A Cumulative Approach to Crystalline Structure Characterization in

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C: Physical Processes in Nanomaterials and Nanostructures

A Cumulative Approach to Crystalline Structure Characterization in Atomistic Simulations Ali Radhi, Vincent Iacobellis, and Kamran Behdinan J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b02608 • Publication Date (Web): 04 Jun 2018 Downloaded from http://pubs.acs.org on June 4, 2018

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The Journal of Physical Chemistry

A Cumulative Approach to Crystalline Structure Characterization in Atomistic Simulations Ali Radhi1, Vincent Iacobellis*1, Kamran Behdinan1 1

Advanced Research Lab for Multifunctional Lightweight Structures, Department of Mechanical & Industrial Engineering, University of Toronto, 5 King's College Road, Toronto, ON, Canada M5S 3G8 Abstract

Crystalline characterization poses a challenge when atomic deformation and phase transformation are occurring in an atomic simulation. Crystalline solids are typically characterized by parameters used to classify local atomic arrangements in order to extract features such as crack tips, dislocations, and free surfaces. One such characterization parameter, the Common Neighborhood Parameter (CNP), has been used as an approach to characterize those features with an enhanced formulation applicable to non-monoatomic interactions. The present work introduces a novel approach that extends the CNP to characterize crystalline structures by means of cumulative common neighborhood parameterization (CCNP) for arbitrary structures. The method is compared with centrosymmetry parameter (CSP) and common neighborhood parameter (CNA). The methods were applied to a molecular dynamics (MD) simulation of uniaxial tension in an aluminum nanowire. The results showed CCNP’s superior performance in detecting distinct surface features from bulk features with excellent parameter value ranges. The method was also extended to characterize a complex P42/mnm space group, non-monoatomic crystal with no common first nearest neighbors in a type I MD fracture simulation. The data refinement of the proposed method was applied to extract surface features like edges, roughness, corner, and undeformed surface atoms.

I.

Introduction

The current work of atomic simulations in today’s research is dedicated to investigating crystalline structures, computing their physical properties and analysing their resulting atomic features 1. Multiple centrosymmetric space groups can exist within a crystalline structure that may result from external stimuli with different load rates and sample sizes. Numerous materials can also exist in a polymorphic state that requires rigorous analysis in order to discern one state from the other. To that end, multiple characterization methods have been developed to study the underlying mechanisms and phases observed from an atomistic simulation using molecular dynamics (MD) 2. Currently, crystalline characterization methods have mainly revolved around the Centrosymmetry Parameter (CSP) 3-4 and the Common Neighbor Analysis (CNA) 5-6 for the identification of atomic defects, voids, stacking faults, etc. Both of these methods utilize a simple formulation to measure the local lattice disorder around an atom in a crystalline structure. As such, these approaches have been included with known numerical and atomistic visualization software packages such as Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) 7, Open Visualization Tool (OVITO) 8 and AtomEye 9. Most characterization methods aim to utilize the vector sum and its norm in order to describe the level of centrosymmetry around an atom with respect to its atomic neighbors. These methods are usually viable only for a limited set of crystalline structures, meaning experimental methods are sometimes the only option for more complex structures 10-11. However, atomic configurations of simple centrosymmetric structures have been thoroughly analyzed and have shown great results with CSP for Face Centered Cubic (FCC) and Body Centered Cubic (BCC) structures 1, 12. CNA has been used frequently in the literature, especially for crystalline deformation characterization 13 . CNA has also been commonly used in structural characterization of a crystalline solid for crack detection in multiscale adaptive modeling, crystalline phase transformations, and coarse-graining for complex fracture patterns 14-18 . The Common Neighborhood Parameter (CNP) has recently been developed to combine the advantages of both CNA and CSP 19. CSP is attributed to each atom individually but fails at adequately distinguishing between different atomic phases. On the other hand, CNA is capable of extracting atomic phases of simple crystal structures, but is not attributed to each individual atom and is more complex to implement. This limits the use of CNA in extracting deformation and surface features. Moreover, the work of CNP has been extended to more complex atomic systems with non-monoatomic structures 20. More recently, the work of Predominant CNP (PCNP) has been introduced in order to enhance the CNP method to include optimal cross-species interaction in atomic parameterization with a higher level of sensitivity for feature detection in complex centrosymmetric groups 21. Nevertheless, the PCNP is better at distinguishing bulk features while being less optimal for surface feature detection. Applications such as the performance of metallic micropillars 22-23 and energetics found in Rutile grain

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boundaries 24 require accurate detection and identification of surface features. Thus, there is a need for a characterization method which can effectively distinguish both surface and bulk features while being applicable to arbitrary atomic structures. To achieve this, a novel approach towards crystalline characterization by cumulative parameterization of the CNP and PCNP is introduced in this paper. Unlike other identification methods, this approach awards perfect crystalline structures with higher parameter values while atomic defects and features take on lesser values. It is easy to apply with the knowledge of first nearest neighbors for arbitrary structures to extract better information regarding surface configuration and other features within certain crystalline structures with no common first neighbors and no self-species interactions in binary atomic systems. Section II of this paper outlines the formulation of the proposed cumulative parameterization of the CNP and PCNP, while Section III demonstrates the proposed characterization parameter’s application to identifying atomic defect features in an aluminum nanowire as well as for crack growth in Rutile with a comparison to mentioned characterization parameters. II.

Cumulative Framework of the Common Neighborhood Parameter A. Common Neighborhood Parameter CNP based CCNP

As mentioned previously, CNP couples both CSP and CNA to characterize atomic and phase features simultaneously in MD simulations 19. Both CSP and CNP describe atoms as centrosymmetric when the attributed parameter is close to zero. Meanwhile, larger values are attributed to atoms belonging to atomic features such as surfaces, voids, cracks, dislocations, stacking faults, etc. These features become more dominant during deformation, where CSP could potentially overlook some atoms due to lack of opposing pairs or divergence from centrosymmetric structures. CNA would distinguish simple atomic phases, but does not attribute each atom with a distinct value. Also, CNA requires a more complex approach than CSP since it utilizes shells or clusters around each atom to obtain neighbor bond topology necessary for its calculation 1. Alternatively, CNP is able to characterize individual atoms through the following parameter: 1 Qi  ni

ni

nij

  r j 1 k 1

ik

 rjk 

2

(1)

, where ni is the number of first nearest neighbors of atom i . nij represents the number of common nearest neighbors between atom i and current neighbor atom j . The vector rik contains the distance vector between atom i and common neighbor atom k . The two vectors r jk and rik can be viewed in Figure 1 for an atom in FCC crystal. It can be seen that a single atomic pair would have values approaching zero if CNP analysis is used (as shown in Figure 1). For a crystal lattice, parameters approaching zero are considered as perfect lattice sites. In a case where a pair of atoms is present at crack tips or edges, the value may miss the atomic feature and consider it as perfect lattice atoms. In order to offer a better distinction and contrast between surface and bulk atoms, a cumulative approach is proposed. This cumulative framework awards perfect lattice atoms with higher values than other crystalline features like crack tips and surface atoms. The proposed analysis is called Cumulative Common Neighborhood Parameter (CCNP). Taking a cumulative approach modifies the CNP analysis to produce the following expression: 1 AQi  ni

 nij   rik  r jk  j 1  k 1 ni

  

2


(2)

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Figure 1: (a) Common neighborhood and (b) predominant common neighborhood parameterization of for a FCC crystal. Perfect FCC lattice contains 12 first nearest atoms and 4 common atomic neighbors.

This allows individual lumping of each atomic neighbor and their corresponding common neighbors’ vector norms to the central atom i . The method is defined as “cumulative” because the vector norm is attributed to the atom in an additive fashion from each nearest neighbor and their respective common neighbors. This would ensure that an atom’s sum norm is not zero when it contains instances of single atomic pairs or large deformation as discussed earlier. A new parameter BQi is also introduced as: BQi 

1 ni

 ni nij   rik  r jk  j 1 k 1

  

2

(3)

The BQi parameter serves as a means of amplification or demagnification of the calculated parameters using a cumulative framework to provide a better distinction between different features. Hence, a cumulative parameterization results in higher values for high atomic packing locales. B. Predominant Common Neighborhood Parameter PCNP based CCNP The PCNP prioritizes the common neighbors over the first neighbors in an effort to enhance parameter sensitivity to bulk features and deformation detection 21. It is also able to characterize complex atomic structures, such as Zirconium Diboride, that lack in common first neighbors for certain atomic species on cross-species interactions. As it stands, the CNP vector sum of atomic neighbors as presented in Figure 1a shows a vector sum of zero in the direction of rij . Such situations in an FCC crystal are undesirable, especially when atom i is located within or close to edges or corner regions. Alternatively, in Figure 1b the vector sum would have a non-zero value in the direction of rij for the PCNP formulation. This is done by prioritizing the common neighbors between atoms i and j over the current neighbor j in the vector loop sum as shown in Eq.(4) and Eq.(5). This is accomplished by changing the two position vectors in Eq.(1) from rik and r jk to rij and rkj , respectively (indices j and k are switched). PCNP is defined by two parameters. The first is defined as: 1 Pi  ni

ni

nij

  r j 1 k 1

ij

 rkj 

2

(4)

,while the other has the following expression: 1 Ni  ni

ni

nij

 r j 1 k 1

ij

 rkj 

2


(5)


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The first one has been shown to extract crystalline phases and dislocation features, while the second one highlights the centrosymmetry of the atoms with better distinction for surface and crack tip atoms. The proposed cumulative approach, CCNP, also aims to distinguish atomic features for a better understanding of surface deformation mechanics present in more complex structures. The following expression represents a cumulative formalism to the PCNP approach: APi 

1 ni

 nij    rij  rkj    j 1  k 1  ni

2

(6)

Another parameter for the CCNP is also introduced to enhance contrast of certain features, and it is expressed as: 1 BPi  ni

 ni nij   rij  rkj  j 1 k 1

  

2

(7)

Much like the PCNP, the CCNP approach can make use of second common nearest neighbors to account for situations with no common first neighbors by making the index j go over ni( l ) . The value of l specifies either first or second nearest neighbor parameterization (a value of 1 or 2 respectively). Such a situation is vital when considering certain tetragonal crystal systems, like those with space group P42/mnm. A Rutile polymorph of Titanium Dioxide TiO2 has such configuration. Titanium Ti atoms have 6 oxygen O atoms as its first nearest neighbors, while each oxygen atom has 3 first nearest neighbors of Ti atoms as shown in Figure 2. CSP cannot be adequately applied here as it cannot handle an odd number of atomic neighbors for oxygen atoms. Conventional CNA is also not applicable for discerning the specified space group of the crystalline structure. It can also be seen that there are no common first neighbors for all the atoms in such a structure. Moreover, the first neighbors are all involved with cross-species interactions. Using the second nearest neighbor formulation, the pair of atoms fall into the same atomic species (i.e. self-species interactions). This is applicable to non-monoatomic systems since the second neighbors are designated indirectly by the cross-species interaction between first neighbors through the specified interatomic potential in the MD simulation. The second neighbor is defined here as the first nearest neighbor of the first nearest neighbor, excluding the current central atom i . This way, self and cross-species atomic pairs are accounted for using such a definition, albeit indirect for cross-species pairing. However, introducing the CCNP approach would ensure direct cross and self-species interactions between the common neighbors as it considers individual vector norms of the surrounding atoms. As such, the proposed approach can describe local configurations with an enhanced contrast by means of first and second nearest neighbor formulations.

Figure 2: Unit Cell of a Rutile polymorph of TiO2 (silver atoms refer to Ti atoms while red ones indicate oxygen atoms).

III.

Results

Next, we will introduce two case studies to validate the proposed CCNP parameterization. When extracting atomic features, data refinement of the calculated parameters is in order. The first step in the data refinement is to visually locate certain atomic regions that correspond to a selected atomic feature with approximate relative position matching and then subsequently obtain the range of CCNP values for this feature. Since each individual atom will


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have a CCNP value, refinement of the data must be ensured to aid in this feature extraction. In the second step of the data refinement, the maximum and minimum values of the parameter associated with the feature are reduced and increased, respectively, to establish a benchmark range of CCNP values for the desired feature. Since parameter values for CCNP are expected to have a reversed feature values from other parameters, CCNP values are first reversed (i.e. high becomes low and low becomes high) before going through the data refinement stage. A. Feature Identification in Strained Aluminum Nanowire Using CCNP In order to assess the proposed method, a study of an aluminum nanowire is conducted using MD simulation. FCC aluminum was used to set up 40,000 atoms with periodic boundary conditions applied only in the z direction (coinciding with the lattice direction) and a time step of 1 fs was chosen. The initial domain of the study spanned 40×40×400 Å. To account for thermal expansion, the atomic system was heated to 100 K with an NPT ensemble using a Nose-Hoover Thermostat 25 for 20,000 steps. The temperature relaxation was achieved after roughly 10,000 time steps. The deformation stage utilized a canonical ensemble (NVT) using the same thermostat. Three Nose-Hoover chains were utilized for the thermostat and barostat. The nanowire was deformed with a strain rate of 5×108 s-1 for 20,000 time steps. The x and y directions corresponded to and lattice directions, respectively. The interatomic potential chosen for this study was an Embedded Atom Method EAM potential 26. The Young’s modulus for the MD simulation was evaluated at 166 GPa as obtained from the linear segment of the stress-strain curve shown in Figure 3. The obtained value corresponds well to previous DFT studies 27, where the Young’s modulus was obtained at roughly 173 GPa for ultra-thin Al nanowires (with radii of 8 nm and below) along the applied load direction.

Figure 3: Stress-strain curve for the Al nanowire case study

The main focus of this research was to observe surface features, crack surfaces, edges and dislocation cores. However, we first look at performance CCNP compared to the other methods in extracting features within the bulk of the atomistic domain, in particular identifying FCC and HCP phases. The results using CSP, CNA and CNP methods are illustrated in Figure 4 at the pre-fracture stage. The CNA results are obtained using OVITO package 8, where Figure 4b shows atoms belonging to FCC (blue) and hexagonal closed packed HCP (yellow) phases. CSP is capable of showing the free surfaces of the simulation and atoms of high non-centrosymmetry. However, it needs a higher level of refinement in order to distinguish surface atoms, stacking faults, dislocation and other features. It is also unable to distinguish the necking region from surrounding atoms. It is worth mentioning that the stacking faults correspond to HCP phases in an FCC lattice. CNP results are shown in Figure 4c,d after data refinement to extract FCC and HCP locales along with the surface. The refinement is adequate compared to CNA, but still includes some unwanted atoms belonging to the necking region or surface dislocations that can be misinterpreted as HCP locales. Also, the surface refinement could not distinguish necking regions from the rest of the domain or other surface features. Alternatively, Figure 5 shows the characterization of the deformed structure with the PCNP approach using the Pi parameter. The results are suitable for obtaining both crystalline phases when using common second neighbors, while the N i parameter shows a better distinction of surface atoms, as shown in Figure 6, when using



common first neighbors. The first neighbor formulation is more adequate for the N i parameter due to its dependency on the centrosymmetry in the atom’s immediate vicinity.

Figure 4: (a) CSP, (b) CNA and (c) CNP characterization of aluminum nanowire case study. The CNA and CNP results were refined to show FCC and HCP locales. (Blue are FCC and yellow is HCP phases in CNA results).

Figure 5: Structural characterization of aluminum nanowire using the Pi parameter of PCNP analysis. The results show (a) common first neighboring, (b) common second neighboring and (c) refinement of (b) results to show FCC and HCP locales.


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Figure 6: Structural Characterization of aluminum nanowire using the N i parameter of PCNP analysis. (a) and (c) show common first and second neighboring, respectively, while (b) and (d) shows the corresponding refinement to identify surface atoms.

The results for extracting the bulk features corresponding to FCC and HCP locales from the proposed CCNP analysis are shown in Figure 7. The results show excellent extraction of FCC locales both in first and second common neighboring of CCNP. However, HCP locales are observed only in Figure 7b,d,f,h. The CCNP formulation of CNP factors (Eq.(2) and Eq.(3)) have the best extraction of HCP locales found in the system with common second neighboring. Alternatively, the first common neighboring showed a more clear result for FCC locales than other CCNP factors as evident from Figure 7a,c,e,g. Moreover, the CCNP formulation proved better in distinguishing FCC locales than CSP, CNP and PCNP from Figure 4 and Figure 5c. Although the results for HCP locales can still be observed, the PCNP results using the Pi parameter still hold a slightly better distinction for HCP atoms.



Figure 7: Bulk feature extraction using CCNP formulation of (i) CNP and (ii) PCNP factors. (a) and (c) utilize Eq.(2) and Eq.(3), respectively, with common first neighboring, while (b) and (d) are produced with common second neighboring. (e) and (g) utilize Eq.(6) and Eq.(7), respectively, with common first neighboring, while (f) and (h) are produced with common second neighboring. Refer to Table 1 for typical values of the extracted features.


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Although the PCNP Pi parameter holds a slightly better distinction for HCP atoms, the same could not be said when observing surface features, where the CCNP shows superior performance over PCNP in terms of surface feature extraction. Here, surface dislocations indicate dislocation along the interface of two different phases of the crystal 28. Another way to see this is that an edge dislocation would create a tilt interface between two identical boundaries. Identification of surface features was performed based off of the results shown in Figure 8 and Figure 9. Figure 8a-d shows the CCNP results, calculated using Eq.(2), before and after refinements for common first neighboring. Figure 8e-h, calculated using Eq.(3), refers to the same results, but with common second nearest neighboring. All the result value ranges in CCNP are reversed before refinement in order to better compare the results with the other characterization methods which, as discussed previously, associate localized atomic features with higher attributed values as opposed to lower ones. The formulation of CCNP using Eq.(2) showed a better distinction of dislocation cores when employing the first common neighboring (Figure 8a). As an example, the inset figure in Figure 8a shows the typical atomic locales that correspond to a dislocation core. The emanating dislocations were found to lie on the [1 1 1] plane, which is expected in an FCC lattice. Alternatively, Figure 8h showed the best results when refining the characterized structure to show surface atoms, embedded dislocations and the resulting surface roughness during deformation. The FCC and HCP locales were also obtained, with a much higher distinction of FCC but not for bulk HCP. However, the surface features obtained makes this method much better in terms of surface identification compared to the PCNP’s N i parameter. The cumulative aspect of the CCNP can visually be seen in Figure 8a,c,e,g, where surface atoms correspond to the lower range of the CCNP values while perfect lattice atoms have a higher range. CCNP values of dislocation cores are along the high end of the value range due to high concentration of atomic neighboring at that vicinity. The same could be said when applying the CCNP approach to the PCNP factors (Eq.(6) and Eq.(7)). Their results are shown in Figure 9a-d for common first neighboring and Figure 9e-h for common second neighboring. Comparing Figure 8 and Figure 9 indicates that prefect lattice atoms had a better distinction in Figure 8 than Figure 9 (evident from Figure 8e), while surface roughness, dislocations and edge features are more visible in Figure 9 (as seen in Figure 9h). However, Figure 8h still showed the best distinction of surface atoms not belonging to any roughness, edge, dislocation or necking features. Both Figure 8d and Figure 9d showed distinguishable features related to atoms at the necking region. This is of great importance for possible fracture site detection. The PCNP is still better in visualizing HCP locales resulting from bulk dislocations and stacking faults, but it does not effectively distinguish bulk from surface dislocations. It is worth mentioning that all these methods employ different formulations such that feature extraction differs from one factor to the next according to their value ranges. The CSP and CNP value refinement was unable to distinguish surface from necking region atoms as shown in Figure 4b-d. Alternatively, CCNP has shown an affinity in visualizing atomic surface features over the rest of the parameterization. Table 1 shows a description of typical values obtained for the extracted features of the proposed study.



Figure 8: Surface feature refinement using CCNP formulation of CNP factors with common (i) first and (ii) second neighboring of an aluminum nanowire. (a) and (e) utilize Eq.(2), while (c) and (g) correspond to Eq.(3). The other sub-figures show the refinement of surface atoms of each respective result after reversing their value ranges. The inset in (i) shows dislocation cores refined with CCNP results and also highlights a visible section of (a) that corresponds to a core. Refer to Table 1 for typical values of the extracted features.

Figure 9: CCNP formulation of PCNP approach using common (i) first and (ii) second neighboring of an aluminum nanowire. (a) and (e) utilize Eq.(6), while (c) and (g) correspond to Eq.(7). The other sub-figures show the refinement of surface atoms for each respective result after reversing their value ranges. Refer to Table 1 for extracted feature values.


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1 2 3 Table 1: Value ranges of MD structural characterization of Al nanowire features from Figure 4 to Figure 9 (values are in Å2and UI refers to Unidentifiable). 4 5 Characterization 6 FCC HCP Dislocation Edges and Roughness Necking Region Undeformed Surface Factor 7 CSP ~0 5-7 20-25 50-70 UI 30-50 8 CNP ~ 0 5-10 35-45 >50 UI 30-50 9 st Pi (1 Neighboring) 250-350 NA 175-250 UI 5-100 UI 10 nd 11 Pi (2 Neighboring) ~0 5-15 50-65 UI 70-90 UI 12 N i (1st Neighboring) ~0 UI 125-180 UI 150-250 50-150 13 nd N (2 Neighboring) ~0 UI 500-700 UI 600-800 250-400 i 14 15 AQi (1st Neighboring) 350-450 200-250 > 500 150-200 0-120 250-300 16 AQi (2nd Neighboring) 1300-1400 800-900 > 1430 700-800 0-650 850-1000 17 BQi (1st Neighboring) 4000-5500 1500-2000 > 7000 700-1500 0-1000 1500-3000 18 nd 4 19 BQi (2 Neighboring) (× 10 ) 6-7 1.5-2 > 7.3 1-1.5 0-1 2-3.5 st 20 APi (1 Neighboring) 300-400 200-250 > 550 130-200 0-130 200-300 21 nd APi (2 Neighboring) 450-600 270-330 > 720 120-300 0-150 350-450 22 st BPi (1 Neighboring) 4000-6000 1700-2200 > 7000 700-1200 0-1300 2000-3000 23 24 BPi (2nd Neighboring) (× 104) 1.5-2 0.7-1 > 2.3 0.3-0.5 0-0.3 0.7-1.3 25 26 27 B. Feature Identification in TiO2 using CCNP 28 A non-monoatomic structure is investigated in a Rutile polymorph of TiO2. The MD simulation was set up 29 with 5984 Ti and 11968 O atoms with Matsui-Akaogi interatomic potential 29. The value of 1 fs was chosen for the 30 time-step size with a strain rate of 2.5×108 s-1 applied every 100 steps for a total of 220,000 steps in an NVT 31 ensemble. Initially, the system was simulated with an NPT ensemble for 200,000 steps to account for thermal 32 expansion and to set the system’s temperature to 300 K. Again, we utilized three Nose-Hoover chains 25 for the 33 thermostat and barostat. The temperature relaxation was obtained after 100,000 time steps. The strain was applied on 34 the z axis, which is oriented in the direction (x and y directions correspond to and , 35 respectively). A hole of 10 Å is introduced at the center along the y direction with periodic boundary conditions in 36 all directions. The simulation domain spanned 101×18.5×101 Å. The stress-strain curve was also obtained for this 37 study along the [0 0 1] plane, as shown in Figure 10. The value of Young's modulus obtained from the stress-strain 38 curve was approximately 464 GPa. This value is comparable with both the experimental value of approximately 484 39 GPa 30-31 and the value obtained from DFT simulations of approximately 472 GPa 32-33. From this, it was concluded 40 that the interatomic potential used in this simulation was adequate for the MD simulation of such metal oxide 41 ceramics. 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 Figure 10: Stress-strain curve of the Rutile fracture case study. 57 58 59 ACS Paragon Plus Environment 60


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Figure 11a shows the structure after 53,000 load steps. It can be seen that CSP results in Figure 11b are harder to interpret due to the lack of opposing neighbors necessary for the calculation of the CSP, while CNA analysis is unable to characterize the embedded features as it recognizes the entire domain as an unidentifiable crystalline phase. The structure does not contain any common first neighbor, so the CNP formalism with common first neighbors would produce all zero values for those atoms. Figure 11c includes a CNP analysis with common second neighbors, where it can be seen that each atomic species has different values within the bulk structure. This is undesirable as surface and bulk features become harder to distinguish when multiple atomic species are present in the system. Ideally, the parameter value for each atom relates to the surface or bulk features where the atom is located and is independent of the atomic species. The presented results are all normalized in order to achieve feature extraction attributed to both atomic species. When applying the PCNP formalism, it showed better results than CNP characterizing underlying features regardless of the atomic species. Both PCNP parameters are computed with common second neighboring, where the N i parameter is able to highlight corner and crack tip atoms, but not all surface atoms as shown in Figure 12c,d. The Pi parameter has better performance for determining bulk atoms and some dislocation propagations, but not corner lines and crack tip atoms as shown in Figure 12a,b. Figure 13 depicts the structural characterization of a Rutile structure when applying the CCNP approach on CNP and PCNP methods. The CCNP results using Eq.(2) showed improved surface-crack atom distinction compared to CNP and PCNP, with normalized values close to 0 referring to corner and chain-suspended atoms. Eq.(3) showed a better contrast of surface atoms from bulk phase with indications of dislocation propagation around crack tips. The method showed an enhanced distinction between undeformed surface, crack tip and edge atoms. Additionally, the results from Figure 13a,b were able to distinguish atoms that are not on the surface, but close to it. This could highlight the ensuing dislocations that emanate from earlier dislocation propagation around the crack. As for CCNP results obtained from Eq.(6), there was lesser distinction of surface atoms as shown in Figure 13c. On the other hand, it does show a clear cut distinction between surface and bulk atoms more clearly than previous results with higher cumulative values accredited to the bulk than all the rest of the features. Figure 13d corresponds to values obtained from Eq.(7), but had a lesser performance than the previous results. The simulation was able to show the initial erratic propagation of the crack from the introduced hole, but it settled down to a lateral crack with the preferred cleavage plane at [1 1 0]. All of the presented results suggest a large normalized value range corresponding to the undeformed bulk phase of the lattice while low values indicate embedded atomic features in a cumulative parameterization (as the CCNP definition suggests). The Rutile ceramic adapts a brittle-like behavior, presenting a more unique characterization of crack atoms with fewer nucleations of dislocations compared to what is typically found in ductile materials. Table 2 gives a refined value range to the normalized parametrization for the extracted atomic features found from Figure 12 and Figure 13. Table 2: Normalized parametric value ranges of Rutile fracture geometry with PCNP and CCNP approaches applied to Figure 11 to Figure 13. (UI refers to an unidentifiable feature).

Characterization Parameter Pi

Tetragonal (undeformed) 0.6-0.8

Ni AQi BQi

Surface

Corner

Crack tip

Dislocation

0.5-0.6

0-0.1

0.2-0.4

0.4-0.5

0-0.1

0.14-0.2

0.1-0.12

0.4-0.5

UI

> 0.75

0.2-0.4

0-0.25

0.25-0.4

0.4-0.6

> 0.8

0.2-0.3

0-0.1

0.3-0.45

0.4-0.6

APi

0.5-0.6

0.25-0.35

0-0.1

0.35-0.45

0.4-0.55

BPi

0.5-0.65

0.2-0.35

0-0.1

0.25-0.4

0.4-0.55


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Figure 11: (a) Deformed Rutile structure after 53,000 load steps (silver and red refer to Ti and O atoms, respectively). (b) CSP analysis of deformed Rutile and (c) CNP analysis of deformed Rutile in MD simulation.

Figure 12: PCNP analysis on the fractured Rutile structure using (a) Pi and (c) N i parameters. (b,d) Data refinement of the two respective parameters. All the values are normalized for cross-species assessment.



Figure 13: CCNP analysis on a MD study of Rutile fracture with (a) Eq.(2), (b) Eq.(3), (c) Eq.(6) and (d) Eq.(7). All the values are normalized for cross-species assessment.

IV.

Conclusion

In the presented study, a cumulative approach is introduced to characterize complex crystalline systems and embedded surface features by modifying both the CNP and the PCNP methods. The approach relies on rewarding perfect crystalline phases and large atomic neighboring with higher parametric values than other crystalline features existing in the atomic system. The method is able to characterize non-monoatomic systems with complex centrosymmetric crystals and/or space group configurations. The proposed CCNP method is able to identify crystalline features with the simple knowledge of first nearest neighbors, and it is much simpler to implement than the CNA. CCNP was able to illustrate a higher sensitivity as to the type of surface atoms, which are categorized as perfect surface, surface dislocation or edges, and roughness atoms. Also, it was able to distinguish FCC locales in the aluminum nanowire case study, but the PCNP Pi parameter was slightly better in extracting HCP atoms. Additionally, the common second neighboring in CCNP can quantify cross-species interaction for the presented Rutile case study by lumping each displacement vector of pair atoms to the parameter in an additive fashion, generating an accumulating value that increases with the existing neighboring locales. The original work of CNP was atomic-species dependent, which is highly undesirable in geometric feature detections. The method is applicable to arbitrary structures, where the CCNP can benchmark the extracted features in order to identify one phase/feature from the rest of the domain. The CCNP based on the CNP formulation (Eq.(2) and Eq.(3)) showed a significant improvement in the distinction of dislocations and surface atoms from crack tip and corner atoms, but the CCNP based on the PCNP formulation (Eq.(6) and Eq.(7)) proved better at isolating surface atoms from the rest of the simulation domain. The work has potential in adaptive multiscale simulations, where coarsening through techniques like coarse-graining (CG) hold a better computational performance in given atomistic models. Some adaptive CG simulations still relies on simplistic CSP and potential energy variations in crack tip detection 18. Future work may include the use of such methods in order to predict polymorphic behavior of materials, much like Brookite and Anatase reverting to a more stable Rutile phase under elevated thermal and/or mechanical loading 34-36.


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Author Information Corresponding Author *E-mail: [email protected] . Phone: +1 (416) 946-3631, Fax: +1 (416) 978-7753. ORCID Ali Radhi: 0000-0002-6074-2263 Vincent Iacobellis: 0000-0001-7297-8495

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Figure 1: (a) Common neighborhood and (b) predominant common neighborhood parameterization of for a FCC crystal. Perfect FCC lattice contains 12 first nearest atoms and 4 common atomic neighbors. 53x27mm (300 x 300 DPI)


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Figure 2: Unit Cell of a Rutile polymorph of TiO2 (silver atoms refer to Ti atoms while red ones indicate oxygen atoms). 62x50mm (300 x 300 DPI)



Figure 3: Stress-strain curve for the Al nanowire case study. 53x41mm (600 x 600 DPI)


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Figure 4: (a) CSP, (b) CNA and (c) CNP characterization of aluminum nanowire case study. The CNA and CNP results were refined to show FCC and HCP locales. (Blue are FCC and yellow is HCP phases in CNA results). 54x16mm (600 x 600 DPI)



Figure 5: Structural characterization of aluminum nanowire using the parameter of PCNP analysis. The results show (a) common first neighboring, (b) common second neighboring and (c) refinement of (b) results to show FCC and HCP locales. 55x17mm (600 x 600 DPI)


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Figure 6: Structural Characterization of aluminum nanowire using the parameter of PCNP analysis. (a) and (c) show common first and second neighboring, respectively, while (b) and (d) shows the corresponding refinement to identify surface atoms. 125x103mm (600 x 600 DPI)



Figure 7: Bulk feature extraction using CCNP formulation of (i) CNP and (ii) PCNP factors. (a) and (c) utilize Eq.(2) and Eq.(3), respectively, with common first neighboring, while (b) and (d) are produced with common second neighboring. (e) and (g) utilize Eq.(6) and Eq.(7), respectively, with common first neighboring, while (f) and (h) are produced with common second neighboring. Refer to Table 1 for typical values of the extracted features. 210x570mm (600 x 600 DPI)


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Figure 8: Surface feature refinement using CCNP formulation of CNP factors with common (i) first and (ii) second neighboring of an aluminum nanowire. (a) and (e) utilize Eq.(2), while (c) and (g) correspond to Eq.(3). The other sub-figures show the refinement of surface atoms of each respective result after reversing their value ranges. The inset in (i) shows dislocation cores refined with CCNP results and also highlights a visible section of (a) that corresponds to a core. Refer to Table 1 for typical values of the extracted features. 83x40mm (600 x 600 DPI)



Figure 9: CCNP formulation of PCNP approach using common (i) first and (ii) second neighboring of an aluminum nanowire. (a) and (e) utilize Eq.(6), while (c) and (g) correspond to Eq.(7). The other sub-figures show the refinement of surface atoms for each respective result after reversing their value ranges. Refer to Table 1 for extracted feature values. 66x35mm (600 x 600 DPI)


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Figure 10: Stress-strain curve of the Rutile fracture case study. 52x39mm (600 x 600 DPI)



Figure 11: (a) Deformed Rutile structure after 53,000 load steps (silver and red refer to Ti and O atoms, respectively). (b) CSP analysis of deformed Rutile and (c) CNP analysis of deformed Rutile in MD simulation. 53x15mm (600 x 600 DPI)


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Figure 12: PCNP analysis on the fractured Rutile structure using (a) and (c) parameters. (b,d) Data refinement of the two respective parameters. All the values are normalized for cross-species assessment. 120x95mm (600 x 600 DPI)



Figure 13: CCNP analysis on a MD study of Rutile fracture with (a) Eq.(2), (b) Eq.(3), (c) Eq.(6) and (d) Eq.(7). All the values are normalized for cross-species assessment. 118x92mm (600 x 600 DPI)


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36x25mm (600 x 600 DPI)


A Cumulative Approach to Crystalline Structure Characterization in

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