11022
J. Phys. Chem. 1992, 96, 11022-11029
Intermolecular Potential Energy of Water Clathrates: The Inadequacy of the Nearest-Neighbor Approximation Kevin A. Sparks and Jefferson W.Tester* Chemical Engineering Department and Energy hboratory, Room E40-455, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307 (Received: June 12, 1992)
A rigorous treatment of multiple water shell and guest-guest potential energy effects in water clathrates is presented. A
modified lattice sum approach is used to characterize the quantitative extent of these effects on the configurational partition function and the three-dimensional Langmuir constant. Results are presented for both structure I and I1 gas hydrate systems for an extensive range of intermolecular potential energy distance dependence, from r-5 to r-30.
1. General Water Clathrate Properties and Structures The water clathrate structure is a polymeric threedimensional crystalline lattice connected by nearly tetrahedral hydrogen bonds. Although clathrate hydrates are known to form several different types of structures, including a recently reported hexagonal form by (Ripmeester et al.,' they generally crystallize in one of two cubic structures. The unit cell of a structure I water clathrate is cubic with space group Pm3n and a lattice constant of 12 A. For every 46 water molecules, there are 2 pentagonal dodecahedral cavities and 6 tetrakaidecahedral cavities. The unit cell of a structure I1 water clathrate is cubic with space group Fd3m and a lattice constant of 17 A. For every 136 water molecules, there are 16 pentagonal dodecahedral cavities and 8 hexakaidecahedral cavities. The polyhedra of these two distinct structures are shown in Figure 1. The unit cells for each of the structure types are shown in Figures 2 and 3. A detailed description of structural characteristics of the two water clathrate types is given elsewl~ere.~-~ Clathrate networks consisting of hydrogen-bonded host water molecules are in fact unstable by themselves unless a number of the voids or cavities are filled by guest molecules. It is the interaction of these enclathrated guest molecules with the host lattice that ensures the stabilization of the host lattice structure. The diameters of the voids formed by the lattice are such that the attractive intermolecular forces between the host water molecules are strong enough to collapse the hydrogen-bonded host structure. It is the relatively weak van der Waals interactions between the host water molecules and the entrapped guest molecules that ultimately stabilizes the compound. Several of the larger hydrate forming compounds, although capable of stabilizing the larger cavities within the overall clathrate structure, require the presence of a second hydrate forming component, often regarded as a hilfgars (help gas), to complete the stabilization of the structure. Water clathrates are generally regarded as nonstoichiometric compounds since all available cages within the lattice structure need not be occupied to ensure stability. A pure gas water clathrate can be treated thermodynamically as a two-component system consisting of water and a particular guest component. Multicomponent gas hydrates can be treated in a similar fashion if the composition of the gas phase is fixed. When three equilibrium phases are present, the system will be monovariant, and fixing the temperature should specify the pressure. These equilibrium vapor pressures are commonly measured as a function of temperature for various three-phase, monovariant systems. For example, when either ice or liquid water, solid gas hydrate, and vapor are present in equilibrium, the measured pressure is referred to as the dissociation pressure. A phase diagram for water and various natural gas guest components is shown in Figure 4. Strictly speaking, the dashed vertical line representing the ice-liquid boundary should curve *Address correspondenceto this author. Telephone: (617) 253-3401. Fax: (617) 253-8013.
0022-365419212096-11022$03.00/0
to the left at the higher pressures shown.
2. Previous Theoretical Work In 1959, van der Waals and Platteeuw6 proposed that the thermodynamic properties of clathrates could be derived from a simple model cOrrespOnding to the threedimensional generalization of ideal localized adsorption. The model assumes the empty host lattice to be thermodynamically unstable. The difference between p t , the chemical potential of H20in the unstable empty lattice, and p!, the chemical potential of H20in the occupied lattice, is given by
where k is Boltzmann's constant, T i s the absolute temperature, and vi is defined as $e number of type i cavities per water molecule in the host lattice, f J is the fugacity of guest component J, and CJiis the Langmuir constant for a type J guest component encaged within a type i cavity and is defined by Cji
Zji/kT
where the "free volume" or configurational integral, ZJi,is given by ZJi =
8r2
S e - u ( r , R , ~ , a , ~ , sin ~ ) l8k T dB$d$ dr d a sin j3 dj3 dy (3)
where U is the total interaction potential between the guest molecule and all host molecules defined in spherical coordinates r, 8, and 4 and Euler orientation angles a, 8, and y for the guest molecule. Unfortunately, the asymmetries of the host lattice cavities and of the guest molecule itself makes analytical integration intractable. Generally, a Lennard-Jones and Devonshire liquid cell theory approach6 has been adopted for the quantitative evaluation of the configurational partition function of the guest %elute" molecule within the host lattice cavity. It is generally assumed that the host water molecules are uniformly distributed on a spherical surface corresponding to an average cavity radius. This spherical cell model simplifies the integration of eq 3 considerably. (4)
Van der Waals and Platteeuw used a Lennard-Jones (6-12) potential in the development of the spherically symmetric cell potential model
where r is the usual distance between molecular centers, u is the collision diameter, and c is the characteristic energy. The actual Lennard-Jones parameters for the guest-host interactions were 0 1992 American Chemical Society
Potential Energy of Water Clathrates
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 11023
Pentagonal Dodecahedron (Structure I)
Tetrakaidecahedron (Structure I) Figure 2. Structure I unit cell.
Pentagonal Dodecahedron (Structure 11)
Hexakaidecahedron (Structure 11) Figure 1. Structure I and I1 polyhedra.
determined using the Berthelot geometric mean approximation for e, and the hard-sphere approximation for 6. e
= (eguestChost)1’2
(6)
= (‘guest + bhost)/2 (7) The discrepancy between theory and experiment later directed McKoy and Sinanoglu7to study the Lennard-Jones (6-12), (7-28), and Kihara potentials in the spherical cell model. The Kihara potential is represented by U(r) = rI2a
where 2a is the molecular hard core diameter, 6 is the collision diameter, and c is the characteristic energy. They also attempted to account for the general shape of the guest molecule by considering two cases, specifically a molecule with a thin rod core such as N2 or C2H6,and a molecule with a spherical core such as CH4 or Ar. The host molecules were modeled as point molecules having no hard core diameter.
Figure 3. Structure I1 unit cell.
-
Nagata and Kobayashi* extended the method to the prediction of dissociation pressures of mixed gas hydrates from data for hydrates of pure gases with water. They used the Kihara potential for spherical and rodlike molecules to describe the interaction between the encaged guest and the host lattice. Parrish and Prausnitz9 later extended the use of the van der Waals and Platteeuw hydrate model to the prediction of the dissociation pressures of gas hydrates formed by gas mixtures both above and below the ice point. They also chose to use the Kihara potential with a spherical core to model gas-water interactions in the clathrate cavity. Recently, John and (1985) examined the validity of the spherical cell approximation. Using the Kihara potential in all of their calculations, they proposed several modifications to original van der Waals and Platteeuw6 treatment: (1) the choice of cell size used in the model;” (2) the addition of terms to account
11024 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
Sparks and Tester TABLE I: Structure I and Structure 11 Hvdrate Lattice Pro~erties structure I structure I1 water molecule wr unit cell 46 136 cavities per uni;cell small 2 16 large 6 8 average cavity radius, A small 3.905 3.902 large 4.326 4.682 space group Pm3n Fd3m lattice constant, A 12.03 0.01 17.31 i 0.01 typical guest compounds methane argon ethane krypton ethylene nitrogen oxygen co2 xenon propane ~~
*
j
cyclopropaned
G€&0
ideal composition
cyclopropane"
isobutane H2S MI-3M2*23H,0b 2Ml*My17H,0b
M " Forms both types. b M I= molecules occupying small cavities; M2
0.1 200.00
225.00
250.00
275.00
30(
0
Temperature (K) Figure 4. Phase diagram for water and various natural gas guest components. for the contribution of second and subsequent water shells to the potential energy of the guest-host interactions;I2 and (3) the addition of an empirical corresponding states correlation to correct the results of the smoothed Lennard-Jones Devonshire model.'O9I3 These modifications attempted to remove the inadequacies of the spherical cell approximation but unfortunately to some extent tend to cloud the significance of the van der Waals and Platteeuw physical model. Although John and Holderlomaintain that their potential parameters are consistent with thwe observed for virial coefficient data, they have effectively introduced new empirically fitted parameters such as the cell radius into the model. Almost without exception, the interaction potential parameters used in these lattice models are determined ad hoc by fitting experimental phase equilibrium data such as along various univariant, three-phase dissociation pressure curves (treatments by Parrish and Prausnitz9 and Nagata and Kobayashi).* The parameters obtained in this manner are not uniquely defined. Often, as described by Tse and Davidson,I4 agreement between intermolecular parameters obtained from fitting hydrate dissociation pressure data and to those obtained from gas-phase second virial coefficient or viscosity measurements is poor. Since the macroscopic properties of water clathrates are determined to a large degree by the molecular structure of the host lattice and the nature of the interaction between the host and guest molecules, the complete characterization of these intramolecular and intermolecular interactions is essential if we are to accurately predict the thermodynamic properties of clathrate compounds. To date, however, the models used to evaluate the configurational properties of these gas hydrates have for the most part utilized the spherically-symmetric Lennard-Jones Devonshire (LJD) cell theory approach and have therefore neglected the asymmetries within the clathrate structure. These asymmetries arise from the structure of the guest molecules as well as from the geometry of the host lattice cages that contain the guest molecules. For example, a linear guest such as CO, would be expected to behave differently from that of spherically symmetric guests such as Ar or CH,. Large discrepancies could result if branched guests such as i-C4HI0or cyclopropane were treated as being spherically symmetric. In fact, Anderson and PrausnitzI5 recently claimed that most of the disagreement between experiment and theory is inherently due to symmetry assumption of the van der Waals and Platteeuw clathrate model. The inadequacies of the spherical cell model have been under scrutiny for some time, yet it is still the theory of choice for many investigators. In addition, these early models only treat nearest-neighbor guest-host interactions exclusively, with the first shell host water molecules interacting with a simple entrapped guest molecule. The
= molecules occupying large cavities.
only notable exception to this approximation was presented by John and Holder1, as mentioned earlier. Their approach was to decompose the total guest-host potential U(r,...) given in cq 4 into separate contributions for the first, second, and third water shells. They expressed this decomposition in terms of the configurational integral as
(9) where U(1)= f i r ) and U(2)and U(3)are assumed to be independent of r and therefore can be removed from the integration in the spirit of a "mean field approximation". John and Holder', selected a LJD approach to estimate the U(2)and U(3) contributions. Although this represents a pioneering effort, their choice of water shell structural characteristics is somewhat ill-defined, therefore motivating us to examine the effects of subsequent water shells in a much more complete fashion. The work presented here therefore represents an extensive evaluation of the van der Waals and Platteeuw theory and its underlying assumptions. The property parameters used in this modeling effort are summarized in Table I. Given the crystallographic data of the two water clathrate structures we were able to accurately account for the asymmetries which arise from the guest-host interactions while maintaining the physical significance of the potential parameters that are used to characterize the intermolecular forces between guest and host molecules. This we considered an important requirement, especially since the spherical cell model uses nonunique potential parameters regressed from experimental dissociation pressure data. Molecular dynamics simulations also were used to study the motion of guests within the host lattice cavities. Additionally, this enabled us to quantitatively estimate the lattice distortions associated with larger more asymmetric guest molecules. 3. Objective and Lattice Sum Approach Our main objective in this study was to quantitatively characterize the guest-host potential interaction energy in the framework of an infinite clathrate lattice for both structure I and I1 gas hydrates. With this approach, we could rigorously calculate the contributions that multiple water shells have on the interaction potential. In addition, we could also calculate the contribution of guest-guest interactions. Basically, our approach followed the lattice summation method as described by Hirshfelder, Curtiss, and Bird.I6 When the potential energy V(r,) between two atoms can be expressed as an inverse power series in separation, r,/
where the A i s are potential energy constants which can represent
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 11025
Potential Energy of Water Clathrates
1
0.1
2,
0
4 0.001
Unit Cell
L.C.- Lattice Constant Figure 5. Two-dimensional lattice summation diagram.
0.0001
both attractive (-) and repulsive (+) contributions to U(r& In practice, the Xis are set by the choice of the potential energy function or by theoretical considerations. In general, the total potential energy of an atom within an infinite cubic crystal is given by
5 5 5
it-
jt-m
k=-m
"-1 ~
+ k)2]-42 + ... (11)
where the subscript c denotes the atom of interest and the subscript n denotes the remaining interacting atoms within the unit cell of the cubic crystal. The number of atoms within the unit cell is given by p. Recasting this expression into a more compact form yields
where the potential energy constant, A,,, is defined by
C C C
i=-o j=-m
0.0000 1 1
R
/
10
Lattice Constant
100
Figure 6. Lattice sum convergence-structure I pentagonal dudecahedron.
~~[(x~-x,+i)~+~y,-y,+~)~+(z,2,
A,,=
0.01
Cdos,[(xc-x,+i)2+Cyc-~n+j)2+
&=-m
(2, - Z,
+ k)2]-S,/2 (13)
and do is a normalizing distance, usually defined as the minimum nearest-neighbor distance. The errors associated with the finite limits imposed on the direct summation of the potential energy series can be estimated by the conversion of the residual discrete summations to continuous integrations beyond a certain radius R. With this approach, the error can be estimated by 4rp9
dr = dOsLm4rpr2-, dr (14)
where A,- is the potential energy constant associated with a truly infinite lattice sum (eq 13) and A, is the potential energy constant associated with the truncation of the series at a radial value of R, depicted in the tw+dimensional analog of the lattice summation diagram illustrated in Figure 5. The integration of eq 14 yields R3-S AA, A,_ - A,, = do'hps>3 (15) s-3 or equivalently using base 10 logarithms
+ log [R3-']
(16)
+ (3 - S ) lOgR
(17)
log AA, = log [doS4rp/(s - 3)] which further simplifies to log AA,
log [do'&p/(~ - 3)]
This simple linear expression relating the logarithm of the potential energy constant error (AA,) is shown graphically in Figure 6. In this case, the AA, given is associated with the truncation of the series at a finite limit for a spherical guest molecule located at the center of a pentagonal dodecahedron in a structure I water clathrate. It is obvious that for integral values of s > 6 the
convergence of the series is rapid and the direct summation of the series is easily done. However, for a value of s = 4, the direct summation is completely intractable; in fact, using the Cray 2 supercomputer, we estimated that it would require more than lo6 years of CPU time to converge the potential energy constant to five decimal precision. Hirschfelder, Curtiss, and Bird16tabulated the potential energy constants for several simple crystalline structures, specifically, the face-centered cubic, body-centered cubic, and simple cubic structures. These potential energy constants were first calculated in 1924 and 1925 by Lennard-J~nesl~ and Lennard-Jones and Inghaml* using elaborate transformations involving the EulerMaclauren sum formula with the Riemann zeta function. Quite remarkably, without the aid of an electronic computer, they were able to calculate the potential energy constants to five decimal precision for the three cubic structures for values of s ranging from 4 to 30. Unfortunately, the structure I.and I1 water clathrate structures are considerably more complex and thus do not easily lend themselves to a similar analytical treatment. These points are discussed in more detail in a forthcoming paper.19
4. Clathrate Lattice Sum Results The potential energy constants, A,, for the water clathrate guest-host interactions are given in Table 11. These constants were calculated from the direct summation of the series using double precision arithmetic. For these calculations, the guest molecules were always located at the center of their respective cavities and were assumed to be spherical. The host water molecules were positioned at the crystallographiclocations of the oxygen atom^.^.^ The hydrogen atoms were not considered. Potential Energy Constants/Guest-Guest Interactions. As mentioned earlier, we also calculated guest-guest potential energy interactions. The resulting A, constants are given in Tables I11 and IV. Again, the guest molecules were always located at the center of their respective cavities, which were assumed to be at full occupancy. Only single-site spherical guests were considered. The guest-guest interactions were subdivided into four types per clathrate structure, specifically the following. Structure I: 1. The interaction of a guest within a pentagonal dodecahedron with guests within other pentagonal dodecahedrons. 2. The interaction of a guest within a pentagonal dodecahedron with guests within tetrakaidecahedrons. 3. The interaction of a guest within a tetrakaidecahedron with guests within pentagonal dodecahedrons. 4. The interaction of a guest within a tetrakaidecahedron with guests within other tetrakaidecahedrons. Structure II:
11026 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 TABLE II: Clathrate Potential Energy Constants (A,)/Cuest-Host Interactions
Sparks and Tester TABLE III: Clathrate Potential Energy Constants (A,)/Structure I CuestGuest Interactions Structure I: Water Clathrate Guest-Guest Interactions
Water Clathrate Guest-Host Interactions A.
structure I pentagonal tetrakaidodecahedron decahedron 22.725 71 21.937 21 19.199 66 19.385 96 18.205 75 17.387 25 16.218 92 17.508 93 17.01950 15.355 67 16.630 77 14.662 20 16.29618 14.076 25 15.99366 13.565 98 13.11345 15.712 20 12.707 66 15.446 0 1 12.341 25 15.191 91 12.00888 14.94808 14.71339 1 1.706 45 14.487 12 1 1.430 64 11.178 66 14.268 75 14.057 89 10.948 13 10.73694 13.854 20 10.543 25 13.657 42 13.467 28 10.365 42 13.28354 10.201 98 10.051 61 13.105 98 9.913 14 12.93441 9.785 49 12.768 60 9.667 71 12.608 36 9.558 92 12.45350 9.458 36 12.30385 0.337 17 0.31800
s
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 do'
structure I1 pentagonal hexakaidodecahedron decahedron 19.872 55 34.693 43 17.243 31 30.064 53 15.892 27 27.99541 14.997 68 26.871 50 14.305 43 26.17033 13.717 73 25.679 64 13.191 91 25.301 35 12.707 60 24.986 35 12.254 39 24.708 74 24.454 31 11.826 53 24.215 04 1 1.420 63 23.986 34 1 1.034 47 23.765 46 10.666 49 23.55077 10.315 49 23.341 31 9.980 48 23.136 43 9.660 58 22.935 73 9.355 01 22.738 95 9.063 07 22.545 91 8.78407 22.35645 8.51739 22.17048 8.262 43 21.98790 8.01865 21.808 63 7.785 50 21.63261 7.56249 21.45977 7.349 13 21.29005 7.144 97 0.267 70 0.21651
s
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 do"
do E fractional distance between nearest neighboring cavities.
do E fractional distance between nearest neighboring host water molecules. 1. The interaction of a guest within a pentagonal dodecahedron
with guests within other pentagonal dodecahedrons. 2. The interaction of a guest within a pentagonal dodecahedron with guests within hexakaidecahedrons. 3. The interaction of a guest within a hexakaidecahedron with guests within pentagonal dodecahedrons. 4. The interaction of a guest within a hexakaidecahedron with guests within other hexakaidecahedrons. The first interaction type given under the structure I category in Table 111 for pentagonal dodecahedrons corresponds to a body-centered cubic type structure. For this situation, the potential energy constants tabulated by Hirschfelder, Curtis, and BirdI6 in Table 13.9-2 on p 1040 can be compared with those calculated here by direct summation. Discrepancies exist only in the last one or two decimal values for some of the constants particularly at values of s less than 10. Based on this comparison and upon examination of the convergence properties of our numerically evaluated lattice sum series, eq 13 verifies that the values cited in Tables 11-IV are.correct to the number of significant figures shown. 5. Discussion
If the Lennard-Jones (6-12) intermolecular potential is used to model the various guest-type interactions within the water clathrate system, then the total potential energy of a guest molecule located at the center of a given cavity within a rigid host lattice is given by =
(
4ew-gatf_,A12 -
do1,
(
4aw-ga6,-pA6
do6 4eg-$,,*
I +) guest-host
- 4cg-ga36 do6
where XI, = 4e8-ga~!gand X6 = 4e,,~;-~
(18)
guest-guest
for guest-guest interac-
A, pentagonal dodecahedron tetrakaidecahedron pentagonal tetrakaipentagonal tetrakaidodecahedron decahedron dodecahedron decahedron 14.75840 15.116 60 5.038 86 6.777 33 12.25367 13.415 25 4.47 1 75 5.14048 12.720 28 1 1.054 24 4.24009 4.286 68 12.389 94 10.355 20 4.12998 3.749 16 12.219 37 9.894 59 4.073 12 3.37400 12.126 65 9.56440 4.042 22 3.095 8 1 12.07446 9.3 13 26 4.024 82 2.881 83 12.04436 9.114 18 4.01479 2.713 51 12.026 68 8.951 81 4.008 89 2.579 26 12.016 17 8.81677 4.005 39 2.471 25 12.009 85 8.702 98 4.003 28 2.383 89 12.006 02 8.606 25 4.002 01 2.31298 12.003 70 8.523 53 4.001 23 2.255 31 12.002 27 8.452 50 4.000 76 2.208 33 12.001 40 8.391 35 4.000 47 2.170 03 12.000 86 8.33860 4.000 29 2.13880 12.00053 8.29305 4.000 18 2.11331 12.00033 8.235 68 4.000 11 2.092 51 12.000 20 8.21962 2.075 53 4.00007 12.000 13 8.190 16 2.061 66 4.000 04 12.000 08 8.164 65 2.050 35 4.000 03 12.00005 8.14258 2.041 11 4.00002 12.00003 8.12347 4.00001 2.033 56 12.000 02 8.10692 4.000 01 2.027 40 12.00001 8.092 59 4.000 00 2.022 38 12.00001 8.080 19 2.018 27 4.000 00 0.55901 0.866 02 0.500 00 0.55901
TABLE IV Clathrate Potential Energy Constants (A,)/Structure I1 GuestGuest Interactions Structure 11: Water Clathrate Guest-Guest Interactions A.
pentagonal dodecahedron pentagonal hexakais dodecahedron decahedron 5 8.096 72 7.889 52 6.927 68 6.91 143 6 6.454 06 6.492 68 7 6.233 92 6.28271 8 6.16805 9 6.124 19 6.102 14 6.067 24 10 6.062 99 11 6.036 90 6.039 24 12 6.02045 6.024 60 13 6.01 142 6.015 50 14 6.00641 15 6.009 80 6.003 62 16 6.006 21 6.002 05 6.003 94 6.001 16 17 6.002 5 1 18 6.000 66 6.001 60 6.000 38 19 20 6.001 02 6.000 22 21 6.000 65 6.000 12 22 6.00041 6.000 07 23 6.000 26 6.000 04 24 6.000 17 6.000 02 25 6.000 11 6.00001 6.000 07 26 6.00001 21 6.000 04 6.000 00 28 6.000 03 6.000 00 6.000 02 6.000 00 29 30 6.00000 6.00001 0.414 57 0.353 55 doa a
do
hexakaidecahedron pentagonal hexakaidodecahedron decahedron 15.77903 6.31274 13.82287 5.1 16 77 12.985 37 4.59448 12.565 42 4.331 91 12.336 10 4.19037 12.204 28 4.11102 12.125 99 4.065 47 12.078 47 4.038 90 12.04921 4.023 25 12.031 01 4.01396 12.01960 4.008 41 12.012 42 4.005 08 12.007 89 4.003 07 4.001 86 12.00501 12.003 19 4.001 13 12.002 03 4.000 69 12.001 29 4.000 42 12.00083 4.000 26 12.00053 4.000 16 12.000 34 4.000 09 12.000 2 1 4.000 06 12.000 14 4.000 04 12.000 09 4.000 02 12.00006 4.00001 12.00004 4.000 01 12.000 02 4.000 00 0.41457 0.43301
fractional distance between nearest neighboring cavities.
tions and XI, = 4ew-ga$a and
= 4tw-ga~6w-g for guest-host interactions. Given the inherent differences in size and interaction
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 11027
Potential Energy of Water Clathrates energies between polar water molecules and typical apolar guest molecules, such as methane, one would not normally expect the potential parameters to be the same for guest-host and guestmest interactions. Of course, for the unlike guest-host molecular interactions, tW and uq need to be estimated from either the fitting of ab initio quantum mechanical calculations or from some suitable mixing rule. The first term in eq 18 represents the usual guest-host interactions, with the exception that all of the subsequent water shell contributions are included. The second term has been added to represent the usually ignored guest-guest interactions. Since there are two types of cavities in the different water clathrate structures, the guest-guest interaction term actually consists of two separate terms, specifically one to model the interactions of guest molecules within like cavities and one to model the interactions with guest molecules in unlike cavities.
In the case of a structure I water clathrate, the first term could represent either the interactions of a guest within a pentagonal dodecahedron with all the neighboring guests also located in pentagonal dodecahedrons or similarly a guest within a tetrakaidecahedron with all the neighboring guests also located in tetrakaidecahedrons. The second term would represent the remaining unlike interactions. For a guest molecule that only fills the large cavities within a water clathrate structure, the second term involving these unlike interactions would disappear leaving only those with like interactions. The guest-host interactions can also be similarly modeled by the sum of two terms Ugurst-host
- (4t.-2?'12 -
- 4tw-g&-&k do6
I
+
(20) where the first-shell potential energy constants, A\, are given in Table V. In this case, the first term involves the interactions of a guest molecule situated at the center of a given cavity with the first shell of neighboring water molecules, while the second term captures the residual contributions to the total potential energy of the subsequent water shells which have been neglected in most earlier treatments. These results are not restricted just to Lennard-Jones systems. Other intermolecular potential functions such as the Kihara potential or those more appropriately derived from ab initio quantum mechanical calculations could also have been used to illustrate these same effects. Using the results presented earlier for the various guest-host lattice summation configurations, the contributions to the total potential energy by the inclusion of the subsequent water shell interactions and the various guest-guest interactions were examined for a number of systems. As a means of estimating the effect these potential energy changes had on the guest configurational integral, or equivalently the Langmuir constant, the mean value theorem was used (see eqs 2 and 3). Specifically, a general Langmuir constant, C, was expressed as C = (kT)-'(e-U/kT)Vf = (k7')-1e(-U/kT)Vf (21) where Vf is the free volume which enables us to write the ratio of two Langmuir constants as, for example c,,,,, e(-Umtal/kT) Vf,total -iT (22) Cshcll 1 "kT)Vf,shc]] I Assuming the free volumes associated with the different potential
TABLE V: First Shell Potential Energy Const.pts (A:)/Cuest-Host Intenctione Water Clathrate Guest-Host Interactions A! ~~
structure I pentagonal tetrakai-
s
5 6 '1 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
do"
structure I1
pentagonal hexakaidodecahedron decahedron dodecahedron decahedron 18.11488 17.77499 17.44653 17.129 11 16.82235 16.52589 16.23940 15.96254 15.69498 15.43641 15.18653 14.94505 14.71168 14.48615 14.26820 14.057 57 13.85402 13.657 32 13.467 22 13.283 51 13.10597 12.93439 12.76859 12.60835 12.45350 12.30385 0.31800
18.09507 17.23449 16.45563 15.75069 15.111 57 14.531 77 14.00533 13.52689 13.091 69 12.69545 12.33435 12.00496 11.70421 11.42936 11.17793 10.947 70 10.73670 10.543 11 10.365 34 10.201 93 10.051 59 9.913 12 9.785 48 9.667 70 9.558 92 9.458 36 0.337 17
16.43662 15.821 05 15.234 17 14.67457 14.14091 13.631 91 13.14637 12.683 13 12.241 12 11.81928 11.41664 11.03227 10.66527 10.31482 9.980 11 9.66037 9.354 90 9.063 00 8.78403 8.51737 8.26242 8.018 65 7.785 50 7.56249 7.349 13 7.14497 0.216 51
26.61284 26.35025 26.09239 25.839 18 25.59054 25.34638 25.10663 24.871 20 24.64001 24.41299 24.19007 23.971 17 23.75621 23.545 12 23.337 84 23.13430 22.934 42 22.738 15 22.54541 22.356 14 22.17029 21.987 78 21.808 56 21.63257 21.45974 21.29003 0.267 70
a do 3 fractional distance between nearest neighboring host water molecules.
energies to be approximatelyequal, the Langmuir constant ratio was approximated as
where the average potential energies, ( U / k T ) ,are for the sake of comparison assumed to be equal to the value of the potential energy at the center of the assorted cavities. To further enhance the presentation of the various effects subsequent water shells and the inclusion of guest-guest interactions have on the value of the configurational integral, or equivalently, the Langmuir constant, the total guest-host interaction potentials at the center of each of the various cavities have been tabulated for typical structure I and I1 hydrate systems. For illustrative purposes the Lennard-Jones 6-12 potential function was selected to model guest-guest and guest-host interactions. Potential parameters for guest-host interactions were derived using standard mixing rules, namely the hard-sphere approximation for 0 -, and the Berthelot geometric mean approximation for c, , wit1 pure component parameters taken from viscosity data fits.-80 The pure component parameters for water were taken from the simple point charge (SPC) model.21Specifically, the results for the structure I pentagonal dodecahedron are given in Table VI while the results for the structure I tetrakaidecahedron are given in Table VII. Similarly, Tables VI11 and IX present results for the appropriate structure I1 cavities. The results for the largersized hydrate formers, namely those that can only occupy the larger structure I and structure I1 cavity types, are somewhat misleading due to the increase in the range of the interaction potential imposed by the use of a simple singkite Lennard-Jones 6-12 potential function. The zero entries in Tables VI1 and I X under guest-guest potential energies correspond to those guests that are too large to occupy the smaller pentagonal dodecahedral cavities. 'The format of Tables VI. VII. VIII. and IX is self-evident. One notes that the contribution of the subsequent water shells to the
11028 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
Sparks and Tester
TABLE VI: Lattice Su"ation R e d & Structure I Pentagonal Dodecahedron (T= 273.15 K)
water clathrate guest component species uI (A) t,/k (K) Ar 3.542 93.3 Kr 3.655 178.9 3.798 71.4 NZ 3.467 106.7 0 2 148.6 CH4 3.758 Xe 4.047 231.0 3.623 301.1 HIS 3.941 195.2 CO2 "u,
= 3.120 A, e,/&
Lennard-Jones (6-12) Potential potential energy, cell center guest-hat guest-guest shell 1 shell >1 small cavity large cavity -5.969 -0.902 -0,026 -0.384 -8.480 -1.380 -0.060 -0.885 -5.413 -0,986 -0.030 -0.442 -6.233 -0.902 -0.026 -0.387 -7.809 -1.375 -0.059 -0.865 -9.108 -2.185 -0.142 -2.063 -10.937 -1.741 -0.095 -1.414 -8.757 -1.840 -0.102 -1.497
~~
total WILT -7.281 -10.804 -6.871 -7.548 -10.107 -13.499 -14.1 88 -1 2.196
Langmuir constant ratio total/shell 1 3.71 10.22 4.30
total WILT -6.280 -9.648 -6.413 -6.396 -9.320 -1 4.140 -12.596 -12.113 -14.928 -16.376
Langmuir constant ratio total/shell 1 3.1 1 7.77 3.49 3.14
total UIkT -7.249 -10.844 -6.899 -7.501 -10.176 -13.877 -14.278 -12.427
Langmuir constant ratio total/shell 1 3.85 11.39 4.48 3.87 11.04 105.53 3 1.23 37.72
total U/kT -4.940 -7.740 -5.273 -4.978 -7.612 -12.288 -10.065 -10.271 -13.418 -17.450 -1 9.104 -23.867
Langmuir constant ratio total/shell 1 2.68 5.92 3.00 2.68 5.80 3 1.04 12.33 14.41 15.55 64.16 191.26 1771.75
3.73 9.96 80.67 25.81 31.15
= 78.2 K.
TABLE VII: Lattice Summation Results: Structure I Tetnkaidecahedron (T = 273.15 K)
water clathrate guest component species ug (A) talk (K) Ar 3.542 93.3 3.655 178.9 71.4 3.798 3.467 106.7 148.6 3.758 4.047 23 1 .O 3.623 301.1 3.941 195.2 4.443 215.7 4.807 248.9
Lennard-Jones (6-12) Potential potential energy, cell center guest-h-t guest-guest small cavity large cavity shell 1 shell >1 -5.144 -0.722 -0.128 -0.286 -7.598 -1.095 -0,295 -0.660 -0.774 -5.162 -0.147 -0.329 -5.253 -0.725 -0.129 -0.289 -7.305 -1.082 -0,288 -0.644 -1.676 -0,688 -1.534 -10.242 -1.385 -0.471 -9.685 -1.055 -1.426 -0.499 -1.114 -9.075 -2.119 -2.41 1 -10.398 0.000 -2.822 -9.344 O.Oo0 -4.210
7.50 49.30 18.38 20.88 92.75 1132.51
= 3.120 A, r,/k = 78.2 K.
Ou,
TABLE VIII: Lattice Summation Results: Structure I1 Pentagonu1 dodecahedron ( T = 273.15 K)
water clathrate guest component us (A) talk (K) species Ar 3.542 93.3 Kr 3.655 178.9 3.798 71.4 N2 106.7 02 3.467 148.6 CH4 3.758 231.0 Xe 4.047 301.1 H2S 3.623 195.2 COZ 3.941 "uW=
3.120 A, t,lk
Lennard-Jones (6-12)Potential potential energy. cell center guest-host guest-guest shell 1 shell >1 small cavity large cavity -0.344 -0.135 -5.900 -0.869 -8.411 -1.331 -0.791 -0.311 -5.398 -0.952 -0.393 -0,156 -6.149 -0.869 -0.348 -0.136 -7.774 -1.326 -0,771 -0.305 -9.218 -2.111 -1.817 -0.731 -10.837 -1.678 -1.266 -0.497 -8.797 -1.777 -1.325 -0,529
78.2 K.
TABLE I X Lattice Summation Results: Structure I1 Hexakaidecabedron ( T = 273.15 K)
water clathrate guest componentspecies u8 (A) t,/k Ar 3.542 93.3 3.655 178.9 Kr 71.4 N2 3.198 3.467 106.7 0 2 148.6 CH4 3.758 Xe 4.047 231.0 301.1 H2S 3.623 3.941 195.2 CO2 4.443 215.7 C2H6 C3H6 4.807 248.9 C3H8 5.118 237.1 i-C4Hlo 5.278 330.1 "u,
Lennard-Jones (6-12)Potential potential energy, cell center guest-host guest-guest shell 1 shell >1 small cavity large cavity -3.955 -0.638 -0.270 -0.077 -0.622 -0.178 -5.962 -0.971 -0.089 -0.31 1 -4.173 -0,699 -0.272 -0.078 -3.990 -0.638 -0.609 -0.175 -5.853 -0.974 -1.462 -8.852 -0.420 -1.553 -1.232 -0.995 -0.285 -7.553 -1.306 -1.058 -0.304 -7.603 -10.674 0.000 -2.067 -0.677 -13.288 -2.935 -1.226 0.000 0.000 -13.851 -3.597 -1.657 -4.754 -2.725 -16.388 0.000
= 3.120 A, e,lk = 78.2 K.
total potential energy is quite significant in that it ranges from 10 to 15% of the total potential energy for a number of different
hydrate systems. The contributions of the various guest-guest interactions is again quite significant in that it ranges from 0 to
J. Phys. Chem. 1992, 96, 11029-11038 10% of the total potential energy. The effect these additional contributions have on the actual Langmuir constant can be very large, ranging from 1 to 2 orders of magnitude or more. Thus the inclusion of these "additional" interactions is essential to obtain a rigorous, quantitative characterization of the guest-host configurational partition function. The lattice sum approach utilized in this paper serves to illustrate the importance of the effect of multiple shells and guest-guest interactions which are so frequently neglected in estimating values for the configurational integral or Langmuir constant following the van der Waals-Platteeuw modeling approach. In fact, with the exception of the treatment of John and Holder,I2 these effects have been essentially ignored. In actual practice, of course, Monte Carlo integration or numerical quadrature methods are used to evaluate the configurationalpartition function (eq 3). Although the lattice positions of the host water molecules would be fixed, all possible guest positions and guest-host interaction distances need to be examined within the integration. The implications of these efforts are discussed in the context of the van der WaalsPlatteeuw statistical mechanical model in a separate paper.19 6. Conclusions A rigorous, numerical method of calculating guest-host and guest-guest intermolecular potential energy contributions for an infinite water clathrate lattice has been presented. Results are given for both structure I and I1 gas hydrate systems for a complete set of potential energy constants with distance dependence ranging from r+ to r-30. As originally described by John and Holder,12subsequent water shell interactions do indeed have a significant effect on the total guest-host interaction potential and thus on the value of the calculated Langmuir constant (CJt).Guest-guest interactions can be as important, as well. Thus, in performing rigorous partition function calculations using a molecular model of the clathrate structure, the inclusion of multiple shell and guest-guest inter-
11029
actions is necessary to obtain realistic potential parameters from phase equilibrium data. Acknowledgment. The authors would like to thank Professors T. A. Hatton, D. Blankschtein, U. Suter, I. Oppenheim, and S. Yip for their helpful input. Partial support for this research was provided by Norsk Hydro. The authors would also like to acknowledge the generous grant provided by the Massachusetts Institute of Technology Supercomputer facility. Thanks are also due to A. Carbone for her assistance with the manuscript prep aration. References and Notes (1) Ripmeester, J. A.; Tse, J. S.;Ratcliffe, C. I.; Powell, B. M. Nature 1987, 325, 135. (2) Sloan, E. D. Clathrate Hydrates of Natural Gases; Marcel Dekker: New York, 1990; Chapter 2. (3) McMullan, R. K.; Jeffrey, G.A. J . Chem. Phys. 1965, 42, 43. (4) Mak, T. C. W.; McMullan, R. K. J . Chem. Phys. 1965, 42, 2735. (5) Stackelberg, M. von; Muller, H. R. J . Chem. Phys. 1951, 19, 1319. (6) van der Waals, J. H.; Platteeuw, J. C. Adu. Chem. Phys. 1959, 2, 1. (7) McKoy, V.;Sinanoglu, 0. J. Chem. Phys. 1963, 38, 2946. (8) Nagata, I.; Kobayashi, R. Ind. Eng. Chem. Fundam. 1966, 5, 466. (9) Parrish, W. R.; Prausnitz. J. M. Ind. Eng. Chem. Process Des. Deu. 1912, 11, 26. (10) John, V. T.; Holder, G.D. J . Phys. Chem. 1985,89, 3279. (11) John, V. T.; Holder, G. D. J . Phys. Chem. 1981, 85, 1811. (12) John, V. T.; Holder, G. D. J . Phys. Chem. 1982, 86, 455. (13) John, V. T.; Padadopoulos, K. D.; Holder, G. D. AIChE J. 1985,31, 252. (14) Tse, J. S.;Davidson, D. W. Proc. 4th Can. Permafrost Conf. 1982, 329. (15) Anderson, F. E.; Prausnitz, J. M. AIChE J. 1986, 32, 1321. (16) Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B. Molecular Theory oj Gases and Liquids; Wiley: New York, 1954; Chapter 13. (17) Lennard-Jones, J. E. R. SOC.Proc. 1924, A106,709. (18) Lennard-Jones, J. E.; Ingham, A. E. R. Soc. Proc. 1925, A107,636. (19) Sparks, K. A.; Tester, J. W. To be submitted for publication, 1992. (20) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. Properties of Gases & Liquids, 4th ed.; McGraw-Hill: New York, 1987; Appendix B. (21) Berendsen, H. J. C.; Postma, J. P. M.; Van Gunsteren, W. F.; Hermans, J. In Intermolecular Forces; Pullman, B., Ed.; Reidel: Dordrecht, 1981.
Microscopy, X-ray Dlffraction, and NMR Studies of Lyotropic Liquid Crystal Phases in the C,,EO,/Water System. A New Intermediate Phase Sergio S. Funan,+Michael C. Holmes,* Department of Physics and Astronomy, University of Central Lancashire, Preston, PR1 2TQ. U.K.
and Gordon J. T. Tiddy Unilever Research, Port Sunlight Laboratory, Bebington, Wirral, Merseyside, M 3 3JW, U.K. (Received: July 13, 1992;In Final Form: September 28, 1992) Mixtures of the nonionic surfactant hexaethylene glycol cis- 13-docosenylether (C22EO6) with water (2H20)were studied by optical microscopy, small-angleX-ray scattering,and 2HNMR spectroscopy. A schematic phase diagram has been obtained, which shows two high-temperature bicontinuous cubic phases (V,) and a low-temperature fluid intermediate phase (Fl) in addition to the usual hexagonal (HI) and lamellar (L,) phases. Although thermodynamically stable and separated from the L, phase by a first-order transition, the Fl phase has a lamellar structure. It differs from the L, phase structure in that the lamellae are broken by water-filled defects, which exhibit no correlation in position between layers. Possible structures for the defects are examined. The occurrence of this phase rather than a V, cubic phase is attributed to decreased alkyl
chain flexibility.
* Author to whom correspondence should be addressed. 'Present address: Fachbereich Physik, Universitiit Leipzig, Linnbstr. 5, D 0-7010, Leipzig, Germany.
In the nohonic surfacGnt system, hexaethylene glycol monon-ddecyl ether (C12E06)/water,6 the hexagonal, HI phase shows cubic fluctuations on approaching the cubic phase from low
0022-3654/92/2096-11029$03.00/00 1992 American Chemical Society
