^{1,a)}, H. Xu

^{2}, P. Polińska

^{1}, F. Weysser

^{1}and J. Baschnagel

^{1}

### Abstract

The shear modulus *G* of two glass-forming colloidal model systems in *d* = 3 and *d* = 2 dimensions is investigated by means of, respectively, molecular dynamics and Monte Carlo simulations. Comparing ensembles where either the shear strain γ or the conjugated (mean) shear stress τ are imposed, we compute *G* from the respective stress and strain fluctuations as a function of temperature *T* while keeping a constant normal pressure *P*. The choice of the ensemble is seen to be highly relevant for the shear stress fluctuations μ_{F}(*T*) which at constant τ decay monotonously with *T* following the affine shear elasticity μ_{A}(*T*), i.e., a simple two-point correlation function. At variance, non-monotonous behavior with a maximum at the glass transition temperature *T* _{g} is demonstrated for μ_{F}(*T*) at constant γ. The increase of *G* below *T* _{g} is reasonably fitted for both models by a continuous cusp singularity, *G*(*T*)∝(1 − *T*/*T* _{g})^{1/2}, in qualitative agreement with recent theoretical predictions. It is argued, however, that longer sampling times may lead to a sharper transition.

H.X. thanks the CNRS and the IRTG Soft Matter for supporting her sabbatical stay in Strasbourg and P.P. thanks the Région Alsace and the IRTG Soft Matter and F.W. the DAAD for funding. We are indebted to A. Blumen (Freiburg) and H. Meyer, and O. Benzerara and J. Farago (all ICS, Strasbourg) for helpful discussions. We are grateful to A. Zaccone and E. M. Terentjev for communicating their recent theoretical work on disorder-assisted glass transition in amorphous solids.^{23}

I. INTRODUCTION

II. THEORETICAL CONSIDERATIONS

A. Reminder of thermodynamic relations

B. Shear modulus at imposed shear strain

C. Shear stress fluctuations in different ensembles

D. Impulsive corrections for truncated potentials

III. ALGORITHMIC AND TECHNICAL ISSUES

IV. COMPUTATIONAL RESULTS

A. High-temperature liquid limit

B. Dynamical matrix harmonic network

C. Low-temperature glass limit

D. Scaling with temperature

V. CONCLUSION

### Key Topics

- Elastic moduli
- 49.0
- Elasticity
- 35.0
- Monte Carlo methods
- 23.0
- Glass transitions
- 20.0
- Colloidal systems
- 13.0

##### B01J13/00

## Figures

Setup, notations, and ensembles investigated numerically: (a) Schematic experimental setup for probing the shear modulus *G*. (b) Expected shear modulus *G*(*t*) as a function of measurement time *t* at *T* ≪ *T* _{g} (top curve), *T* slightly below *T* _{g} (middle curve) and in the liquid limit for *T* ≫ *T* _{g} (bottom curve). (c) We use periodic simulation boxes with deformable box shapes where we impose τ = 0. The shear modulus *G* may be computed using Eqs. (6) or (9) from the instantaneous strains and stresses . (d) Using Eq. (3) *G* may also be computed using the affine shear elasticity μ_{A} and the shear stress fluctuations μ_{F} in ensembles of fixed strain (γ = 0). (e) Combined volumetric and shear strain fluctuations allowing the computation of both compression modulus *K* and shear modulus *G* from the respective strain fluctuations.

Setup, notations, and ensembles investigated numerically: (a) Schematic experimental setup for probing the shear modulus *G*. (b) Expected shear modulus *G*(*t*) as a function of measurement time *t* at *T* ≪ *T* _{g} (top curve), *T* slightly below *T* _{g} (middle curve) and in the liquid limit for *T* ≫ *T* _{g} (bottom curve). (c) We use periodic simulation boxes with deformable box shapes where we impose τ = 0. The shear modulus *G* may be computed using Eqs. (6) or (9) from the instantaneous strains and stresses . (d) Using Eq. (3) *G* may also be computed using the affine shear elasticity μ_{A} and the shear stress fluctuations μ_{F} in ensembles of fixed strain (γ = 0). (e) Combined volumetric and shear strain fluctuations allowing the computation of both compression modulus *K* and shear modulus *G* from the respective strain fluctuations.

Density ρ(*T*) and determination of glass transition temperature *T* _{g}. Inset: Plotting log (1/ρ) vs*.* *T* for the KA model as suggested in Ref. ^{ 8 } reveals a linear low-*T* (solid line) and a linear high-*T* regime which confirms *T* _{g} ≈ 0.41. ^{ 49 } Main panel: Number density ρ for pLJ systems at constant normal pressure *P* = 2 as a function of temperature *T* for different quench rates *R* in units of MCS. The crosses refer to a quench in the NPτT-ensemble where the box shape is allowed to fluctuate at τ = 0. All other data refer to the NPγT-ensemble with γ = 0. The solid line and the dashed line are linear fits to the best density estimates (stars) obtained for, respectively, low and high temperatures. By matching both lines one determines *T* _{g} ≈ 0.26. A similar value is obtained if log (1/ρ) is plotted as a function of *T*.

Density ρ(*T*) and determination of glass transition temperature *T* _{g}. Inset: Plotting log (1/ρ) vs*.* *T* for the KA model as suggested in Ref. ^{ 8 } reveals a linear low-*T* (solid line) and a linear high-*T* regime which confirms *T* _{g} ≈ 0.41. ^{ 49 } Main panel: Number density ρ for pLJ systems at constant normal pressure *P* = 2 as a function of temperature *T* for different quench rates *R* in units of MCS. The crosses refer to a quench in the NPτT-ensemble where the box shape is allowed to fluctuate at τ = 0. All other data refer to the NPγT-ensemble with γ = 0. The solid line and the dashed line are linear fits to the best density estimates (stars) obtained for, respectively, low and high temperatures. By matching both lines one determines *T* _{g} ≈ 0.26. A similar value is obtained if log (1/ρ) is plotted as a function of *T*.

Energy per particle *e* (main panel) and acceptance rate *A* (inset) obtained for pLJ systems quenched at constant pressure *P* = 2 at a given quench rate *R* as indicated. The crosses refer to a quench at imposed τ = 0. The solid line and the dashed line are linear fits to the energy for, respectively, low and high temperatures confirming the value *T* _{g} ≈ 0.26 for the pLJ model. Only local MC moves with one fixed (non-adaptive) maximum particle displacement δ*r* _{max} for all temperatures *T* have been used for the MC data shown in the main panel. The acceptance rate decreases thus with decreasing *T* as shown in the inset for δ*r* _{max} = 0.01 (upper data) and δ*r* _{max} = 0.1.

Energy per particle *e* (main panel) and acceptance rate *A* (inset) obtained for pLJ systems quenched at constant pressure *P* = 2 at a given quench rate *R* as indicated. The crosses refer to a quench at imposed τ = 0. The solid line and the dashed line are linear fits to the energy for, respectively, low and high temperatures confirming the value *T* _{g} ≈ 0.26 for the pLJ model. Only local MC moves with one fixed (non-adaptive) maximum particle displacement δ*r* _{max} for all temperatures *T* have been used for the MC data shown in the main panel. The acceptance rate decreases thus with decreasing *T* as shown in the inset for δ*r* _{max} = 0.01 (upper data) and δ*r* _{max} = 0.1.

Rescaled shear modulus *G*(*t*)/μ_{A} vs*.* sampling time *t* in the high-*T* liquid limit of the pLJ model comparing various ensembles and observables. The filled squares show the bare shear modulus obtained using Eq. (3) for the NVγT-ensemble *without* the impulsive correction Δμ_{B} ≈ 0.2 (horizontal line). The corrected modulus for the NVγT-ensemble (open squares) and the NPγT-ensemble (open diamonds) vanishes as it should. Note that *G* _{ττ} decreases slightly faster than *G* _{γτ} ≈ *G* _{γγ}. As shown by the dashed line, all operational definitions of *G* decay inversely with *t*. The dashed-dotted line indicates the expected power-law exponent −1/2 for the correlation coefficient *c* _{γτ} (stars).

Rescaled shear modulus *G*(*t*)/μ_{A} vs*.* sampling time *t* in the high-*T* liquid limit of the pLJ model comparing various ensembles and observables. The filled squares show the bare shear modulus obtained using Eq. (3) for the NVγT-ensemble *without* the impulsive correction Δμ_{B} ≈ 0.2 (horizontal line). The corrected modulus for the NVγT-ensemble (open squares) and the NPγT-ensemble (open diamonds) vanishes as it should. Note that *G* _{ττ} decreases slightly faster than *G* _{γτ} ≈ *G* _{γγ}. As shown by the dashed line, all operational definitions of *G* decay inversely with *t*. The dashed-dotted line indicates the expected power-law exponent −1/2 for the correlation coefficient *c* _{γτ} (stars).

Rescaled shear modulus *G*(*t*)/μ_{A} vs*.* sampling time *t* for systems of permanent linear springs corresponding to the dynamical matrix of pLJ systems at *P* = 2, τ = 0, and *T* = 0.001. As indicated by the bold line, we obtain *G*/μ_{A} ≈ 0.67 in the long-time limit. In agreement with Eq. (19) *G* _{ττ} vanishes with time if it is computed in an ensemble where γ fluctuates as shown for the NVτT-ensemble (filled squares).

Rescaled shear modulus *G*(*t*)/μ_{A} vs*.* sampling time *t* for systems of permanent linear springs corresponding to the dynamical matrix of pLJ systems at *P* = 2, τ = 0, and *T* = 0.001. As indicated by the bold line, we obtain *G*/μ_{A} ≈ 0.67 in the long-time limit. In agreement with Eq. (19) *G* _{ττ} vanishes with time if it is computed in an ensemble where γ fluctuates as shown for the NVτT-ensemble (filled squares).

Shear stress fluctuations μ_{F}(*t*)/μ_{A} with μ_{A} = 33.9 for the same systems as in Fig. 5 . We observe μ_{F}(*t*)/μ_{A} → 1 (dash-dotted line) for imposed τ, while μ_{F}(*t*)/μ_{A} → 1 − *G*/μ_{A} ≈ 0.32 (dashed line) for imposed γ. Note the broad time-dependence of μ_{F}(*t*) in the latter cases.

Shear stress fluctuations μ_{F}(*t*)/μ_{A} with μ_{A} = 33.9 for the same systems as in Fig. 5 . We observe μ_{F}(*t*)/μ_{A} → 1 (dash-dotted line) for imposed τ, while μ_{F}(*t*)/μ_{A} → 1 − *G*/μ_{A} ≈ 0.32 (dashed line) for imposed γ. Note the broad time-dependence of μ_{F}(*t*) in the latter cases.

*G* _{γτ}(*t*)/μ_{A}, *G* _{γγ}(*t*)/μ_{A} and *G* _{ττ}(*t*)/μ_{A} for pLJ beads in the low-temperature limit for *P* = 2, τ = 0 and *T* = 0.001 where μ_{A} ≈ 33.9. Qualitatively, the data compares well with the various shear moduli presented in Fig. 5 . The filled triangles show *G* _{ττ} for the NPτT-ensemble which is seen to vanish rapidly with sampling time *t*. The dashed line indicates a 1/*t*-decay.

*G* _{γτ}(*t*)/μ_{A}, *G* _{γγ}(*t*)/μ_{A} and *G* _{ττ}(*t*)/μ_{A} for pLJ beads in the low-temperature limit for *P* = 2, τ = 0 and *T* = 0.001 where μ_{A} ≈ 33.9. Qualitatively, the data compares well with the various shear moduli presented in Fig. 5 . The filled triangles show *G* _{ττ} for the NPτT-ensemble which is seen to vanish rapidly with sampling time *t*. The dashed line indicates a 1/*t*-decay.

Reduced shear stress fluctuation μ_{F}(*t*)/μ_{A} for pLJ beads using the same representation as in Fig. 6 . The small filled squares refer to data obtained for *T* = 0.010 using the NVγT-ensemble, all other data to *T* = 0.001. The reduced Born coefficient μ_{B}(*t*)/μ_{A} (thin line) and the affine shear elasticity μ_{A}(*t*)/μ_{A} (dashed-dotted line) are essentially time independent while μ_{F}(*t*)/μ_{A} approaches its large-*t* limit (dashed line) from below.

Reduced shear stress fluctuation μ_{F}(*t*)/μ_{A} for pLJ beads using the same representation as in Fig. 6 . The small filled squares refer to data obtained for *T* = 0.010 using the NVγT-ensemble, all other data to *T* = 0.001. The reduced Born coefficient μ_{B}(*t*)/μ_{A} (thin line) and the affine shear elasticity μ_{A}(*t*)/μ_{A} (dashed-dotted line) are essentially time independent while μ_{F}(*t*)/μ_{A} approaches its large-*t* limit (dashed line) from below.

Stress fluctuation μ_{F} (open symbols) and affine shear elasticity μ_{A} (small filled symbols) vs*.* temperature *T* for both models. The indicated values have been obtained from the longest simulation runs available for a given temperature. μ_{A} decays roughly linearly with *T* as shown by bold solid lines for the low-*T* and by dashed line for the high-*T* regime. For constant-τ systems and for all systems with *T* ⩾ *T* _{g} we confirm that μ_{F} ≈ μ_{A}, while μ_{F}|_{γ} (open spheres and squares) is seen to be non-monotonous with a maximum at *T* _{g}.

Stress fluctuation μ_{F} (open symbols) and affine shear elasticity μ_{A} (small filled symbols) vs*.* temperature *T* for both models. The indicated values have been obtained from the longest simulation runs available for a given temperature. μ_{A} decays roughly linearly with *T* as shown by bold solid lines for the low-*T* and by dashed line for the high-*T* regime. For constant-τ systems and for all systems with *T* ⩾ *T* _{g} we confirm that μ_{F} ≈ μ_{A}, while μ_{F}|_{γ} (open spheres and squares) is seen to be non-monotonous with a maximum at *T* _{g}.

Unscaled shear modulus *G* as a function of temperature *T* for both models. For the pLJ model the values *G* _{ττ} obtained in the NVγT ensemble (squares) are essentially identical to the moduli *G* _{γτ} (crosses) and *G* _{γγ} (triangles) obtained for constant τ. As shown by the small filled triangles, *G* _{ττ} vanishes in the NPτT-ensemble for all *T* in agreement with Eq. (19) . The solid and the dashed line indicate the cusp-singularity, Eq. (65) , for, respectively, the KA model and the pLJ beads. ^{ 30,31 }

Unscaled shear modulus *G* as a function of temperature *T* for both models. For the pLJ model the values *G* _{ττ} obtained in the NVγT ensemble (squares) are essentially identical to the moduli *G* _{γτ} (crosses) and *G* _{γγ} (triangles) obtained for constant τ. As shown by the small filled triangles, *G* _{ττ} vanishes in the NPτT-ensemble for all *T* in agreement with Eq. (19) . The solid and the dashed line indicate the cusp-singularity, Eq. (65) , for, respectively, the KA model and the pLJ beads. ^{ 30,31 }

Rescaled shear modulus *y* = *G*/μ_{A} vs*.* reduced temperature *x* = *T*/*T* _{g} for the KA model at *P* = 1 (NVγT-ensemble) and the pLJ model at *P* = 2 (NVτT-, NPτT-, NVγT-, and NPγT-ensembles). The bold line indicates again a *continuous* cusp-singularity. Inset: Squared correlation coefficient for NVτT- and NPτT-ensembles.

Rescaled shear modulus *y* = *G*/μ_{A} vs*.* reduced temperature *x* = *T*/*T* _{g} for the KA model at *P* = 1 (NVγT-ensemble) and the pLJ model at *P* = 2 (NVτT-, NPτT-, NVγT-, and NPγT-ensembles). The bold line indicates again a *continuous* cusp-singularity. Inset: Squared correlation coefficient for NVτT- and NPτT-ensembles.

Unscaled shear modulus *G* _{γτ} as a function of sampling time *t* in MCS obtained for the pLJ model in the NPτT-ensemble for different temperatures as indicated. The thin line indicates the power-law slope −1 for large times in the liquid regime. A clear shoulder (on the logarithmic scales used) can only be seen below *T* ≈ 0.2 and only below *T* = 0.01 one observes a *t*-independent plateau over two orders of magnitude. We emphasize that we present here a sampling time effect and *not* a time correlation function. The same sampling time effect is observed after additional tempering.

Unscaled shear modulus *G* _{γτ} as a function of sampling time *t* in MCS obtained for the pLJ model in the NPτT-ensemble for different temperatures as indicated. The thin line indicates the power-law slope −1 for large times in the liquid regime. A clear shoulder (on the logarithmic scales used) can only be seen below *T* ≈ 0.2 and only below *T* = 0.01 one observes a *t*-independent plateau over two orders of magnitude. We emphasize that we present here a sampling time effect and *not* a time correlation function. The same sampling time effect is observed after additional tempering.

Reduced shear modulus *G*(*T*)/μ_{A}(*T*) as a function of reduced temperature *x* = *T*/*T* _{g} for different sampling times *t* as indicated: (a) *G* _{ττ}(*T*) for the KA model (NVγT-ensemble), (b) *G* _{γτ}(*T*) for the pLJ model (NPτT-ensemble). For both models the glass transition is seen to become sharper with increasing *t*. The continuous cusp-singularity is indicated by the bold line. More points in the vicinity of *x* ≈ 1 and much larger sampling times are needed in the future to decide whether the shear modulus is continuous or discontinuous at the glass transition.

Reduced shear modulus *G*(*T*)/μ_{A}(*T*) as a function of reduced temperature *x* = *T*/*T* _{g} for different sampling times *t* as indicated: (a) *G* _{ττ}(*T*) for the KA model (NVγT-ensemble), (b) *G* _{γτ}(*T*) for the pLJ model (NPτT-ensemble). For both models the glass transition is seen to become sharper with increasing *t*. The continuous cusp-singularity is indicated by the bold line. More points in the vicinity of *x* ≈ 1 and much larger sampling times are needed in the future to decide whether the shear modulus is continuous or discontinuous at the glass transition.

Affine dilatational elasticity η_{A}(*T*) and compression modulus *K*(*T*) for both models. Inset: η_{A} *vs.* reduced temperature *x* = *T*/*T* _{g}. The small triangles indicate η_{F} for the pLJ model using the NPτT-ensemble confirming that Eq. (A7) holds. Main panel: The Rowlinson formula *K* _{ pp }, Eq. (A14) , for the NVγT-ensemble is given for both the KA model (spheres) and the pLJ model (squares). For the pLJ model we indicate in addition *K* _{ vp } (crosses), *K* _{ vv } (large triangles) and the rescaled correlation coefficient (stars) which collapses on the other pLJ data as suggested by Eq. (A8) .

Affine dilatational elasticity η_{A}(*T*) and compression modulus *K*(*T*) for both models. Inset: η_{A} *vs.* reduced temperature *x* = *T*/*T* _{g}. The small triangles indicate η_{F} for the pLJ model using the NPτT-ensemble confirming that Eq. (A7) holds. Main panel: The Rowlinson formula *K* _{ pp }, Eq. (A14) , for the NVγT-ensemble is given for both the KA model (spheres) and the pLJ model (squares). For the pLJ model we indicate in addition *K* _{ vp } (crosses), *K* _{ vv } (large triangles) and the rescaled correlation coefficient (stars) which collapses on the other pLJ data as suggested by Eq. (A8) .

## Tables

Some properties for the KA model at normal pressure *P* = 1 and shear stress τ = 0: the mean density ρ, the mean energy per volume *e*ρ, the impulsive correction term Δμ_{B} (Sec. II D ), the (corrected) Born-Lamé coefficient μ_{B}, the affine shear elasticity μ_{A} = μ_{B} − *P* _{ex}, the shear modulus *G* _{ττ} obtained in the NVγT-ensemble, the hypervirial η_{B} and the affine dilatation elasticity η_{A} = 2*P* _{id} + η_{B} + *P* _{ex} discussed in the Appendix and the compression modulus *K* _{ pp } obtained for the NVγT-ensemble. The affine elasticities μ_{A} and η_{A} are the natural scales for, respectively, the shear modulus *G* and the compression modulus *K*.

Some properties for the KA model at normal pressure *P* = 1 and shear stress τ = 0: the mean density ρ, the mean energy per volume *e*ρ, the impulsive correction term Δμ_{B} (Sec. II D ), the (corrected) Born-Lamé coefficient μ_{B}, the affine shear elasticity μ_{A} = μ_{B} − *P* _{ex}, the shear modulus *G* _{ττ} obtained in the NVγT-ensemble, the hypervirial η_{B} and the affine dilatation elasticity η_{A} = 2*P* _{id} + η_{B} + *P* _{ex} discussed in the Appendix and the compression modulus *K* _{ pp } obtained for the NVγT-ensemble. The affine elasticities μ_{A} and η_{A} are the natural scales for, respectively, the shear modulus *G* and the compression modulus *K*.

Some properties of the pLJ model at *P* = 2 and τ = 0: the mean density ρ, the mean energy per volume *e*ρ, the impulsive correction term Δμ_{B}, the affine shear elasticity μ_{A}, the shear moduli *G* _{γτ} and *G* _{γγ} obtained in the NPτT-ensemble and *G* _{ττ} obtained in the NVγT-ensemble, the affine dilatation elasticity η_{A}, the compression modulus *K* _{ vp } obtained in the NPτT-ensemble, and the Rowlinson formula *K* _{ pp } for the NVγT-ensemble. All data have been obtained for a sampling time *t* = 10^{7} MCS.

Some properties of the pLJ model at *P* = 2 and τ = 0: the mean density ρ, the mean energy per volume *e*ρ, the impulsive correction term Δμ_{B}, the affine shear elasticity μ_{A}, the shear moduli *G* _{γτ} and *G* _{γγ} obtained in the NPτT-ensemble and *G* _{ττ} obtained in the NVγT-ensemble, the affine dilatation elasticity η_{A}, the compression modulus *K* _{ vp } obtained in the NPτT-ensemble, and the Rowlinson formula *K* _{ pp } for the NVγT-ensemble. All data have been obtained for a sampling time *t* = 10^{7} MCS.

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