hydrate systems Gránásy Research Institute for Solid State Physics & Optics H-1525 Budapest, POB 49, Hungary László

Size: px

Start display at page:

Download "hydrate systems Gránásy Research Institute for Solid State Physics & Optics H-1525 Budapest, POB 49, Hungary László"

Penelope Hood
5 years ago
Views:

1 Phase field theory of crystal nucleation: Application to the hard-sphere and CO 2 Bjørn Kvamme Department of Physics, University of Bergen Allégaten 55, N-5007 N Bergen, Norway László Gránásy Research Institute for Solid State Physics & Optics H-1525 Budapest, POB 49, Hungary hydrate systems

3 Crystalline solidification: nucleation & growth Phase field theory

Nucleation Heterophase Heterophase fluctuations: fluctuations: Wolde & Frenkel,, 1996 Gasser et al.

) picture: picture: cluster cluster free free energy energy (W)( (W) ) = = volumetric volumetric + +

4 Nucleation Heterophase Heterophase fluctuations: fluctuations: Wolde & Frenkel,, 1996 Gasser et al., 2001 Auer & Frenkel,, 2001 Yonezawa,, 1991 Classical Classical (sharp (sharp intf.) intf.) picture: picture: cluster cluster free free energy energy (W)( (W) ) = = volumetric volumetric + + interfacial interfacial contributions contributions Crystal-liquid Crystal-liquid liquid interface interface Crit. Crit. fluct. fluct.. typically typically ~ ~ molecules molecules Diffuse interface model is is needed!!!!!! (e.g. (e.g. phase phase field field theory) theory) Davidchack & Laird, 1998

Warren & Boettinger 1995, adopted to HS) F bt 2 = d r ( m ) + f ( m, φ ) 2 where f(m,f) ) = w T g(m) ) + [1 - p(m)] f S (f) + p(m) f L (f) g(m) )

5 II. Phase-field theory of nucleation in hard sphere system Free energy functional: (standard phase field theory, see e.g. Warren & Boettinger 1995, adopted to HS) F bt 2 = d r ( m ) + f ( m, φ ) 2 where f(m,f) ) = w T g(m) ) + [1 - p(m)] f S (f) + p(m) f L (f) g(m) ) = ¼ m 2 (1 - m) 2, p(m) ) = m 3 (10-15m + 6m6 2 ), f L,S =polynomials reproducing the MD results, Model parameters: w & b intf. intf.. free free energy energy & intf. intf.. thickness they they can can be be fixed fixed in in equilibrium & predictions be be made made for for nucleation without adjustable parameters

6 Critical fluctuations: Extremum of grand potential functional (unstable equilibrium): W = F - m dr r Euler-Lagrange eqs. for phase field m (non-conserved) conserved): δω ω ω ω 0 = = = bt δm m m m 2 m and for volume fraction f (conserved): 0 = δω δφ = ω φ ω φ = ω φ ω φ

8 Critical fluctuations: spherical symmetry Phase-field: m + 2/r m = Dm(m)/( /(bt) Volume fraction: f = f(m) implicit equation Boundary cond.: r fi : m fi1, 1, f fi f ; r = 0: = 0: m, f = 0 W*

9 Hard sphere system: - Thermodynamics known accurately (Hall, 1970) - Interface profiles for density available 10% - 90% interface thickness, d (Davidchack & Laird, 1998) - Interface free energy from MD (Davidchack & Laird, 2000) orientation g s 2 /kt (111) (100) (110) g & d b & w

10 Free energy surface: For σ = 890 nm & (111) interface

11 (X-X min )/(X max -X min ) Equilibrium profiles: z/σ (111) Normalized distributions: (X-X( min )/(X max red: MD peak height blue: MD filtered density black solid: PFT [1-m(z)] black dashed: PFT volume fraction max -X min ) good agreement for (111) & (100) less satisfactory for (110) (X-X min )/(X max -X min ) (X-X min )/(X max -X min ) Compare solid line with solid, dashed with dashed z/σ (100) z/σ (110)

12 Nucleation : : Radial order parameter profiles: solid line: 1-m dashed line: (φ-φ )/(φ S -φ ) 0.6 X r/σ calculated with average interface properties

13 Nucleation barrier: No adjustable parameters Classical droplet model overestimates J by 3 to 5 orders of magnitude PFT upper curve: (110) central curve: average lower curve: (111) W*/kT CNT Φ calculated assuming spherical shape triangles: MC (Auer( & Frenkel,, 2001)

14 Tolman length: Curvature correction by Tolman (1949): g(r)» g / (1+ 2d2 T /R) δ T = Tolman length Route (i): δ T = R e -R p (exact) Route(ii): δ T = (R( p /2)[γ /γ 1] Route(ii): δ T = (R e /2)[γ /γ 1] R p & R e : radius of surf. of tension & of equimolar surface. solid line: route (i) dashed line: route (ii) dotted line: route (iii) δ T /σ Φ calculated assuming spherical shape

15 III. Phase-field theory of nucleation of hydrate nucleation Free energy functional: (Warren & Boettinger 1995) F bt 2 = d r ( φ ) + f ( φ, c) 2 where f(f,c) = w T g(f) ) + [1 - p(f)] f + p(f) f S L g(f) ) = ¼ f 2 (1 - f) 2, p(f) ) = f 3 (10-15f + 6f6 2 ), f L,S = polynomials for aqueous solution of CO 2 and CO 2 -hydrate Model parameters: w & b intf. intf.. free free energy energy & intf. intf.. thickness calculation of of nucl. nucl.. barrier barrier can can be be made made without adjustable parameters

17 Thermodynamics A thermodynamic description of the hydrate phase according to the van der Waal Platteuw approach requires the chemical potential of CO2 as well as the impact of CO2 on the inclusion in the cavity.

18 Note that: - the use of the DFT (Density Functional Theory) or PFT (Phase Field Theory) requires continuous description of the thermodynamics in the transitions from one phase to the other (no sharp interface). - consistency is required for all components in all phases - Stochiometry of the hydrate is an implicit function that is proportional to the exponent of the chemical potential of the guest molecule through the cavity partition function I.e.: the most proper way to describe the guest molecule chemical potential is so that it is consistent with the same model that describes the interactions with the cavity water molecules

19 a g = e x l n p[ ( β h ( µ ω ) h ( ω g ) d ω u )] β = 1 / k B T a= exp ([ βµ gu )] g= ln( hhd βω )() ω β= / 1kTB Extended adsorbtion Theory 0 µ = µ ν RT ln( 1 + aij ) w w i i j Single cavity integration (small guest molecules) Harmonic oscillator approach (large guest molecules) π a b V cage [ ( 1 β µ β ln b) ] = exp + = ( m / β 2πh ) [ βw v ] exp ( ) 2 3/ 2 dv ( ) [ β µ ] a = exp g u g = ln( βhω ) h( ω ) dω β = 1 / k T B f µ + x µ S Hydrate Hydrate = x CO 2 CO 2 Hydrate w Hydrate w

20 θ ij a 1 + ij a ij where θ ij is the filling fraction of type i guest in cavity type j Assuming that CO 2 only fills large cavity: x Hydrate CO 2 = 3 23 θ ij I.e.: The stochiometry of the hydrate Is an implicit function of the progress of The nucleation and has to be treated as such

21 Lquid solution description, CO2 µ aqueous CO 2 r aqueous r ( T, P, x) = µ CO 2 ( T, P, x) + RT ln( x CO 2γ CO 2 ) where superscript aqueous denotes liquid water phase and infinity symbol denote infinite dilution as reference state (activity coefficient of CO 2 goes to unity as the molefraction of CO 2 approaches zero). Infinite dilution chemical potential for model system From Molecular Dynamics simulations using Thermodynamic integration. Activity coefficients fitted to equilibrium solubility data.

22 Lquid solution description, H2O aqueous r aqueous r µ H O ( T, P, x) = µ H 2O pure ( T, P, x) + RT ln( xh 2Oγ H 2 2 O ) Activity coefficient for water calculated using Gibbs-Duhem f L = x CO aqueous r aqueous r 2µ CO2 ( T, P, x) + xh2 Oµ H2O ( T, P, x)

23 Free energy surface: b & w γ ice = 29.1 mj/m 2 & d = 1 nm 10-90

24 Radial profiles: Nucleation : : W* = J W* CNT = J φ, c CO red: 1-φ blue: c CO2 dashed: CNT Initial CO 2 content: 3.3% (saturated) r (A) b & w γ ice = 29.1 mj/m 2 & d = 1 nm 10-90

25 Plans: (1) Interface profiles & interface free energy from Molecular Dynamics (2) Full dynamics phase field theory for nucleation and growth to model hydrate formation under realistic conditions

26 Work in progress Interface free energy using a thermodynamic cycle and thermodynamic integration (Mezei s approach). Cycle sismilar to the approach of Broughton and Gilmer (1986), as also applied by several other groups. Direct calculation of absolute chemical potential of CO2 using thermodynamic integrations (Overheads describing the approaches is available)

27 Summary: (1) The phase field theory: - predicts the height of the nucleation barrier reasonably for the hard sphere liquid (without adjustable parameters); - the Tolman length decreases with increasing size of the fluctuations. (2) We formulated a phase field theory for CO 2 hydrate nucleation in aqueous CO 2 solution Financed by ESA Prodex No /00/NL/SFe Forms part of ESA MAP Project AO

29 Hydrate melting Surface tension? γ=70 ±9 mn/m Interface thickness?

30 INTERFACIAL TENSION Interface Run, ns T, K P t, P n, atm h z, Å γ, mn/m γ, mn/m atm Sim liquidliquid Sim liquidliquid Sim liquidliquid Exp a liquidliquid 275 < Sim liquid-gas NVT sat NVT sat 39 6 Exp a liquid-gas 275 < Exp b liquid-gas Exp b liquid-gas Sim liquid-gas NVT half NVT half Exp b liquid-gas Sim this work; NVT sat stands for NVT simulation of liquid-gas interface corresponding to saturated CO 2 density; NVT half, for simulation at half saturated density; a experimental data [13]; b experimental data [12].

31 CO2 hydrate Water System: Characteristic Density Profiles at 22 and 48 ps

32 Trajectories of CH 4 guests Liquid-phase CO 2 molecule trapped near the interface

33 TI Cleaving aproach to interfacial free energy estimation: Reversible path Broughton & Gilmer: Molecular Dynamics of Interface. VI J. Chem. Phys., Vol. 84, No. 10, 15 May 1986

34 POLYNOMIAL PATH FOR THERMODYNAMIC INTEGRATION Mezei MJ. Comp Chem, 13, 651 (1992) E(λ, X N ) = f 1 (λ)*e 1 (X N ) + f 0 (λ)*e 0 (X N ) f 0 (0) =0 and f 0 (1) = 1 f 1 (0) =1 and f 1 (1) = 0 A = A = 1 < E ( λ λ > d λ λ 1 A ) / 0 0

35 E = ( ulj + ues ) = ij i E i Polynomial path, Mezei (1992), introduces different exponents for potential terms differing in inverse distance powers: A = 1 [ k 1 < ] i k i i > k i 0 λ E d λ λ

36 Chemical potential of CO 2 in structure I hydrate 1. Conversion of CO 2 to CH 4 via TI a) scale down charges b) convert LJ parameters 2. Single-guest scaling in CH 4 hydrate 3. Steps 1+2 CO 2 chemical potential

Nucleation rate (m -3 s -1 ) Radius of water nano droplet (Å) 1e+00 1e-64 1e-128 1e-192 1e-256

Nucleation rate (m -3 s -1 ) Radius of water nano droplet (Å) 1e+00 1e-64 1e-128 1e-192 1e-256 Supplementary Figures Nucleation rate (m -3 s -1 ) 1e+00 1e-64 1e-128 1e-192 1e-256 Calculated R in bulk water Calculated R in droplet Modified CNT 20 30 40 50 60 70 Radius of water nano droplet (Å) Supplementary