docWall-liquid and wall-crystal interfacial free energies via
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Wall-liquid and wall-crystal interfacial free energies via
WallLiquid and WallCrystal interfacial excess free energies via Thermodynamic Integration: A Molecular Dynamics simulation study Ronald Benjamin and Jürgen Horbach Theoretical Physics II: Soft Matter Universität Düsseldorf, Germany Crystal growth from melt Melt CL Crystal WC WL Wall Young's Equation WL −WC cos = CL Goal is to determine wl and wc via Thermodynamic Integration (TI) using MD simulation. Computational methods to determine 1.) Pressure anisotropy (PA) Henderson & van Swol (1984, HS, WL) Miguel & Jackson (2006, HS, WL) L G 1 Nijmeijer et al. (1990, LJ, '=lim = [ P z−P z ] dz ∫ A0 T Tang & Harris (1995, LJ , A 2 0 N ) WL ) WL Varnik et al. (2000, LJ, WL ) 2.) Thermodynamic integration (TI) Deb et al. (2011, 2012, HS, A0, WL & WC ) Heni & Loewen (1999, HS, WL & WC) 1 1 G 〈 〉 ∂G 1 ∂U Fortini & Djikstra (2006, HS, & ) 1 if = = d = d WL WC A A ∂ A ∂ Laird & Davidchack (2007, HS, & ) 0 0 WL WC Leroy et al. (2009, LJ, WL) ∫ ∫ 3.) Nonequilibrium work method (BAR) NF ∑ i=1 NR 1 NF [1 expW F −G] NR −∑ i=1 i Mu & Song (2006, LJ, CL ) 1 NR [ 1 exp−W R −G] NF i 4.) GibbsCahn integration (GC) d /T =− e N P v N T 2 =0 Davidchack (2010, HS, CL) vN dT dP T Laird et al. (2009, LJ, CL ) Laird & Davidchack (2010, HS, WL) Which Model? Hard Spheres vs. Lennard Jones U r = 0 r ∞ r≤ 1.) Hard Spheres purely repulsive entropic Number of parameters limited (packing fraction) predicts complete wetting of a hard wall [Deb et al. (2011, 2012 )] 2.) Lennard Jones U r=4 repulsive and attractive forces [ ] r 12 − r vary different parameters (density and interaction strength). 6 from Thermodynamic Integration Step 1: Bulk LJ system = 0 0< <1 LJ + Flat Wall (fw) = 1 from Thermodynamic Integration Step 2 : LJ + Flat Wall LJ + Structured Wall Uwall=Ufw = 0 0< <1 2 2 U wall =1− U fw U struct. wall = 1 Uwall=U struct. wall from Thermodynamic Integration Step 1 Step 2 Tcrystal =0.5 Tcrystal =0.5 Tliquid =2.0 Tliquid =2.0 from Thermodynamic Integration Computational aspects of PA method (Deb at al., IJMPC, 2012) c WC WL Contact Angle Flat Wall (WCA) WL −WC cos = CL CL from Laird et. al. (JCP, 2008) Structured Wall (WCA) Structured Wall (LJ) from NonEquilibrium Work Method 〈W 〉≥ G Second Law 〈W 〉=G〈W diss 〉 Quasistatic process W rev =G 〈W diss 〉≥0 TI Jarzynski (1997, PRL) Jarzynski Equality 〈exp −W 〉=exp −G Ns 1 G=−K B T ln exp−W i ∑ N s i=1 from NonEquilibrium Work Method 〈W 〉=∫ dW W P W 〈exp −W 〉=∫ dW exp − W P W W = G P(W) exp(W) Number of trajectories needed increases exponentially with system size from NonEquilibrium Work Method Crook's NonEq. Fluct. Thm. (PRE, 2000 ) Combine Forward and Reverse trajectories −W F −G 〈 f W e −W R − G 〉 F =〈 f −W 〉 R , 〈 f W 〉 F =〈 f −W e f W =1 〉R Jarzynski Equality 1 f W = [1 N F / N R expW −G ] Bennett Acceptance Ratio (BAR) (Shirts et al., PRL, 2003) NF ∑ i=1 NR 1 NF [1 expW F −G ] NR −∑ i=1 i 1 NR [1 exp−W R − G] NF =0 i Least statistical error from NonEquilibrium Work Method <W>F /A <W>F /A GFJarz /A GFJarz /A + BAR X BAR -G Jarz B TI <W>B /A /A <W>B /A More Complex potentials (EAM) Simulations at coexistence, requiring long equilibration times -GBJarz /A Difficult to find a reversible path from GibbsCahn integration Idea: Apply Cahn's generalization of Gibbs's interfacial thermodynamics to obtain a differential equation for along pressure, temperature or coexistence curve. (Cahn, Interf. Seg., 1979) G = E TS + PV, Gbulk=N A.) Bulk system: ETS+PVN=0 Bulk system with interface: ETS+PVN=A Taking the differential, B.) Bulk system: 0=SbdT+VdPNbd Bulk system with interface: d(A)=SdT+VdPNd (Gibb'sDuhem Equation) Applying Cramer's rule to (B), we get, ∣ ∣ 1 Y [Y / X ]= b X X Ad=[S/X]dT+[V/X]dP[N/X]d Ad=[S/N]dT+[V/N]dP Choosing X=N, d /T =− At constant T, d=vN dP e N P v N T2 vN dT dP T ∞ [ v N =∫ 1− 0 ] z dz b b Y b X from GibbsCahn (GC)integration Liquid in contact with flat structureless wall Gibb'sCahn integration works well for WL . Summary A novel thermodynamic integration scheme to calculate the interfacial free energy of liquid/crystal in contact with a wall. Non-equilibrium Work measurements in good agreement with TI results and offer improvement over TI in terms of computational speed besides being an additional check of our TI scheme. Gibbs-Cahn integration method useful when WL is needed at many points along the pressure, temperature or coexistence curve. Reference R. Benjamin and J. Horbach, J. Chem. Phys. 137, 044707 (2012) Thank You!