Glassy binary mixture with large size ratio: interdiffusion and rheology Vinay Vaibhav The Institute of Mathematical Sciences Chennai, India Jürgen Horbach Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany Pinaki Chaudhuri The Institute of Mathematical Sciences Chennai, India arXiv:2202.12612 arXiv:2202.12189 

Binary mixture with large bidispersity # Materials with large size dispersity are ubiquitous # Microscopic understanding of dynamics and rheology; developing applications and design materials # Structural changes with inclusion of small particles: softening and then hardening # Regime of interest: fl uid of smaller particles in the glassy matrix of bigger species # Smaller particles also form glass at higher packing fraction 

# Large separation in relaxation timescales of bigger and smaller species # Bigger particles show glass transition: described by mode-coupling theory # Smaller particles moving in the glassy matrix of bigger species: localization transition Binary mixture with large bidispersity 

Sheared binary mixture with large bidispersity External timescale introduced via shear competes with the widely separated relaxation timescales of bigger and smaller species Our goal Detailed microscopic understanding of macroscopic observations 

Model System details # WCA interaction: Lennard-Jones interaction with only repulsive part # 50-50 mixture; A: large and B: small # Mean diameter ratio: # Bigger species mostly occupy the volume # Bigger particles are polydisperse: # Weaker cross-coupling: # Temperature of the system is fi xed (T = 2/3) # High density: hybrid swap MC-MD 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 V↵ (r) = 4✏↵ ⇥ ( ↵ r )12 ( ↵ r )6 ⇤ + ✏↵ AAACAnicdVC7SgNBFJ31GeMraiU2g1GwWmciZhUsRBtLBROFJIbZyawZMvtg5q4YlmDjr9hYKGLrV9j5N04egoqe6nDOvdxzj58oaYCQD2dsfGJyajo3k5+dm19YLCwtV02cai4qPFaxvvSZEUpGogISlLhMtGChr8SF3znu+xc3QhsZR+fQTUQjZNeRDCRnYKVmYXWjDuIWsiDWuIexPsClq4xul3sbzUKRuPseKVOCiet5tLy/a0nJoztkD1OXDFBEI5w2C+/1VszTUETAFTOmRkkCjYxpkFyJXr6eGpEw3mHXomZpxEJhGtnghR7etEoL91MEcQR4oH7fyFhoTDf07WTIoG1+e33xL6+WQrDXyGSUpCAiPjwUpApDjPt94JbUgoPqWsK4ljYr5m2mGQfbWt6W8PUp/p9USy4tu/SsVDw8GtWRQ2toHW0hijx0iE7QKaogju7QA3pCz8698+i8OK/D0TFntLOCfsB5+wQLGJXs for r < 21/6 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 AA/ BB ⇡ 2.85 AAACBHicdVBLTwIxGOziC/G16pFLI5h42rQYWEg8gF48YiJgAoR0S4GG7iNt14RsOHjxr3jxoDFe/RHe/DeWh4kanaTJdGa+tN94keBKI/RhpVZW19Y30puZre2d3T17/6CpwlhS1qChCOWNRxQTPGANzbVgN5FkxPcEa3nji5nfumVS8TC41pOIdX0yDPiAU6KN1LOzeeSUi/AMdhQf+qSX1GpTc8MOLuZ7dg45FReVMILIcV1cqhQNKbj4FJVNBM2RA0vUe/Z7px/S2GeBpoIo1cYo0t2ESM2pYNNMJ1YsInRMhqxtaEB8prrJfIkpPDZKHw5CaU6g4Vz9PpEQX6mJ75mkT/RI/fZm4l9eO9aDcjfhQRRrFtDFQ4NYQB3CWSOwzyWjWkwMIVRy81dIR0QSqk1vGVPC16bwf9IsOLjk4KtCrnq+rCMNsuAInAAMXFAFl6AOGoCCO/AAnsCzdW89Wi/W6yKaspYzh+AHrLdP+g2VIg== 0.85 < AA < 1.15 AAAB/XicdVDJSgNBEO1xjXEbl5uXxkTwNPQkJJMchBAvHiOYBZJh6Ol0kiY9C909QhyCv+LFgyJe/Q9v/o2dRVDRBwWP96qoqufHnEmF0Iexsrq2vrGZ2cpu7+zu7ZsHhy0ZJYLQJol4JDo+lpSzkDYVU5x2YkFx4HPa9seXM799S4VkUXijJjF1AzwM2YARrLTkmcf5nmTDAHtpvT6FFxBZxVLeM3PIqjqobCMtOI5drpY0KTh2EVWgbaE5cmCJhme+9/oRSQIaKsKxlF0bxcpNsVCMcDrN9hJJY0zGeEi7moY4oNJN59dP4ZlW+nAQCV2hgnP1+0SKAyknga87A6xG8rc3E//yuokaVNyUhXGiaEgWiwYJhyqCsyhgnwlKFJ9ogolg+lZIRlhgonRgWR3C16fwf9IqWHbZsq8LuVp9GUcGnIBTcA5s4IAauAIN0AQE3IEH8ASejXvj0XgxXhetK8Zy5gj8gPH2Ce7ik5k= BB = 0.35 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 ✏AB = 0.1; ✏AA = ✏BB = 1.0 LAMMPS: Molecular dynamics simulation ! Density of the system is varied ! Ref: Voigtmann and Horbach, PRL (2009) 

Interdiffusion 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 h r2(t)icm = h[RA(t) RA(0)]2i AAAB+nicbVDLTgIxFL3jE/E16NJNI5i4msyQ+NiYIG5cYiKPBCaTTinQ0Hmk7Shk5FPcuNAYt36JO//GArNQ8CT35uSce9Pb48ecSWXb38bK6tr6xmZuK7+9s7u3bxYOGjJKBKF1EvFItHwsKWchrSumOG3FguLA57TpD2+mfvOBCsmi8F6NY+oGuB+yHiNYackzC6WRd42u0Mir6m5bZyXPLNqWPQNaJk5GipCh5plfnW5EkoCGinAsZduxY+WmWChGOJ3kO4mkMSZD3KdtTUMcUOmms9Mn6EQrXdSLhK5QoZn6eyPFgZTjwNeTAVYDuehNxf+8dqJ6l27KwjhRNCTzh3oJRypC0xxQlwlKFB9rgolg+lZEBlhgonRaeR2Cs/jlZdIoW8655dyVi5VqFkcOjuAYTsGBC6jALdSgDgQe4Rle4c14Ml6Md+NjPrpiZDuH8AfG5w82KZFf xA = xB = 0.5 AAACBHicbVC7TsMwFHXKq5RXgLGLRYvEVCUdgLEtHZhQkehDakPkuE5r1U4i20Gqogws/AoLAwix8hFs/A1umwFajmTp6Jx7dX2OFzEqlWV9G7m19Y3Nrfx2YWd3b//APDzqyDAWmLRxyELR85AkjAakrahipBcJgrjHSNebXM387gMRkobBnZpGxOFoFFCfYqS05JrFctNN6o0UDiIRRiqEN7Dp1u8TzNOya5asijUHXCV2RkogQ8s1vwbDEMecBAozJGXftiLlJEgoihlJC4NYkgjhCRqRvqYB4kQ6yTxECk+1MoR+KPQLFJyrvzcSxKWcck9PcqTGctmbif95/Vj5l05CgyhWJMCLQ37MoA47awQOqSBYsakmCAuq/wrxGAmEle6toEuwlyOvkk61Yp9X7NtqqdbI6siDIjgBZ8AGF6AGrkELtAEGj+AZvII348l4Md6Nj8Vozsh2jsEfGJ8/s0WW2w== DAB / NDcm A # Interdiffusion: relaxation of concentration fl uctuation # Track centre-of-mass motion of bigger species # Center of mass MSD: # Interdiffusion coef fi cient: # For our system: Interdiffusive precocess in the context of binary mixture with large size ratio? Ref: V. Vaibhav, J. Horbach, and P. Chaudhuri; arXiv: 2202.12189 

10°7 10°5 10°3 10°1 101 hdR2 A (t)icm (a) 2.100 2.205 2.296 2.500 3.000 3.500 10°4 10°2 100 102 hdr2 A (t)i (b) 10°1 101 103 105 t 10°4 10°2 100 102 104 hdr2 B (t)i (c) 2.00 2.25 2.50 2.75 3.00 r 10°7 10°5 10°3 10°1 Diffusion coefficient DA DB Dcm A DAB Diffusion and interdiffusion MSD: center of mass (A) MSD: bigger species MSD: smaller species Interdiffusion Smaller species Bigger species # Bigger particles show fi nite diffusion at higher densities beyond MCT # At higher densities, bigger particle diffusion is same as COM diffusion # Interdiffusion follows the motion of smaller species Ref: V. Vaibhav, J. Horbach, and P. Chaudhuri; arXiv: 2202.12189 

10°7 10°5 10°3 10°1 101 hdR2 A (t)icm (a) 2.100 2.205 2.296 2.500 3.000 3.500 10°4 10°2 100 102 hdr2 A (t)i (b) 10°1 101 103 105 t 10°4 10°2 100 102 104 hdr2 B (t)i (c) 2.00 2.25 2.50 2.75 3.00 r 10°7 10°5 10°3 10°1 Diffusion coefficient ˜ DA DA DB Dcm A DAB Solid line: Usual de fi nition Dashed line: Modi fi ed de fi nition; quantity wrt respective centre of mass Dynamics with centre of mass correction Interdiffusion Smaller species Bigger species # Recti fi cation: measurement in respective centre of mass frame Ref: V. Vaibhav, J. Horbach, and P. Chaudhuri; arXiv: 2202.12189 Center of mass: bigger species MSD: bigger species MSD: smaller species 

Snapshot diameter of A-species Rheology: imposing fi xed shear rates 0.0 0.2 0.4 0.6 0.8 1.0 g 0.0 0.4 0.8 1.2 sXY ˙ g = 4e ° 6 ˙ g = 1e ° 5 ˙ g = 1e ° 4 ˙ g = 1e ° 3 10°4 10°2 100 102 hdr2 A (t)i 1.050 1.087 1.095 1.103 1.113 1.125 1.148 1.250 1.500 1.750 10°1 101 103 105 t 10°4 10°2 100 102 104 hdr2 B (t)i Bigger species Smaller species rho_A Stress-strain Quiescent MSD # Shear XY plane along X with different shear-rates # Nonequilibrium steady state: system yields and shear-stress becomes steady; measurements are done in y directions # Interplay of different time-scales: timescale introduced by shear competes with the relaxation timescales associated with two species Control parameter: shear-rate and density Timescale separation Ref: V. Vaibhav, J. Horbach, and P. Chaudhuri; arXiv: 2202.12612 

# Small density and small shear rates: Newtonian viscosity # High density: behaves like yield stress fl uid; emergence of yield stress near MCT density of 1.115 # Shear thinning: Viscosity decreases with increasing shear-rate Macro-rheology: fl ow curves 10°6 10°5 10°4 10°3 ˙ g 10°3 10°1 101 sxy( ˙ g) 10°6 10°5 10°4 10°3 ˙ g 102 104 106 h( ˙ g) 1.050 1.079 1.087 1.095 1.103 1.113 1.125 1.148 1.250 1.500 1.750 rhoA Stress vs shear-rate Viscosity vs shear-rate 

10°6 10°5 10°4 10°3 ˙ g 0 5 10 15 sxy( ˙ g) 1.250 1.500 1.750 1.1 1.4 1.7 rA 0 3 6 s0 AAACDHicbVDLSgMxFM3UV62vqks3wVYQhDLThboRim4ENxXsAzq13EnTNjTJDElGKEM/wI2/4saFIm79AHf+jWk7C209EDicc26Se4KIM21c99vJLC2vrK5l13Mbm1vbO/ndvboOY0VojYQ8VM0ANOVM0pphhtNmpCiIgNNGMLya+I0HqjQL5Z0ZRbQtoC9ZjxEwVurkC0Vfs74AfIFnpOOe3Pjd0CR+H4SA8b0s2pRbcqfAi8RLSQGlqHbyX/YGEgsqDeGgdctzI9NOQBlGOB3n/FjTCMgQ+rRlqQRBdTuZLjPGR1bp4l6o7JEGT9XfEwkIrUcisEkBZqDnvYn4n9eKTe+8nTAZxYZKMnuoF3NsQjxpBneZosTwkSVAFLN/xWQACoix/eVsCd78youkXi55pyXvtlyoXKZ1ZNEBOkTHyENnqIKuURXVEEGP6Bm9ojfnyXlx3p2PWTTjpDP76A+czx80MJp3 = 0 + K ˙n Herschel-Bulkley rheology Herschel-Bulkley rheology Stress vs shear-rate rhoA Ref: V. Vaibhav, J. Horbach, and P. Chaudhuri; arXiv: 2202.12612 Yield-stress 

rhoA = 1.050 rhoA = 1.1125 rhoA = 1.750 Mean-squared displacement Density dependent response of the particles 10°5 10°3 10°1 101 103 hdy2 A (t)i (a) 1.0e ° 3 1.0e ° 4 1.0e ° 5 0.0 (b) (c) 10°1 101 103 105 t 10°5 10°3 10°1 101 103 hdy2 B (t)i (d) 10°1 101 103 105 t (e) 10°1 101 103 105 t (f ) Shear-rate Micro picture: MSD # Smaller densities below MCT: dynamics under shear is similar to equilibrium dynamics # Densities close to MCT: bigger species respond to shear; smaller species continues to follow equilibrium dynamics # Higher density above MCT: both species respond to shear rhoA_MCT = 1.115 Bigger species Smaller species 

Micro: Diffusion coef fi cient vs shear-rate # Very small densities: Newtonian liquid <=> shear independent diffusivity of both species # Intermediate densities: Non-Newtonian regime <=> shear dependent diffusivity of only bigger species # High densities: Emergence of fi nite yield stress at vanishing shear-rates <=> shear dependent diffusivity of both species rhoA 10°6 10°5 10°4 10°3 DA 1.050 1.087 1.113 1.148 1.250 1.500 1.750 10°6 10°5 10°4 10°3 ˙ g 10°4 10°3 10°2 10°1 DB 10°3 10°1 101 sxy( ˙ g) 10°6 10°5 10°4 10°3 ˙ g 102 104 106 h( ˙ g) 1.050 1.079 1.087 1.095 1.103 1.113 1.125 1.148 1.250 1.500 1.750 Macro-micro connection Macro: stress and viscosity vs shear-rate rhoA Stress Viscosity Bigger species Smaller species 

MSD Stress Viscosity No shear Under shear # Systematic insertion of smaller species into the mixture lowers the viscosity i.e., softening of the material # Similar observations in colloidal and granular systems Symbol Composition A-B circle 100-0 square 75-25 triangle 50-50 10°1 101 103 105 t 10°4 10°2 100 102 hdr2 A (t)i 1.113 1.050 1.05 1.08 1.11 rA 10°6 10°4 10°2 DA 100-0 75-25 50-50 10°2 10°1 100 sxy( ˙ g) 1.113 1.087 103 104 105 h( ˙ g) 1.113 1.087 10°6 10°5 10°4 10°3 ˙ g 10°5 10°4 10°3 DA 1.087 1.113 Diffusion coef fi cient Varying composition: role of smaller particles Ref: V. Vaibhav, J. Horbach, and P. Chaudhuri; arXiv: 2202.12612 

# Quiescent dynamics and shear response of system of colloidal binary mixture with large size disparity # Finite interdiffusion beyond glass formation by bigger species ==> single particle data needs centre-of mass correction # Timescale separation between species: density dependent response to the shear # Shear: connecting microscopic observations with macro response # Varying the composition of the mixture Finite-size effects in the diffusion dynamics of a glassforming binary mixture with large size ratio Vinay Vaibhav, Juergen Horbach, and Pinaki Chaudhuri; arXiv: 2202.12189 Rheological response of a glass-forming liquid having large bidispersity Vinay Vaibhav, Juergen Horbach, and Pinaki Chaudhuri; arXiv: 2202.12612 Conclusions 2.00 2.25 2.50 2.75 3.00 r 10°7 10°5 10°3 10°1 Diffusion coefficient ˜ DA DA DB Dcm A DAB 10°6 10°5 10°4 10°3 ˙ g 10°3 10°1 101 sxy( ˙ g) 10°1 101 103 105 t 10°4 10°2 100 102 hdr2 A (t)i 1.113 1.050 1.05 1.08 1.11 rA 10°6 10°4 10°2 DA 100-0 75-25 50-50 

# Quiescent dynamics and shear response of system of colloidal binary mixture with large size disparity # Finite interdiffusion beyond glass formation by bigger species ==> single particle data needs centre-of mass correction # Timescale separation between species: density dependent response to the shear # Shear: connecting microscopic observations with macro response # Varying the composition of the mixture Finite-size effects in the diffusion dynamics of a glassforming binary mixture with large size ratio Vinay Vaibhav, Juergen Horbach, and Pinaki Chaudhuri; arXiv: 2202.12189 Rheological response of a glass-forming liquid having large bidispersity Vinay Vaibhav, Juergen Horbach, and Pinaki Chaudhuri; arXiv: 2202.12612 Conclusions 2.00 2.25 2.50 2.75 3.00 r 10°7 10°5 10°3 10°1 Diffusion coefficient ˜ DA DA DB Dcm A DAB 10°6 10°5 10°4 10°3 ˙ g 10°3 10°1 101 sxy( ˙ g) 10°1 101 103 105 t 10°4 10°2 100 102 hdr2 A (t)i 1.113 1.050 1.05 1.08 1.11 rA 10°6 10°4 10°2 DA 100-0 75-25 50-50 ThankYou