International Conference on Gas Hydrates (ICGH) (6th : 2008)


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eedings of the 6th International Conferen
e on Gas Hydrates (ICGH 2008), Van
s and Te
hnology University of Bergen Allegaten 55, 5007 Bergen NORWAY ABSTRACT Hydrates in some 
rude oils have a smaller tenden
y to form plugs than in others, and lately this is be
oming a fo
us of resear
h. To study this and the a
tion of hydrate antiagglomerants in general, hydrate surfa
e properties must be known. To help in 
terizing the surfa
e properties by simulation, the 
apillary waves of 
lathrate hydrate surfa
es in va
uum are examined in all unique 
rystal fa
es by Mole
ular Dynami
s, and an attempt is made to estimate the surfa
e energies in the respe
rystal fa
es from the wave u
tuations [1℄. We also attempt to estimate solid/liquid surfa
e energies of hydrate/oil and hydrate/water for a spe
e, for 
omparison. The for
eeld OPLS_AA is used for the organi
ompounds, while TIP4P/i
e is used for the water framework. The anisotropy of the surfa
e energy is then estimated and the result 
ompared to the initial growth rate of dierent 
rystal fa
es as found in experiment. [2℄ Keywords: gas hydrates, plug prevention, surfa
e energy, mole
ular dynami
s LIST OF SYMBOLS Symbol Denition a Capillary length [m℄ A Interfa
ial area [nm2℄ c1 c2 Cubi
oe. [-℄ g Gravitational a

eleration [m/s2℄ H[z] Hamiltonian of z[r] [kJ/mol℄ kB Boltzmann 
onstant [J/K℄ k, ki wavenumber [nm −1 ℄ l Min. surfa
e wavelength [nm℄ L Max. surfa
e wavelength [nm℄ p Pressure [Bar℄ P Probability [-℄ R2 Sample 
oe.[-℄ sij (Surfa
e) stress tensor [mN/m℄ dS Element of surfa
e [nm2℄ T Temperature [K℄ V Volume [nm3℄ r = (x, y) surfa
oordinates 〈X〉 Average of X z(x, y) Model for real interfa
e α Fourier expansion 
ient γ Surfa
e free energy κ Interfa
ial stiness [mN/m℄ ǫ (Surfa
e) Strain tensor µ Chemi
al potential [kJ/mol℄ ρ Density [kg/m3℄ θ Angle (xyz) Normal to 1D-strip [uvw] Tangent to 1D-strip INTRODUCTION Hydrates formed by water and light hydro
ar- bon mole
ules are in
reasingly be
oming the fo
us of resear
h in the energy industries and a
ademia. One reason for this is that the energy in- *Corresponding author: Phone: +47 55 582869 Fax: +47 55 589440 E-mail: dustry is be
oming aware of the potential energy resour
e that naturally o

urring hy- drate reservoirs 
onstitute, and methods for extra
ting methane from su
h reservoirs are being sought [39℄. Another reason, whi
h is the fo
us of this paper, is that plugging of pipelines, parti
u- larly subsea pipelines, by hydrates is be
om- ing an in
ostly problem. As oil and gas exploration is expanding into the ar
- ti
onditions in subsea pipelines are getting further into the hydrate-stable region, giving rise to extra 
osts in hydrate preven- tion. While natural hydrate reservoirs often 
on- tain almost pure methane, and therefore mainly are stru
ture I, hydrates in pipelines form in the presen
e of a variety of light hydro
arbons, and are therefore often stru
- ture II, sin
e this is the thermodynami
ally favored 
onguration in all but the purest methane gas environments [10℄. At present, thermodynami
 inhibitors are added, often in large amounts [11℄, to avoid plugging by pipeline hydrates. In the HYPE- RION proje
t, of whi
h this study is a part, the fo
us is on managing the risk of plug for- mation in the presen
e of hydrate parti
les, rather than on suppressing hydrate parti
le formation in the rst pla
e. This is a rela- tively new fo
us in hydrate resear
h [12℄. In this eld of study, quanti
ation of the plug- ging tenden
y of hydrate parti
les in various pipeline-liquid environments is important, as is the identi
ation of anti-agglomerants for hydrates and the issue of why some oils tend to form plugs under hydrate-stable 
onditions and others not [1317℄. This latter is thought to be related to the o

e in some oils of natural inhibiting 
omponents (NICs). In this approa
h to hydrate management, the 
ohesivity of hydrate parti
les, and there- fore their surfa
e energy, in the presen
e of various pipe-line liquids and also in the pres- en
e of a variety of tra
omponents, poten- tially a
ting as anti-agglomerants, is 
ial. The surfa
e energy of hydrates is also needed to predi
t the rate of growth of hy- drate parti
les, and knowing the dieren
e be- tween the surfa
e energies on the 
rystal fa
es aids in predi
ting the shape of the parti
les formed, whi
h may again impa
t the forma- tion of hydrate plugs. In this paper, surfa
e energies of hydrates are studied using 
al Mole
ular Dynam- i
s. THEORY It is well known that the surfa
e free-energy of interfa
es with elasti
 solids is not in general equal to the interfa
ial stress (or the measured surfa
e tension of 
ting uids). This is due to the fa
t that the surfa
e of solids, in 
ontrast to that of liquids, 
an deform, e.g. expand, by elasti
 deformation, a

umulating shear stress but without exposing more par- ti
les (atoms, ions or mole
ules) from bulk to the surfa
e. Mathemati
ally this is embodied in the Shuttleworth relation [18℄. sij = γδ i j + ( ∂γ ∂ǫ ) T,V,µ (1) This ne
essitates another route to the surfa
e- free energy of solid-uid interfa
es than the 
onventional method used for uids, i.e. 
ulating the dieren
e between the normal and tangential 
omponents of the stress ten- sor, whi
h only works for solids in the absen
e of elasti
 stresses in the surfa
e. This latter 
ondition is near impossible to a
hieve in the small timespans and spatial s
ales of mole
u- lar modelling. Capillary wave u
tuations and their rela- tion to interfa
ial stiness We 
an over
ome the di
ulties presented by possible elasti
 stress by modelling the inter- fa
e as a u
tuating membrane a

ording to the theory of 
apillary waves [19,20℄. We 
on- sider the interfa
e as a mathemati
al 2D sur- fa
e z = z(r) = z(x, y). The Hamiltonian is: H[z(r)] =∫∫ A dS ( κ √ 1 + (∇z(r))2 + ∫ z 0 dh∆ρgh ) (2) The Hamiltonian fun
ontains two terms, the rst 
omes from the work of 
reat- ing surfa
e by extending the membrane, and the se
ond term 
omes from the work against gravity. This se
ond term is only needed for obtaining an analyti
 solution to the problem, and will be removed by a limiting pro
edure in our 
ase of zero gravity. κ is the interfa
ial stiness, and ∆ρ is the density dieren
e be- tween the phases separated by the interfa
e. We 
an linearize Eq. (2) into: H[z(r)] =∫∫ A dS ( κ (1 + 1 2 (∇z)2) + 1 2 ∆ρgz2 ) (3) This is to be examined under the periodi boundary 
onditions in the simulation box: z(0, y) = z(L, y) z(x, 0) = z(x,L) (4) Now we perform a spe
tral analysis of the energy fun
tional. To that end we 
onsider the possible dis
rete os
illation modes of the periodi
e: z[k] = ∑ k α(k)eik·r (5) By substituting Eq. (5) in Eq. (3) and us- ing the known relation for the Fourier trans- form of the derivative, we obtain: H[z(k)] = κA + 1 2 κA ∑ kx ∑ ky α(kx)α(ky) (∫ x ∫ y dS ei(kxx+kyy)(2a−2 − kxky) ) (6) Where a is a generalized 
apillary length, a2 = 2κ/(∆ρg) Imposing the boundary 
onditions, Eqs. (4), kx and ky have the allowed values 2πn/L, n ∈ Z. With the above bound- ary 
onditions, the fun
tional vanishes for all kx,ky unless kx = −ky. This gives the follow- ing energy spe
trum upon integration: H[z[k]] = κA(1 + 1 2 ∑ k (α(k)α(−k) × (2a−2 + k2)) (7) where k is the absolute value of kx and ky. The mean square u
tuations of the inter- fa
e position for a given set of modes are: 〈z2〉 = A−1 ∫ ∫ A z2(r)dS = ∑ k>0 α(k)α(−k) (8) where the last equality 
omes from the well- known Parseval's relation. Now, the probability of nding a mode with energy E[k] in the 
al ensemble is P ∝ e− H[z[k]] kBT In our linear approximation it means that the spe
trum of modes has a Normal distribution. >From the properties of the Normal distribu- tion we know that the expe
tation value of a squared Normal variable (in our 
ase z2) is equal to the Varian
e of the Normal dis- tribution. The varian
an be easily read o the Normal probability density fun
tion as the quadrati
 part of P[H[k]]. That is, the expe
ted u
tuations in our linearized model be
ome: 〈z2〉 = ∑ k>kmin 〈α(k)α(−k)〉 = kBT κA ∑ k>0 1 2a−2 + k2 (9) Obviously the dominant 
ontribution to the expe
tation value 
omes from the low- wavenumber ripples, whi
h the boundary 
on- ditions di
tate to be kmin = 2π/L. The maximum wavenumber is more 
hallenging to 
t to a measurable quantity, but it has to be an integer multiple of kmin, let us say kmax = 2π/l where l is a length of mole
ular dimensions. Transferring to a 
ut-o in 2D wave-ve
tor spa
e, and using the 
ontin- uum approximation we have: ∑kmax k=kmin 7→∫ kmax kmin k/(2π) dk. Integrating equation (9) out we arrive at the following expression 〈z2〉 = kBT 4πκ ln [ 1 + 2(πa/l)2 1 + 2(πa/L)2 ] . where l 
orresponds to the minimum, and L to the maximum lengths
ale of 
apillary waves respe
tively. Now letting gravity: g → 0, the 
apillary length goes to innity, and the expression above be
omes: 〈z2〉 = kBT 4πκ ln [ L2 l2 ] = kBT 2πκ ln [ L l ] . (10) In 
ase of a 1D strip, the derivation is sim- ilar, and the result is [20℄: 〈z2〉 = 1√ 2π2 kBT κ L(1−O[l/L]). (11) As 
an be seen the u
tuation s
ales linearly with the linear size of the system in this 
ase, as opposed to logarithmi
ally whi
h would lead to larger signal-to-noise ratio. This, and the singling out of a preferred dire
tion on the surfa
e (see next se
tion), is the main reason for adopting this geometry. On interfa
ial stiness and surfa
e free en- ergy and a means to de
ouple them As noted above, both surfa
e stiness and surfa
e free energy in elasti
 materials are di- re
tion dependent. The u
tuation spe
trum therefore depends not only on the magnitude of γ. but also on the energy required for lo
al orientation-u
tuations. [21℄ The anisotropy 
an be quantied with the help of the param- eter θ, dened to be the angle between the instantaneous fa
e normal, and the normal of a 
e fa
e. The surfa
e stiness κ(θ) is then related to the surfa
e free energy γ(θ) through κ(θ) = γ(θ) + d2γ dθ2 (12) To de
ouple the interfa
ial free energy γ from the surfa
e stiness κ we have to mea- sure the stiness in several dierent interfa- 
ial orientations, and the free energy 
an then be determined indire
tly if we 
an t γ(θ) to a suitable fun
tion des
ribing the u
tu- ation anisotropy. Su
h a fun
tion must 
on- form to the symmetries of the 
rystal. As hy- drates have 
 symmetry, a natural 
e would be 
s. Like David
k et. al. [22℄ we start with the produ
t expan- sion [23℄: γ(n) γ = c1 (∑ i n4i − 3 5 ) + c2 ( 3 ∑ i n4i + 66(n1n2n3) 2 − 17 7 ) (13) where c1 and c2 are expansion 
ap- turing the anisotropy. We see that we have 3 undetermined pa- rameters, γ, c1, and c2. To x these we need to measure the surfa
e u
tuations in at least the 3 unique fa
es: (001), (110) and (111) of the 
rystal, preferably we should overdeter- mine the system by measuring u
tuations in more than one dire
tion on ea
h fa
e. We obtain the following formulas for the di- re
tional interfa
ial free energy and stiness in terms of the averaged interfa
ial free en- ergy and the anisotropy parameters c1 and c2 [22℄ (Observe that we dier by using the tangent-ve
tor and not the binormal-ve
tor for the surfa
e strip as referen
e): (Fa
e)[t℄ γ γ κ γ (001)[100℄, [010℄ 1 + 25c1 + 4 7c2 1− 185 c1 − 807 c2 (110)[110] 1− 110c1 − 1314c2 1 + 3910c1 + 15514 c2 (110[001] 1− 110c1 − 1314c2 1− 2110c1 + 36514 c2 (111)[112] 1− 415c1 + 6463c2 1 + 125 c1 − 128063 c2 METHOD We perform a series of Mole
ular Dynami
s simulations over O(106) steps with a system size of O(10000) atoms. Constru
ting the 
lathrate hydrates stru
ture I and II, we fol- low the pro
edure outlined in our previous work [24℄ employing the spa
e groups Pm3n and Fd3m for respe
tively stru
ture I and II hydrate. Our setup 
onforms essentially to that of David
k et. al [22℄. However, we em- ploy a slightly dierent analysismeasuring density-proles and dire
t u
tuations in real spa
e (Eq. 10) and furthermore we also 
om- pare with measurements using the full 2D model. Sin
e we do not have a truly 1D sys- tem, we found it suitable to employ a purely 2D analysis, even when our pseudo-1D geome- try were adopted to enhan
e the surfa
e-wave u
tuations over the bulk u
tuations. To de
ouple the free energy from the measured surfa
e stiness we perform va
uum simula- tions of the hydrate slabs. We then use the va
uum simulations to 
alibrate the geomet- ri
tion-dependent fa
tors of the 
ubi harmoni
 expansion for our parti
ular geom- etry. Having obtained the 
 ex- pansion, we 
an then dire
onvert the hy- drate/uid interfa
ial stinesses to surfa
e en- ergy values. Sin
e we are only able to do this rigorously for the pseudo-1D surfa
e geome- try at present. We 
an only give interfa
ial stinesses for the 2D simulations. Figure 1: Constru
tion (110)-fa
e - hydrate II The proto
ol is as follows: Geometri
tion We repli
ate and sta
k the unit 
ell to a large 
ube, then rotate the 
rystal, so that ea
h of the planes (100) (110) and (111) are su

es- sively brought into alignment with the xy-plane of the simulation box to make three separate starting 
ongurations. Figure 2: Constru
tion (111) - fa
e - hydrate II Clipping - 2D We 
rop the start-up 
on- gurations prepared above by planar 
uts to make an elongated re
tangular prism with square 
ross se
tion in the Z- dire
tion of the simulation box. The 
uts are not sharp, no mole
ules are split up. Clipping - 1D We 
rop the start-up 
ong- urations in a preferred dire
tion, making a strip of surfa
e Energy minimization A brief steepest de- s
ent minimization was used to 
t for edge-ee
ts Equilibration We equilibrate the systems using the Berendsen thermostat and barostat [25℄. The pressure s
aling is done with independent box s
aling in the 3 orthogonal 
oordinate dire
tions. Va
uum or uid addition - We expand the box in the z-dire
tion to 
reate roughly 1/3 of total volume to be lled with va
uum, or one of the pro
ess uids to be investigated. Produ
tion simulations The produ
tion runs are performed in the NVT-ensemble using the Noose-Hoover thermostat. [26℄ Analysis We t the interfa
ial density- prole of waters averaged over the whole traje
tory to the expe
ted Gaussian form in 
apillary wave-theory [27℄: ρ(z) = 1/2 × (ρ1 + ρ2)− 1/2×(ρ1−ρ2)×erf ( (z − h0)/ √ 2〈z2〉 ) We extra
t the thi
kness from the aver- age of our two equivalent interfa
es. The details of the general MD te
alities (treatment of 
ut-os, neighbourlists, ele
tro- stati
s and 
onstraints) are given in table 4 Parameters used in the energy minimiza- tion and equilibration are given in tables 2 and 3 respe
tively. Table 1: General MD parameters Neighboursear
h & PBC Algorithm Verlet list [28℄ Verlet list 
ut-o 0.9 nm Update freq. 5 steps PBC all dire
tions Ele
s & vdW Ele
s PME [29℄ PME interpol. order 5 FFT-grid spa
ing ∼ 0.1nm Ewald sum dir./re
ip. 10 −5 vdW. for
e Twin range 
ut-o neigbourlist-
uto 0.9 nm vdW 
ut-o upper 1.4 nm Bond 
onstraints Constraints All bond lengths Algorithm - Waters SETTLE [30℄ SETTLE parameters (TIP4P/i
e model) Algorithm - Others LINCS [31℄ LINCS order 4 LINCS Iter. - SIM. 1 LINCS Iter. - EM 6 ANALYSIS AND RESULTS The setups of the surfa
e u
tuation simula- tions are given in tables 5 and 6. Table 2: Energy minimization parameters Algorithm Steepest-des
ent Steps 300 Maximum step-size 0.1 nm Toleran
e 25 kJ (mol nm) −1 Table 3: Equilibration parameters Mole
ular Simulation Algorithm Leap-frog(Vel. Verlet) EQ steps 50000 Stepsize 2 fs Center of mass motion removal Type Trans'l momentum Frequen
y Every step Re
ording periods Coordinate 500 steps (1 ps) Velo
ity 5000 steps (10 ps) For
e 5000steps (10 ps) Energy 10 steps ( 0.02 ps) Thermostat Algorithm Berendsen Coupling time 0.1 ps Range All mole
ules Barostat Algorithm Berendsen Coupling time 0.5 ps Init. 
ompressibility 4×10−5 Bar−1 The u
tuations obtained by tting to den- sity proles are given in tables 7 and 8 and the proles for stru
ture II are graphed in gure 4 for the pseudo-1D geometry. The proles were obtained by dividing the box, normally of length ≈10nm, into 1000 bins to obtain a binwidth of 0.01 nm. We see that our present simulations are plagued by quite large un
ertainties with this methodology, showing the urgent need for a rigorous, more pre
ise way to dene the ex- tent and position of the interfa
e, or alterna- tively for studying the prole's development in time. We are still able to make some qual- itative inferen
es from the pseudo-1D simu- Table 4: Simulation parameters Mole
ular Simulation Algorithm Leap-frog (Vel. Verlet) EQ steps 5×106 Stepsize 2 fs Re
ording periods Coordinate 1000 steps (2 ps) Velo
ity 10000 steps (20 ps) For
e 105 steps (200 ps) Energy 500 steps (1 ps) Thermostat Algorithm Nose-Hoover [26℄ Coupling time 0.1 ps Range All mole
ules Table 5: Simulation matrixPseudo 1D geom., hydrate/va
uum Hyd. (Fa
e)[s℄ Box [nm℄ No. HI (001)[010℄ 2.88×7.27×10.8 19224 HI (110)[-110℄ 5.99×3.10×9.00 12233 HI (111)[11-2℄ 6.98×3.10×10.5 14116 HII (001)[100℄ 6.92×3.45×10.3 15168 HII (110)[001℄ 3.11×6.08×9.50 14751 HII (110)[-110℄ 6.97×3.15×10.0 18289 HII (111)[11-2℄ 3.22×6.05×12.1 14675 lations. Expanding the stinesses in 
al harmoni
s we estimate the surfa
e energies in the pseudo 1D geometry, shown along with the surfa
e stress in table 9. Having obtained the expansion parameters of the 
 expansion from the pseudo-1D t, we now utilize it to 
rudely es- timate the surfa
e energies of the stru
ture II hydrate (001)-fa
e with the uid phases from the stinesses obtained in the the 2D simula- tion results in table 10. DISCUSSION First we note that in all our simulations there is not an obvious relation between the sur- fa
e stresses, obtained from the uid-uid ap- proximation, and the surfa
e energies as in- ferred via surfa
e-stiness measurements us- Table 6: Simulation matrix2D geom, hy- drate/va
uum. System (Fa
e) Box [nm℄ No. HII/va
 (001) 3.40×3.40×10.0 10112 HII/va
 (110) 3.54×3.45×10.2 9783 HII/va
 (111) 3.56×3.49×10.5 10923 HII/wat (001) 3.42×3.45×6.34 9854 HII/oil (001) 3.37×3.37×8.22 10672 HI/va
 (001) 3.60×3.61×10.7 12096 HI/va
 (110) 3.68×3.57×10.9 12390 HI/va
 (111) 3.68×3.71×10.9 12824 a) b) 
) Figure 3: Hydrate II/va
uum strip 
ongura- tion - a) (001)[100℄, b) (110)[001℄, 
) (111)[- 110℄ ing 
apillary wave theory. By 
tion we do not see negative values for the sur- fa
e energy, as we do with stresses. It is seen that the values of surfa
e tension for stru
ture II hydrate 
onforms qualitatively to the ex- perimentally determined relation between the growth rate of fa
es in the stru
ture II 
rystal [2℄: (100)>(110)>(111), however our results are mostly for
ed by the t to the 
 har- moni
s and therefore not well established yet, sin
e the un
ertainties are substantial. Fur- thermore, only the relative values 
an be said to have importan
e sin
e we have employed an ad-ho
 assumption of minimum wavelength- Table 7: Surfa
e u
tuations from density prolespseudo-1D strip Interfa
e T p. Flu
t. κ [K℄ [Bar℄ [10−3nm2℄ [mN/m℄ Hydrate I / va
uum (001)[010℄ 254 10 7.6(1.8) 155(40) (110)[-110℄ 254 10 8(1.5) 140(30) (111)[11-2℄ 254 10 11(2) 103(17) Hydrate II / va
uum (100)[001℄ 254 3 4.9(5) 253(8) (110)[001℄ 254 3 8.2(3) 127(4) (110)[-110℄ 254 3 8.1(1) 138(2) (111)[11-2℄ 254 3 37(13) 28(10) 0 500 1000 1500 D en sit y (kg /m 3 ) Density profiles - hydrateII pseudo-1D strips (001)[100] 0 500 1000 1500 D en sit y (kg /m 3 ) (110)[001] 0 500 1000 1500 D en sit y (kg /m 3 ) (110)[-110] Normal Coord. - Z [nm]0 500 1000 1500 D en sit y (kg /m 3 ) Density profiles - hydrateII pseudo-1D strips (111)[-110] Figure 4: Density proles - Hydrate II strips/va
uum (Water framework only) 
uto being of the order of the largest-
age diameter in our hydrate stru
ture. (We have as yet not been able to estimate the bulk- 
orrelation length in hydrates from our simu- lations.) In the both 
ases the results are nu- meri
ally somewhat large, and for stru
ture I up to now all too un
ertain to make any infer- en
es of ordering. As seen from the above the results for stru
ture I do not at all 
onform to the results of Smelik et. al: [2℄ whi
on- rms our reservations as to the un
ertainties inherent in the present method of analysis. Although our method shows a greater anity for hydrate to the water phase than to the oil phase. The magnitude of the intera
tion with water is quite hard to as
ertain due to the smooth transition to the water phase seen in density proles. Espe
ially after long simula- tion times the hydrate water-stru
ture is 
on- Table 8: Surfa
e u
tuations from density proles2D geometry (Fa
e) T. p. Flu
t. κ [K℄ [Bar℄ [10−3nm2℄ [mN/m℄ Hydrate I / va
uum (001) 254 10 5(4) 160(150) (110) 254 10 11(4) 80(30) (111) 254 10 13(2) 61(9) Hydrate II / va
uum (001) 254 3 5.8(0.5) 124(12) (110) 254 3 8(1) 93(13) (111) 254 3 7.4(0.3) 103(4) Hydrate II (001)/oil 253 2 7.7(0.2) 92(18) (001)/wat 253 2 2.9(1.7) 24(12) Table 9: Surfa
e energieshydratespseudo 1D strips System Fa
e T γ Stress (Index) [K℄ [mN/m℄ [mN/m℄ HI/va
 (001) 254 147 56.3(4) HI/va
 (110) 254 148 -43.2(4) HI/va
 (111) 254 152 42.1(4) γavg 149 HII/va
 (001) 254 133 92(1) HII/va
 (110) 254 141 101.4(5) HII/va
 (111) 254 144 66(1) γavg 140(60) Table 10: Surfa
e energieshydrate/uids 2D slabs System T γ Stress [K℄ [mN/m℄ [mN/m℄ HII/va
uum 254 139(60) -443(2) HII/oil 254 65(13) -198(2) HII/water 254 17(8) -17(1) tinued in ordered water-
lusters, and a 
lear identi
ation of the prole thus be
omes di- 
ult. Surfa
e energy for the hydrate/water in- terfa
e in our simulation is therefore near neg- ligible, due to the smoothness of the density transition. Implementing an order parameter for solid phases of water, to more pre
isely determine the position and extent of the in- terfa
e, would be of great help in redu
ing the main sour
e of un
ertainty in the method. REFERENCES [1℄ J. J. Hoyt, M. Asta, and A. Karma. Method for 
omputing the anisotropy of the solid-liquid interfa
ial free energy. Physi
al Review Letters 2001;86(24):55305533. [2℄ E. A. Smelik and H. E. King. Crystal- growth studies of natural gas 
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