The sample size effect in metallic glass deformation

The sample size effect on deformation mode of glasses is one of the most misunderstood properties of this class of material. This effect is intriguing, since materials deemed macroscopically brittle become plastic at small size. We propose an explanation of this phenomenon for metallic glasses. A thermodynamic description of the local rearrangement zones activated under an applied stress is proposed. Using the Poisson distribution to describe the statistics of these zones and the statistical physics to associate entropy, we define a critical sample size for the change in the deformation mode. Predictions are in agreement with experimental observations and reveal hidden structural parameters describing the glassy state.

Scientific RepoRtS | (2020) 10:10801 | https://doi.org/10.1038/s41598-020-67813-w www.nature.com/scientificreports/ specific configurational entropy happens with decreasing sample size. Homogenous deformation should be then allowed when the value at the glass transition is reached. A change in entropy with size has not been considered yet, but was the number of energy minima in the potential energy landscape (PEL) description of glass 11,12 . The PEL is appropriate for modelling the structural local rearrangement under an applied stress and the variation in minima of the energy density supports well an evolution of configuration with size.

Model and experimental evidence
The glass is discretized in non-periodic small regions, we call "clusters" of size l , having various potential energy in the PEL distribution. Figure 1A, is a 2D scheme of a glass in such description, where black clusters are in favorable energy configuration for rearrangement under an external solicitation. A stress is characterized by its spatial orientation and breaks the average spherical symmetry of the glass. It results that only clusters favorably oriented with respect to the applied stress are selected and the resulting atomic displacements are in a close direction to the stress orientation (Fig. 1B). The PEL is heterogeneous and locally non-isotropic 13 . Then, activation of a favorable cluster, (black in Fig. 1B) may trigger atomic rearrangement over many k successive clusters nearby (Fig. 1B). We call ligament of size h ≈ k × l , the successive clusters (black and grey in Fig. 1B) producing the total displacement h . The total number of clusters is very large ( n c ≫ 1 ) and the probability to find a ligament is very small ( p ≪ 1 ). Such rare event configuration is described by a Poisson distribution which gives the probability P k = k e − /k! , of finding ligaments formed of k successive clusters, where = p × n c is the average number of clusters giving the most expected ligament size h = × l . This model description was tested by comparison with experimental data. At first, the predicted displacements were naturally associated to serrations (pop-in) events observed in nano-indentation or nano-pillar compression tests. The mechanical test is probing the local structure of the glass by observing in the displacement extent (serration size) the capacity for atomic rearrangement. It is well known that the serrations events are strain rate dependent 14 , which would mean that the serrations size distribution is not unique. However, it is emphasized that the most faithful structural description obtained from the mechanical probing, necessarily required that the time scale of the experiment is lower than the timescale of the atomic rearrangement. To an experimental point of view, it is then obvious that such experiment must be carried out in quasisatic condition that is at the lower strain rate as possible.
About 7,980 serrations were measured from 320 nano-indentations ( Fig. 2) performed on a Mg based metallic glass 15 (see supplementary materials and methods).
The normalized experimental distribution P e (h) is plotted in the Fig. 3 and compared to a normalized Poisson distribution P(h) with the fitting parameters: h=3.47 ± 0.03 nm, l=0.64 ± 0.03 nm, which gives ≈ 5 . The waiting times, δt distribution (inset of Fig. 3) is also consistent with the Poisson statistic and verifies P(δt) = Ae − δt . It was reported that the activation volume controlling the shear band formation in this Mg glass is of the order of 3 atoms 16 . Similar value was reported by Schall and collaborators 17 and Ju and collaborators 18 for different materials. The present statistical analysis is consistent with that result, considering that an elementary displacement of the order of l , which is about two interatomic distances, needs a rearrangement of at least 3 atoms. It is satisfactory to find analogy between cluster size used for the discretization of the glass and the activation volume controlling shear band formation having a physical meaning.
In a statistical physics approach, as proposed by Gibbs and Adam 9 , the probability to find a ligament formed of k clusters is P k = g k e −u k k b T /Z . u k is the energy of a ligament of k clusters in the PEL, g k is a degeneracy factor and Z is the partition function. If we arbitrarily set for the most probable ligament of clusters, g =1, then . k b is the Boltzmann constant and T is the absolute temperature. Energy of the ligament is unknown but we derive a simple apparent entropy per ligament depending only on ,

The size effect
Among the simplicity of the result, the complexion = !  www.nature.com/scientificreports/ Reducing n means that the statistic is changing from Poisson to binomial, a well-known property for these distributions. In other words, the way events are "drawn" changes with the reduction of the sample size. From this result and considering the Gibbs and Adam analysis 9 , we define the "sample size-temperature" equivalence writing that the entropy of an infinite sample at the glass transition temperature is equal to the entropy of the small size specimen at room temperature. This simple equality needs however, some special care. Entropy from room temperature to glass transition is calculated numerically with heat capacity c p measured for many metallic glasses (Table 1). It is emphasized that entropy of the small size sample is calculated with data obtained from the mechanical testing, then considering only the ligament probed by the mechanical solicitation in its specific direction. Consequently, the entropy from room to glass temperature must be rescaled by the ligaments fraction probed in the mechanical testing. Heat capacity c p is rescaled in number of ligaments (or mole of ligaments), that is by ∼ 3 , assuming cluster formed of about 3 atoms. The balance between the entropy of the small size specimen and the entropy of infinite specimen at the glass transition writes, with the atomic volume and n l n the fraction of ligaments probed: The volume of the specimen is V = n3 and the volume density of ligament is = n l V = n l n3 . After combination, the critical size for the transition in the specimen size effect is obtained: and are structural features of the glass determined from experiments and characterizing the deformation dynamics.
For the Mg based metallic glass used in our experimental work, = 5 is determined from a robust statistical distribution. The ligaments density is estimated of 10 -8 nm −3 , assuming a hemispheric zone, v ≈ 60h 3 max , where h max is the maximum indent depth. An entropy from room temperature to glass transition was numerically evaluated from 19 of about 10 J.mol -1 k -1 . Then the critical size for the transition from brittle to ductile is calculated of about 400 nm, consistent with observations by Lee and collaborators on pillars with diameter smaller than 1,000 nm 20 .

Discussion and concluding remarks
The approach was applied for various metallic glasses tested on nanopillars ( Table 1). The volume of pillar, impacted during deformation is v ≈ ǫV , where ǫ is the strain and V is the pillar volume. The value is estimated from the serration which appears the most (the much probable) in the deformation curves and elementary cluster formed of 3 atoms is assumed. The Table 1 shows rather good estimation of the critical size when comparing experimental observations and the calculated values, .
To come back on initial assumption of a "sample size-T g " equivalence, one relies on the relation of the critical size (3). Observing that the c p variation with temperature are little for the various glasses ( c p is from 25 to 90 J.K -1 .mol -1 ), one derives the size-temperature dependence, from the first terms of a Taylor development: The relation demonstrate the starting point of the approach with the size ( )-temperature ( T g ) or entropy ( C p , T g ) dependence (Fig. 4). The relation (4) also reminds that this equivalence is done between two different property-dependent values, a mechanical one ( ) and thermal one ( T g ) that is why structural parameter, , are necessary to make compatibility between the two. The relation (4) predicts that the critical size is as small as  (Table 1). In this work, the reference temperature is the room temperature; it is interesting to notice that infinite critical size is well predicted for reference temperature of T g and that critical size → 0 when T → 0 . This donnot consider variations of value and with temperature, which is most likely the case.
A dominant parameter impacting the critical size , is the volume density of ligaments as observed from the data in the Table 1. is the concentration of the local zones in the glass where rearrangement is able to occur under stress. One of these zones will evolve forming shear band. It is commonly argued that multiplying shear bands would be much favorable for plasticity while our results would indicate the contrary. The critical size is as large as is small. In other words, the glass would be as robust against brittleness as it is poor in easy rearranging zones where softening happens under stress. This was called "fertile" zones by Guo and collaborators 7 . The argument support well the evidence of large critical size observed for oxide glasses which are probably more "structurally perfect" compared to metallic glasses. This should be examined under the angle of difference in the PEL between oxide and metallic glasses.
In the deformation process, and are novel parameters for describing the glass structure. It has been shown that metallic glass is formed of a distribution of clusters having varied deviations from a perfect icosahedron 21 . The local structure was earlier described by a distribution of atoms and free volume, which is convenient in particular for modeling mechanical behavior as developed by Spaepen 22 and Argon 23 . An alternative to direct or mean field atomic structure is the PEL 11 where atoms are omitted and the properties related to local variation of the system energy. This was used for the modelling glass rheology 24 . Our approach suggests that the glass can be described by the unique average value and the density of ligaments . The first is corresponding to elementary translation in the glass as Burgers vector is in crystals. The second is the density of local "defects" involved in the deformation process analogous to dislocations density in crystals though having different properties.