Pro eedings of the 6th International Conferen e on Gas Hydrates (ICGH 2008), Van ouver, British Columbia, CANADA, July 610, 2008 SURFACE-FLUCTUATIONS ON CLATHRATE HYDRATE STRUCTURE I AND II SLABS IN SELECTED ENVIRONMENTS Bjørn Steen Sæthre*, Alex C. Homann Dept. of Physi 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 hara 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 tive 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 i fa 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 harmoni exp. 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 orrelation oe.[-℄ sij (Surfa e) stress tensor [mN/m℄ dS Element of surfa e [nm2℄ T Temperature [K℄ V Volume [nm3℄ r = (x, y) surfa e oordinates 〈X〉 Average of X z(x, y) Model for real interfa e α Fourier expansion oe 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: bjorn.sathreift.uib.no 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 reasingly ostly problem. As oil and gas exploration is expanding into the ar - ti regions, 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 urren 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 e omponents, poten- tially a ting as anti-agglomerants, is ru 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 lassi 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 onta 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. al- 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 tional 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 surfa 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 anoni 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 e 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 onne 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 ir ular 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 rystallographi referen 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 ubi symmetry, a natural hoi e would be ubi harmoni s. Like David ha 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 oe ients 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 ha 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 dire tion-dependent fa tors of the ubi harmoni expansion for our parti ular geom- etry. Having obtained the ubi harmoni ex- pansion, we an then dire tly 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 onstru 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 orre 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 hni 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 trostati s & vdW Ele trostati 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 ubi 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 ubi -harmoni 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 onstru 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 ubi 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 h 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. 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International Conference on Gas Hydrates (ICGH) (6th : 2008)
SURFACE-FLUCTUATIONS ON CLATHRATE HYDRATE STRUCTURE I AND II SLABS IN SELECTED ENVIRONMENTS Saethre, Bjorn Steen; Hoffmann, Alex C. 2008-07
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Title | SURFACE-FLUCTUATIONS ON CLATHRATE HYDRATE STRUCTURE I AND II SLABS IN SELECTED ENVIRONMENTS |
Creator |
Saethre, Bjorn Steen Hoffmann, Alex C. |
Contributor |
University of British Columbia. Department of Chemical and Biological Engineering International Conference on Gas Hydrates (6th : 2008 : Vancouver, B.C.) |
Date Issued | 2008-07 |
Description | Hydrates in some crude oils have a smaller tendency to form plugs than in others, and lately this is becoming a focus of research. To study this and the action of hydrate antiagglomerants in general, hydrate surface properties must be known. To help in characterizing the surface properties by simulation, the capillary waves of clathrate hydrate surfaces in vacuum are examined in all unique crystal faces by Molecular Dynamics, and an attempt is made to estimate the surface energies in the respective crystal faces from the wave fluctuations [1]. We also attempt to estimate solid/liquid surface energies of hydrate/oil and hydrate/water for a specific face, for comparison. The forcefield OPLS_AA is used for the organic compounds, while TIP4P/ice is used for the water framework. The anisotropy of the surface energy is then estimated and the result compared to the initial growth rate of different crystal faces as found in experiment [2]. |
Extent | 520692 bytes |
Subject |
Gas hydrates Plug prevention Surface energy Molecular dynamics |
Genre |
Conference Paper |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-09-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0041123 |
URI | http://hdl.handle.net/2429/2326 |
Affiliation |
Non UBC |
Citation | Saethre, Bjorn Steen; Hoffmann, Alex C. 2008. SURFACE-FLUCTUATIONS ON CLATHRATE HYDRATE STRUCTURE I AND II SLABS IN SELECTED ENVIRONMENTS. Proceedings of the 6th International Conference on Gas Hydrates (ICGH 2008), Vancouver, British Columbia, CANADA, July 6-10, 2008. |
Peer Review Status | Unreviewed |
Copyright Holder | Saethre, Bjorn Steen |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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