Thermodynamics at Small Scales. Signe Kjelstrup, Sondre Kvalvåg Schnell, Jean-Marc Simon, Thijs Vlugt, Dick Bedeaux

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1 Thermodynamics at Small Scales Signe Kjelstrup, Sondre Kvalvåg Schnell, Jean-Marc Simon, Thijs Vlugt, Dick Bedeaux

2 At small scales: 1. The systematic method of Hill 2. Small systems: surface energies 3. Predicting and verifying Γ and h in small systems using simulations 4. Conclusions and perspectives 1963 Schnell, Vlugt, Simon, Bedeaux, Kjelstrup, Chem.Phys. Letters, ms in review

3 Small systems in equilibrium far from the thermodynamic limit When and how does thermodynamic equations apply? Is there a link between the nanoand macrosize properties?

4 Why Γ and h? 1 dln 1 kt lnn dlnn B h Ts TV, Γ, γ: describes the non-ideality of the system, and diffusivity coefficients h: linked to measurements

5 Custers, colloids, molecular pumps, single-molecules: Snapshots can be made Introduce N : no of independent replicas of the small system:. T 1 2 N E S tot tot NE NS Hill: The small systems can be made extensive using this ensemble!

6 Making small systems extensive, following Hill de TdS pndv dn XdN t t t No. of small systems Total internal energy, entropy chemical potential, particle number Pressure, number of small systems, volume, Idea: Change N, keeping S t,, V and N t constant. The total volume NV changes, and S t and N t redistribute over the new number of subsystems. This changes E t via X.

7 Making small systems extensive. E t is a total differential in S t, V, N t and N: de TdS pndv dn XdN t t t E t X N t SVN,, t pv ˆ XdN: work added due to smallness - a new pressure times the volume of the small system Suitable for grand-canonical ensemble:, VT, constant.

8 E t, S t and N t are linear homogeneous functions in N: E TS N pv ˆ N t t t Ensemble averages: E N E, N NN t The entropy is the same in all parts: S t =NS t With these definitions: E T S N pv ˆ In thermodynamic limit pˆ p

9 Gibbs equation, one small system: de TdS pdv dn Thermodynamic factor and enthalpy of the small system 1 1 lnn pv ˆ p kt B V h TS N TV, Gibbs-Duhem s d pv ˆ SdT pdv Nd T,

10 Statistical mechanics, μ,v,t ensemble pv ˆ N TS E N k T ln t t t B t Ξ t : grand partition function for ensemble of small systems Independence and indistinguishability: Ξ t = Ξ N For the system: pv ˆ TS E N k T ln B Thermodynamic relations for one small system: ln ln ln E k, N k T, p k T B B B 1/ T V, V VT,, T

11 System creation Reservoir: A larger box in equilibrium at μ,t. Side L t Periodic boundary conditions. Small system: Volume element V (side L n ) inside reservoir (snapshot) The μ,v,t ensemble: Sampling N small non-periodic systems inside the μ,t reservoir

12 Calculate Γ and molar enthalpy h from fluctuations in N and E N N N EN E N, h 2 2 N N N N nrt Three systems studied: 1) Lennard-Jones potentials, cutoff (2.5σ) and shifted 2) WCA (Weeks, Chandler, Anderson, 1971) 3) Argon in silicalite-1 (Garcia-Perez et al. JPC C, 2008)

13 Monte Carlo sampling procedure: V is chosen at random locations in the reservoar T* = σ < L < 16 σ L t < 20 σ Particle diameter Volume: V=L n d ( cube, square) Surface: 2dL n d-1

14 Effects of boundary conditions Periodic Non-periodic We know: Periodic systems scale with 1/N (Siepman et al. J Phys. Condes. Matter 1992) We shall see: For non-periodic systems there is a dependence on 1/L!

15 Interactions particles - available space: probability for a particle to be close to the surface differs from the probability to be far from the surface: Surface energy: E s (μ,t) Canonical partition function, one particle: Canonical partition function, N particles:

16 Grand canonical partition function, small system: Small system pressures: Inverse thermodynamic correction factor and enthalpy density:

17 Thermodynamic correction factor of low density fluids, μvt reservoir T* = σ < L < 16 σ

18 Enthalpy density variations in small fluid systems μvt reservoir

19 Thermodynamic factor vs. lattice loading 1 or 2 unit cells; required for reliable results! Compare: T= 2<E>/3k= T(conf.) for cell with dimensions (1,¼,1) Simon, Rubi JPCB 2010

20 Ar-density plots in a unit cell of silicalite-1 The xz plane The xy plane The zy plane

21 Isosteric heat of adsorption

22 Conclusions We have predicted and verified a 1/L dependence of thermodynamic properties, h and Γ, in small systems The limit L gives well defined large-system values The L-corrections is up to 50% of the large system The unit cell: The smallest volume needed to sample thermodynamic properties of Ar in zeolite

23 Perspectives A MD-procedure has been developed for determination of nanoscale thermodynamic properties and macro-scale properties from them

24 Thank you for the attention! Financial support from

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