile behavior of the various diborides reveal, that the metastable structures are more ductile than their stable counterparts (WB2, TcB2, etc.). To design thin film materials, e.g. ternary or layered systems, this study is important for application oriented coating development to focus experimental studies on the most perspective systems.

Au55, a stable glassy cluster: results of ab initio calculations.

Structure and properties of small nanoparticles are still under discussion. Moreover, some thermodynamic properties and the structural behavior still remain partially unknown. One of the best investigated nanoparticles is the Au55 cluster, which has been analyzed experimentally and theoretically. However, up to now, the results of these studies are still inconsistent. Consequently, we have carried out the present ab initio study of the Au55 cluster, using up-to-date computational concepts, in order to clarify these issues. Our calculations have confirmed the experimental result that the thermodynamically most stable structure is not crystalline, but it is glassy. The non-crystalline structure of this cluster was validated by comparison of the coordination numbers with those of a crystalline cluster. It was found that, in contrast to bulk materials, glass formation is connected to an energy release that is close to the melting enthalpy of bulk gold. Additionally, the surface energy of this cluster was calculated using two different theoretical approaches resulting in values close to the surface energy for bulk gold. It shall be emphasized that it is now possible to give a confidence interval for the value of the surface energy.