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Stress correlations in nearcrystalline packings
by Roshan Maharana and Kabir Ramola
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Submission summary
Authors (as registered SciPost users):  Roshan Maharana 
Submission information  

Preprint Link:  scipost_202311_00022v1 (pdf) 
Date submitted:  20231113 14:18 
Submitted by:  Maharana, Roshan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We derive exact results for stresscorrelations in nearcrystalline systems in two and three dimensions. We study energy minimized configurations of particles interacting through Harmonic as well as LennardJones potentials, for varying degrees of microscopic disorder and quenched forces on grains. Our findings demonstrate that the macroscopic elastic properties of such nearcrystalline packings remain unchanged within a certain disorder threshold, yet they can be influenced by various factors, including packing density, pressure, and the strength of interparticle interactions. We show that the stresscorrelations in such systems display anisotropic behavior at large lengthscales and are significantly influenced by the prestress of the system. The anisotropic nature of these correlations remains unaffected as we increase the strength of the disorder. Additionally, we derive the large lengthscale behavior for the change in the local stress components that shows a $1/r^d$ radial decay for the case of particle size disorder and a $1/r^{d1}$ behavior for quenched forces introduced into a crystalline network. Finally, we verify our theoretical results numerically using energyminimised static particle configurations.
Current status:
Reports on this Submission
Strengths
1 detailed analytical analysis of stress correlations in nearcrystalline solids
Report
In this manuscript, the authors introduce a new technique for
calculating stress correlations in nearcrystalline packings. The
theoretical findings exhibit remarkable agreement with the simulation
results, showcasing the efficacy of the developed theory. This
advancement holds promise for investigating systems spanning both
crystalline and amorphous solids. Given its potential impact, I
believe this work is suitable for publication. I would
recommend its publication after the authors address the following
comments.
One main point highlighted by the authors is that the macroscopic
elastic properties of nearcrystalline packings remain unchanged below
a given disorder threshold, which is demonstrated in
Fig. 2(a). However, it will be beneficial to depict the deviation of
<B> or <G> from the theoretical value, B_theory/G_theory, of perfect
crystals on a logarithmic scale, i.e., elucidating whether the
deviations "B_theory  <B>" and "G_theory  <G>" from nonaffine
contribution also follow a powerlaw scaling with the strength of
the disorder \eta. Such analysis
could potentially influence the authors' conclusion that the
macroscopic elastic coefficients are largely independent of the
strength or the disorder.
Minor points:
1. Some of the variables, such as \lambda_{SS}, \lambda_{SL}, and \lambda_{LL} in
Eq. (4) and P_1 and P_2 in Eq. (23), are not defined when they are
first introduced in the manuscript.
2. Not all of the figures are referenced in the manuscript. In
addition, the ordering of the figures in the manuscript does not match
the order in which they are referenced.
Report
The authors have done a very serious job in addressing all the referees' queries. The paper presents a valuable study of stress correlations in partly disordered packings. I recommend publication of the paper as is.
Author: Roshan Maharana on 20240408 [id 4394]
(in reply to Report 2 on 20240215)We thank the Referee for the positive feedback and recommendation of publication.
Strengths
1 detailed analysis of elasticity and stress correlations in nearly crystalline polydisperse solids
2 analytical results partly backed by numerical simulations
3 connection with other results in the literature
Weaknesses
1 some results about moduli are either unclear or in contradiction with previous results for similar systems
2 assumption of affine elasticity to be better justified
Report
I have read this manuscript "Stress correlations in nearcrystalline packings" with much interest. The authors consider nearcrystalline packings with finite polydispersity and compute the pressure tensor, elastic moduli and stress correlations both analytically and in simulations. The results are, overall, interesting and scientifically valid, however there are some problems and points, in the current version, that have to be solved and clarified before I can make a final recommendation.
 the behaviour of the shear modulus in Fig. 2 b is puzzling. Why is the shear modulus decreasing with the volume fraction? This appears in contradiction with what is known for both disordered packings and crystals, where the shear modulus always increases with the packing fraction. This is because the affine shear modulus is proportional to the coordination number, which always increases monotonically with the packing fraction (or at least, it never decreases). Compare e.g. the analytical formulae of shear modulus (including both affine and nonaffine terms) in Phys. Rev. B 83, 184205 (2011) and for disordered crystals in Phys. Rev. B 93, 094204 (2016).
 the assumption of affine elasticity in the analytical derivation should be better motivated. The rootcause of nonaffinity in the elasticity is the lack of inversion symmetry (centrosymmetry). If a particle is not a center of inversion symmetry, the forces coming from the nearest neighbours in the affine position do not balance, and the net force has to be released via the extra nonaffine displacements. The statistical degree of centrosymmetry can be easily evaluated using the order parameter defined in Phys. Rev. B 93, 094204 (2016) or the centrosymmetry command in LAMMPS https://docs.lammps.org/compute_centro_atom.html or by direct evaluation in the numerical simulations. Based on the above guidelines, the authors should provide a more quantitative justification as to why the nonaffine contributions to the moduli can be safely neglected.
This is usually the case for the bulk modulus which is mostly affine even in disordered packings (https://www.nature.com/articles/srep18724) but typically this is not the case for the shear modulus.
 in the context of Refs. 5,7,15 also Phys. Rev. B 83, 184205 (2011) should be mentioned.
 anisotropic correlations in internal stresses (or elastic constants) have been used in Soft Matter, 2020,16, 77977807 within a rigorous field theory to predict the logarithmic Rayleigh scattering observed in Ref. 36. Without the anisotropic correlations in internal stress (or elastic constants), the logarithmic correction cannot be retrieved.
Requested changes
1 clarify or explain why shear modulus decreases with increasing packing fraction in Fig. 2b
2 quantify degree of centrosymmetry to justify assumption of affine elasticity in the derivation of the moduli
3 mention references and results listed in the report
Author: Roshan Maharana on 20240122 [id 4268]
(in reply to Report 1 on 20231218)
We thank the Referee for the positive feedback on our manuscript. We have found the comments extremely useful and incorporated these points into the revised version of our manuscript. Below we address the points raised by the Referee in detail.
Comment 1: "The behaviour of the shear modulus in Fig. 2 b is puzzling. Why is the shear modulus decreasing with the volume fraction? This appears in contradiction with what is known for both disordered packings and crystals, where the shear modulus always increases with the packing fraction. This is because the affine shear modulus is proportional to the coordination number, which always increases monotonically with the packing fraction (or at least, it never decreases). Compare e.g. the analytical formulae of shear modulus (including both affine and nonaffine terms) in Phys. Rev. B 83, 184205 (2011) and for disordered crystals in Phys. Rev. B 93, 094204 (2016)."
Reply:
The referee is certainly correct about the fact that the affine shear modulus increases with increasing packing fraction in disordered solids as the coordination number also increases. But that is not the case in near crystalline systems where the coordination number remains fixed (i.e. $6$ in triangular lattice) with increasing packing fraction. As in nearcrystalline systems, the average distance between two neighbouring particles is inversely proportional to squareroot of the packing fraction (i.e. $R_0 \propto 1/\sqrt{\phi}$), with increasing packing fraction the average separation between particles decreases. Therefore for fixed shear amplitude, the particles are displaced less for higher packing fraction and the corresponding change in the shear stress is also smaller. This can be seen from the expression for the shearstress for small shear amplitude which is given as, \begin{equation} \langle\Delta\Sigma_{xy}\rangle\sim \gamma\frac{3Nk R_0}{2V a_0}\left(\frac{R_0}{2a_0}\frac{3}{4}\right) = \gamma\frac{\sqrt{3}k }{2a_0^2}\left(1\frac{3}{4}\left(\frac{\phi}{\phi_0}\right)^{1/2}\right). \end{equation}
Changes in the manuscript: Based on the Referee's comment, we have included the citations to the analytical formulae for the shear modulus of disordered solids in the revised manuscript.
Comment 2: "the assumption of affine elasticity in the analytical derivation should be better motivated. The root cause of nonaffinity in the elasticity is the lack of inversion symmetry (centrosymmetry). If a particle is not a center of inversion symmetry, the forces coming from the nearest neighbours in the affine position do not balance, and the net force has to be released via the extra nonaffine displacements. The statistical degree of centrosymmetry can be easily evaluated using the order parameter defined in Phys. Rev. B 93, 094204 (2016) or the centrosymmetry command in LAMMPS https://docs.lammps.org/compute_centro_atom.html or by direct evaluation in the numerical simulations. Based on the above guidelines, the authors should provide a more quantitative justification as to why the nonaffine contributions to the moduli can be safely neglected. This is usually the case for the bulk modulus which is mostly affine even in disordered packings (https://www.nature.com/articles/srep18724) but typically this is not the case for the shear modulus.
 in the context of Refs. 5,7,15 also Phys. Rev. B 83, 184205 (2011) should be mentioned."
Reply:
We thank the Referee for raising this important point. Indeed, there should be a better motivation for affine elasticity. In this study, we are calculating the elastic properties numerically with increasing magnitude of disorder. Additionally, we theoretically derive the elastic properties of the elastic properties for a perfect crystal without any disorder. Here the primary aim is to present a comparative analysis between the elastic properties of a crystal and that of a disordered crystal. It is important to note that we are not deducing the elastic properties of the disordered crystal, rather we derive them for the pure crystal and examine the threshold value of disorder where elastic properties begin to diverge.
Based on the Referee's comment, we have now evaluated the centrosymmetry order parameter in all nearcrystalline arrangements, following the above references. This analysis indeed demonstrates the validity of the affine displacement assumption and provides a quantitative justification. We measure the average degree of centrosymmetry ($F_{IS}$) in the configurations which give, $F_{IS} \sim 1$ for $\eta \le 10^{2}$. This indicates that for such nearcrystalline systems, the nonaffine contribution can be neglected.
Changes in the manuscript: Based on the Referee's comment, we have now evaluated the centrosymmetry order parameter and added a discussion regarding the motivation behind the affine displacement assumption in the manuscript. Additionally, we have included the references that were mentioned.
Author: Roshan Maharana on 20240408 [id 4395]
(in reply to Report 3 on 20240401)We thank the Referee for the positive feedback on our manuscript. We provide detailed responses to the Referee below.
Comment 1: One main point highlighted by the authors is that the macroscopic elastic properties of nearcrystalline packings remain unchanged below a given disorder threshold, which is demonstrated in Fig. 2(a). However, it will be beneficial to depict the deviation of $\langle B\rangle$ or $\langle G\rangle$ from the theoretical value, $B_{theory}/G_{theory}$, of perfect crystals on a logarithmic scale, i.e., elucidating whether the deviations "$B_{theory}  \langle B\rangle$" and "$G_{theory}  \langle G\rangle$" from nonaffine contribution also follow a powerlaw scaling with the strength of the disorder $\eta$. Such analysis could potentially influence the authors' conclusion that the macroscopic elastic coefficients are largely independent of the strength or the disorder.
Reply:
We thank the Referee for raising this important point. Based on a linear order perturbation expansion in disorder ($\eta$), our theory predicts that the the average modulus $\langle G\rangle$ and $\langle B\rangle$ remains independent of $\eta$ and are equal in value to those of a crystalline packing below a certain disorder magnitude ($\eta_c$), while the fluctuations in $G$ and $B$ will vary linearly with $\eta$. An earlier numerical work by Tong et. al., Scientific reports 5, 15378 (2015) has also found this behaviour. Our numerical results further suggest that for a small magnitude of disorder, both $B_{theory}  \langle B\rangle$ and $G_{theory}  \langle G\rangle$ do not have any powerlaw dependence on $\eta$.
In the attached figure titled "$Elastic\_moduli\_vs\_disorder.pdf$", we display the variation of the elastic moduli with increasing disorder strength. This shows a constant dependence of the moduli for a range of disorder strength. We also show the variation of the difference $G_{theory}  \langle G\rangle$ and $B_{theory}  \langle B\rangle$ in log scale, showing that the difference is negligible upto a certain threshold disorder value. This threshold value corresponds very closely with the transition point between crystals and disordered crystals. We have now incorporated these points and these plots in the revised version of the manuscript.
Comment 2: Some of the variables, such as $\lambda_{SS}, \lambda_{SL},$ and $\lambda_{LL}$ in Eq. (4) and $P_1$ and $P_2$ in Eq. (23), are not defined when they are first introduced in the manuscript.
Reply:
We thank the Referee for pointing this out.
Based on the Referee's comment, we have now defined all the above parameters in the revised manuscript.
Comment 3: Not all of the figures are referenced in the manuscript. In addition, the ordering of the figures in the manuscript does not match the order in which they are referenced.
Reply:
We thank the Referee for bringing this to our attention.
In the revised manuscript, we have ensured that all figures are referenced correctly and the ordering of the figures now aligns with the order in which they are referred.
Attachment:
Elastic_moduli_vs_disorder.pdf