Emergence of long-range correlations and rigidity at the dynamic glass transition Grzegorz Szamel Department of Chemistry Colorado State University Ft. Collins, CO 80523, USA YITP , Kyoto University Kyoto, July 18, 2013
Outline 1 Statistical mechanical expression for the shear modulus Emergence of rigidity: crystals 2 Goldstone modes and long-range correlations Shear modulus Replica approach 3 4 Emergence of rigidity: glasses Goldstone modes & long-range correlations Shear modulus Numerical results 5 Summary
Statistical mechanical expression for the shear modulus Solid: elastic response to a shear deformation Max Born (1939): “The difference between a solid and a liquid is that the solid has elastic resistance to a shearing stress while a liquid does not.” non-zero shear modulus µ : µ = F / A ∆ x / I
Statistical mechanical expression for the shear modulus Free energy of a deformed system Consider an N -particle system in a box of volume V; particles interact via potential V ( r ) . The non-trivial part of the free energy of this system is � � � d � r 1 ... d � r N − 1 � F = − k B T ln V ( r ij ) . exp V N k B T V i < j Now, let’s deform the box with shear strain γ . Then, one would integrate over a deformed volume, � � � d � r 1 ... d � r N − 1 � F ( γ ) = − k B T ln exp V ( r ij ) . V N k B T V ′ i < j Mathematically, one can change the variables x ′ = x − γ y ; y ′ = y ; z ′ = z and then one integrates over the undeformed box: � �� � r ′ r ′ d � 1 ... d � �� − 1 � ij ) 2 + y 2 N ( x ′ ij + γ y ′ ij + z 2 F ( γ ) = − k B T ln . exp V ij V N k B T V i < j Note: the shear strain γ appears now in the argument of V .
Statistical mechanical expression for the shear modulus General formula for shear modulus Expanding the free energy in the shear strain one gets: 1 2 N µγ 2 F ( γ ) = F ( 0 ) + N σγ + + ... σ - shear stress µ - shear modulus � 2 �� � ��� � 2 � �� ∂ 2 V ( r ij ) ∂ V ( r ij ) ∂ V ( r ij ) µ = 1 1 y 2 − − y ij y ij ij ∂ x 2 ∂ x ij ∂ x ij N k B T ij i < j i < j i < j Squire, Holt and Hoover, Physica 42 , 388 (1969) �� � ∂ 2 V ( r ij ) 1 y 2 ← the Born term ij ∂ x 2 N ij i < j � 2 ��� � 2 � �� ∂ V ( r ij ) ∂ V ( r ij ) �� − � σ � 2 � 1 σ 2 � ≡ N − ← y ij y ij ∂ x ij ∂ x ij N i < j i < j stress fluctuations
Statistical mechanical expression for the shear modulus General formula for shear modulus Expanding the free energy in the shear strain one gets: 1 2 N µγ 2 F ( γ ) = F ( 0 ) + N σγ + + ... σ - shear stress µ - shear modulus � 2 �� � ��� � 2 � �� ∂ 2 V ( r ij ) ∂ V ( r ij ) ∂ V ( r ij ) µ = 1 1 y 2 − − y ij y ij ij ∂ x 2 ∂ x ij ∂ x ij N k B T ij i < j i < j i < j Squire, Holt and Hoover, Physica 42 , 388 (1969) In the thermodynamic limit the free energy density is shape-independent: F ( 0 ) F ( γ ) = lim lim N N ∞ ∞ � ∞ N − 1 ∂ 2 F ( γ ) � However, the shear modulus is finite: � = 0 lim � ∂γ 2 � γ = 0
Statistical mechanical expression for the shear modulus General formula for shear modulus Expanding the free energy in the shear strain one gets: 1 2 N µγ 2 F ( γ ) = F ( 0 ) + N σγ + + ... σ - shear stress µ - shear modulus � 2 �� � ��� � 2 � �� ∂ 2 V ( r ij ) ∂ V ( r ij ) ∂ V ( r ij ) µ = 1 1 y 2 − − y ij y ij ij ∂ x 2 ∂ x ij ∂ x ij N k B T ij i < j i < j i < j Squire, Holt and Hoover, Physica 42 , 388 (1969) This formula is applicable to both crystals and glasses. Can also be evaluated for fluids; computer simulations showed that for fluids this formula gives µ = 0 (as it should). It can be proved that for systems with short range interactions, the above formula gives µ = 0 unless there are long-range density correlations (Bavaud et al. , J. Stat. Phys. 42 , 621 (1986)).
Statistical mechanical expression for the shear modulus General formula for shear modulus Expanding the free energy in the shear strain one gets: 1 2 N µγ 2 F ( γ ) = F ( 0 ) + N σγ + + ... σ - shear stress µ - shear modulus � 2 �� � ��� � 2 � �� ∂ 2 V ( r ij ) ∂ V ( r ij ) ∂ V ( r ij ) µ = 1 1 y 2 − − y ij y ij ij ∂ x 2 ∂ x ij ∂ x ij N k B T ij i < j i < j i < j Squire, Holt and Hoover, Physica 42 , 388 (1969) The above formula was a starting point of a calculation of glass shear modulus by H. Yoshino and M. Mezard (PRL 105 , 015504 (2010)); see also H. Yoshino, JCP 136 , 214108 (2012). Goal: investigate the existence of long range density correlations and derive an alternative formula for the shear modulus.
Emergence of rigidity: crystals Goldstone modes and long-range correlations Broken translational symmetry In crystalline solids translational symmetry is broken The average density n ( � r ) is a periodic function of � r : � G e i � G · � r n ( � r ) = n � � G where � G are reciprocal lattice vectors. Rigid translation: an equivalent but different state By translating a crystal by a constant vector � a we get an equivalent but different state of the crystal. This does not cost any energy/does not require any force. Under such translation the density field changes: G e i � G � = � � G · � a n ( � r ) → n ( � r − � ≡ G → n � a ) n � for 0 Rigid translations ≡ zero free energy cost excitations (Goldstone modes) The existence of such zero-free energy excitations is the reflection of a broken translational symmetry.
Emergence of rigidity: crystals Goldstone modes and long-range correlations Long-range correlations Density fluctuations for wavevectors close to � G diverge � � � e − i ( � k + � n ( � k + � δ n ( � k + � G ) = n ( � k + � n ( � k + � G ) · � r i ; G ) − G ) = G ) i � | A | 2 � � | B | 2 � ≥ | � AB � | 2 = Bogoliubov inequality ⇒ G | 2 � � 2 ( k B T ) 2 | n � n · � ˆ � G � G ) | 2 � 1 ≥ 1 | δ n ( � k + � � � | ˆ k 2 ↔ V k ) · ˆ � σ ( � 1 k · � n | 2 lim � k → 0 V ↔ σ ( � ˆ k ) - microscopic stress tensor � n - an arbitrary unit vector Small wavevector divergence ⇒ long-range correlations in direct space.
Emergence of rigidity: crystals Goldstone modes and long-range correlations Displacement field and its long-range correlations Slowly varying deformation Infinitesimal uniform translation: n ( � r ) → n ( � r ) − � a · ∂ r n ( � r ) � Infinitesimal deformation with a slowly varying � a ( � r ) : n ( � r ) → n ( � r ) − � a ( � r ) · ∂ r n ( � r ) � Microscopic expression for the displacement field � ∂ n ( � � 2 � � r ∂ n ( � r ) r ) k ) = − 1 N = 1 � re − i � u ( � � k · � d � δ ( � r − � r i ) d � r N ∂� ∂� r 3 V r i � �� � microscopic density � � u ( � a ( � If δ n ( � r ) = − � a ( � r ) · ∂ r n ( � r ) , then � k ) = � k ) . � Long-range correlations of the displacement field Bogoliubov inequality = ⇒ ( k B T ) 2 � k ) | 2 � 1 ≥ 1 | ˆ u ( � � n · � � � | ˆ k 2 ↔ V � σ ( � k ) · ˆ 1 k · � n | 2 lim � k → 0 V u ( � This can be used to show that � k ; t ) is a slow (hydrodynamic) mode → G. Szamel & M. Ernst, Phys. Rev. B 48 , 112 (1993).
Emergence of rigidity: crystals Shear modulus Macroscopic force balance equation Macroscopic force balance equation In the long wavelength ( k → 0 ) limit we have the following relation between a a = a x ( k y )ˆ transverse displacement � � e x and the external force (per unit volume) needed to maintain this displacement: F = F x ( k y )ˆ e x = λ xxyy a x ( k y ) k y k y ˆ � � � λ xxyy ≡ µ ← shear modulus e x
Emergence of rigidity: crystals Shear modulus Microscopic force balance equation Transverse non-uniform displacement k ) e i � a ( � k · � r : Infinitesimal transverse deformation with a slowly varying � a ( � r ) = � k ) e i � r · ∂ a ( � a ⊥ � k · � n ( � r ) → n ( � r ) − � a ( � r ) · ∂ r n ( � r ) = n ( � r ) − � r n ( � r ) , � k � � External force needed to maintain deformed density profile External potential needed to maintain the density profile change: � � δ V ext ( � � � r 1 ) � k ) e i � r 2 · ∂ � a ( � k · � d � − � r 2 n ( � r 2 ) r 2 δ n ( � r 2 ) External force on the system (per unit volume): � δ V ext ( � � � � r 1 ) − 1 F ( � r 1 e − i � a ( � k ) e i � � k · � k · � r 1 n ( � r 2 k ) = d � r 1 ) ∂ d � [ − ∂ � r 2 n ( � r 2 )] · � r 2 � r 1 V δ n ( � r 2 ) � � δ V ext ( � � − 1 r 1 ) r 2 e − i � r 1 ( ∂ k ) e i � a ( � k · � k · � r 2 = d � r 1 d � r 1 n ( � r 1 )) [ ∂ � r 2 n ( � r 2 )] · � � δ n ( � r 2 ) V
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