Nucleation and Growth of Hydrates on Ice Surfaces: New Insights from

12338

J. Phys. Chem. B 2001, 105, 12338-12347

Nucleation and Growth of Hydrates on Ice Surfaces: New Insights from Experiments with Hyperpolarized Xenon

129Xe

NMR

Igor L. Moudrakovski, Anivis A. Sanchez, Christopher I. Ratcliffe, and John A. Ripmeester* Steacie institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6 ReceiVed: June 26, 2001; In Final Form: September 13, 2001

The nucleation and growth of hydrate on the surface of ice was followed by NMR spectroscopy using hyperpolarized Xe. From the ice surface area and Xe pressure drop, the thickness of the hydrate film formed was ∼ 500-1000 Å before the reaction became extremely slow. The cage occupancy ratio θL/θS was measured as a function of time from the Xe spectrum and was used to monitor the nature of the material formed. The ratio changed from values close to 1 during the early part of the reaction (induction time) to its equilibrium value of ∼3-4 after nucleation processes were finished and rapid growth commenced. The low value can be seen as evidence of a precursor phase that is quite different from the equilibrium hydrate. The induction time was found to be a reproducible function of temperature and pressure for the conditions studied. The kinetics of gas uptake was analyzed according to the Avrami-Erofeyev equation used to describe solid-gas reactions. A surface memory effect was noted on successive cycles of xenon adsorption-desorption-readsorption, as the induction time was absent on the readsorption cycle. The new results are discussed in terms of models for nucleation and growth of hydrate and the various experiments that have been carried out in the past. It is essential to differentiate between later-stage diffusion-limited hydrate formation processes and the initial steps of hydrate formation at a surface, as they have opposite temperature coefficients.

Introduction Gas hydrates, crystalline materials consisting of frameworks of hydrogen-bonded water molecules forming cages which enclathrate guest molecules, have been the subject of scientific investigation for about 200 years. Although they have been a remarkably difficult subject of study, arguably they constitute the best-understood example of the class of inclusion compounds referred to as clathrates. There are three main families of structures, the two cubic structures I and II and the hexagonal structure H.1,2 The guests range in size from argon up to about methylcyclohexane, with the largest incorporated guest generally determining the structure type. Gas hydrates are more than laboratory curiosities: as early as the 1930s gas hydrates were implicated in the formation of impermeable solid plugs in gas transmission lines.3 Since that time research into prevention of hydrate formation has been a major activity in many engineering laboratories. Currently there is a trend toward controlling hydrate formation by kinetic inhibitors and antiagglomerants rather than antifreezes such as methanol,4-6 although no single solution is capable of handling all situations. Since the speculation and early evidence for the presence of gas-hydrate in nature in the 60’s and 70’s, there has been a remarkable increase in interest in natural gas hydrates. Current estimates make gas hydrates one of the major sources of hydrocarbons on earth, with implications for hydrate deposits both as a resource and as an agent for climate change. A question that is central to the prevention and control of hydrate formation and that remains largely unanswered is this: How do hydrates form? This must be answered in the context of the fact that a solid hydrate phase forms from water and a * To whom all correspondence should be addressed. E-mail: [email protected]. Phone: (613) 993 2011. FAX: (613) 998 7833.

gas that is not nearly soluble enough in water to give the high concentrations required for hydrate formation. There are many examples of kinetic studies of hydrate formation from liquid and gas where the main parameter measured is the uptake of gas and where vigorous stirring takes care of both mass transport of gas into the liquid phase and the transport of heat away from the crystallizing hydrate.7-10 Measurements of gas uptake on ice, either with or without grinding, have also been carried out.11-14 A feature of kinetic experiments includes the presence of an induction period, defined as the time delay between the start of the exposure of water or ice to gas under hydrate-forming conditions, and the first appearance of solid hydrate. There still is considerable controversy about the meaning of the induction period: Does it have intrinsic meaning, or is it strictly related to the stochastics of nucleation? A weak point of kinetic studies that use only gas uptake measurements is that they provide no molecular-scale information. For instance, are precursor states present before nucleation, or what is the nature of the crystalline phase that forms first? Even if the crystal structure is fixed, the composition is unlikely to be that of the equilibrium phase, considering that gas hydrates are nonstoichiometric compounds usually with a significant fraction of the cages unoccupied. There are few techniques capable of giving molecular-scale information on hydrates with sufficient time resolution to carry out kinetic studies. We note recent attempts to use diffraction techniques employing energy-dispersive X-rays15 and intense pulsed neutron sources.16 However, there are distinct advantages to using techniques that are sensitive to local order as well as long-range order, as it would be very useful to obtain information on precursor phases that may be present during nucleation. Spectroscopic techniques such as Raman spectroscopy are therefore attractive.17 A drawback of this techniques is that the

10.1021/jp012419x CCC: $20.00 Published 2001 by the American Chemical Society Published on Web 11/10/2001

129Xe

NMR Experiments with Hyperpolarized Xenon

volume sampled by the beam is very small, and crystallization processes in the beam are likely to change both the direction and amount of scattered radiation as time progresses, thus making quantitative measurements very difficult.18 Another technique that can provide molecular scale information on structure and composition is 129Xe NMR spectroscopy. When spectra are recorded under quantitative conditions, the spectra give relative cage occupancies, and the chemical shift of the encaged xenon atoms provides a definitive indicator of structure type.19,20 The drawback of the NMR experiment is that the low sensitivity and long relaxation times make real-time observation of hydrate formation impossible. Recent experiments have shown that these limitations can be overcome by utilizing the great enhancement in signal intensity offered by using hyperpolarized (HP) xenon.21 We now have refined the experimental technique, giving the NMR experiment single-scan sensitivity with an excellent signal-to-noise ratio and a time resolution limited only by the spin-spin relaxation time of 129Xe. The experiments reported here provide not only kinetic data, but also give new insight into the nature of the surface of ice, the fundamentals of nucleation of hydrates, and the existence of transient states during hydrate formation.

J. Phys. Chem. B, Vol. 105, No. 49, 2001 12339

Experimental Section

Figure 1. Experimental setup for the study of kinetics of xenon hydrate formation on the surface of ice with HP xenon. The polarizing cell (a) with HP xenon is connected to a transfer line, leading to the detection region of the NMR probe containing the sample of ground ice.

HP 129Xe NMR experiments were carried out by using a standard 10 mm high-resolution broad-banded probe on either Bruker AMX 300 (83.03 MHz) or DSX 400 (110.7 MHz) spectrometers. Spectra were acquired on single scans obtained with short (3-5°) pulses. Even long pulse trains did not lead to significant depolarization: after 128 pulses, more than onehalf of the polarization was preserved. Xenon, with the natural isotope distribution, was obtained from Matheson and was hyperpolarized by spin-exchange with optically pumped Rb in the gas phase.22 During optical pumping, the polarization cell was kept at 85-100 °C in a small oven located in the fringe field of the spectrometer magnet. The optical pumping cell was a 6 cm long, 35 mm inner diameter Pyrex cylinder (∼50 cm3 volume) with flat windows, equipped with an Ace 3 mm high vacuum valve. The cells were thoroughly treated with a 10% solution of SurfaSil in pentane in order to reduce the surface relaxation as described in ref 22c. The cell which contained a few droplets of metal Rb could be attached to a HV system with Cajon or Swagelok adapters. Normally, the cell was loaded with xenon at pressures between 500 and 1500 mbar and 200-500 mbar of nitrogen as a quencher gas. A 30 W diode laser array from OptoPower Corporation operating at 795 nm (2.5 nm line width) was used. Typically, the pumping cell was irradiated for 30 min providing Xe polarization of 5-8%. Powdered ice was prepared by cold grinding D2O droplets quick-frozen in liquid nitrogen. The specific surface area of ice prepared in this way was consistently 5.1 ( 0.8 m2/g, as determined from the modified BET technique.23 The powdered ice (0.3-0.5 g) was transferred to a cold NMR tube equipped with a high vacuum fitting (Ace Glass Inc.) and inserted into the cold NMR probe. The gas transfer line with attached sample was connected to a high vacuum system, thus allowing exposure of the sample to vacuum or gas, as needed (Figure 1). Before adsorption of HP xenon, the ice was evacuated for 30-40 min at 10-5 mbar to minimize adsorbed oxygen. The pumping cell with polarized xenon was also attached to the vacuum line, immersed in liquid nitrogen, and evacuated in order to separate xenon from nitrogen. The cell was then closed and left to warm to room temperature to thaw the frozen Xe.

After that, the gas was allowed to contact the ice sample. A VM1500 electronic manometer (EBRO Electronic) was used to monitor pressure changes directly in the line (the volume of the line is approximately 25 mL), allowing an estimate to be made of the quantity of hydrate formed from the pressure difference at the beginning and the end of the experiment. Generally, the pressure drop was small, suggesting complete conversion of the ice surface to hydrate. The experimental apparatus for delivering the HP xenon to the ice used in this work is shown schematically in Figure 1. It should be noted that if the acquisition of data starts before the ice is exposed to the gas, this permits a rather accurate recording of the initial stages of the kinetics. In previous work,21 the HP xenon was supplied in a frozen state to the ice sample, and since it was necessary to wait for the melting of xenon, this contributed significantly to the uncertainty of the starting point for kinetic measurements. Experiments to test the effects of the treatment of the ice surface involved decomposition of hydrates by pumping away the xenon, followed by readmission of HP xenon to the same sample. Further, samples of ice will be referred to as “fresh” (i.e., no xenon was adsorbed on the sample before the experiment in question) or “pre-exposed” (i.e., the sample had been exposed previously to xenon at least once, and before the next exposure it was evacuated for a certain time). “Pre-exposed” samples were evacuated either continuously for an indicated period of time or in a so-called “interrupted” manner, where the sample was initially evacuated for a short time (2 min, just to remove xenon gas from the line); then the line was closed, and the sample was left for a period of time. At the end of the desired period, the line was again exposed to vacuum for a minute to remove any desorbed xenon. We believe that such an “interrupted” evacuation procedure provides milder conditions for hydrate decomposition and without a lot of evaporation of ice from the surface. The residual xenon content of such decomposed hydrate samples was checked by allowing hydrate to melt and measuring the residual gas content with a manometer. For very small quantities of xenon, a residual gas analyzer was used.

12340 J. Phys. Chem. B, Vol. 105, No. 49, 2001

Figure 2. Time development of the xenon spectrum for the reaction of HP xenon with powdered ice (T ) 243 K, PXe ) 580 mbar). The signals are assigned to Xe in the gas (∼0 ppm), the large cage (∼150 ppm), and the small cage (∼242 ppm). Very short RF pulse lengths (1 µs, ∼4°) were used to minimize the effect of depolarization by the pulse. Time between consecutive scans the in plot progressively increases from 1 s to 1 min, as shown on the right side of the spectra.

Moudrakovski et al.

Figure 3. (A) Integrated intensity of the peaks shown in Figure 2, demonstrating the induction period. (B) The ratio of the integral intensities of signals from xenon in large and small cages for the run shown in Figures 2 and 3A.

Results and Discussion The experiments and results discussed in this section address a number of topics related to the formation of clathrate hydrates on the surface of ice. The first one is the kinetics of the hydrate formation as observed by 129Xe NMR of HP xenon. The second topic deals with factors affecting formation and decomposition of the hydrate. We will show that formation of the surface hydrate takes place faster at lower temperatures and also that it depends on the xenon pressure and the history of the sample. The last topic concerns the mechanism of hydrate formation including nucleation, as suggested from our NMR experiments. A. General Observations. Figure 2 shows a typical data set for the time development of the 129Xe NMR spectrum of the structure I hydrate when a powdered ice sample is exposed to HP xenon. The gas-phase signal is seen at ∼0 ppm, and the large (51264) and small (512) cage signals appear at 150 and 242 ppm, respectively, as determined previously.19,20 Although the results are in general agreement with those of the ice-HP Xe gas reaction reported previously,21 better control of gas admission and increased signal-to-noise ratio allow more accurate determination of both the “zero-time” of the experiment and a better definition of the kinetic curve with a time resolution of about 40 ms. The integrated line intensities for the run are shown in Figure 3A. We note that at early times the only peak with observable intensity is the gas-phase signal and that no other signals are observed. From the sigmoid shape of the curve, it is evident that before rapid growth of the hydrate commences there is a period of little or no apparent activity. From gas uptake measurements, this time is known as the induction period and, in general, is thought to have both stochastic and determinative components.6 Our measurements as a function of temperature, pressure, and thermal history of the sample show that the induction time varies systematically with these parameters and that for fresh samples the induction time showed little variation if T and P were kept constant, as shown in Figure 4 and Figure 5. Thus, the induction time becomes a reproducible and determinative parameter for hydrate formation on ice surfaces for the conditions of P and T used. We also note that the

Figure 4. Dependence of induction time on the pressure of xenon at T ) 223 K.

minimum pressure for which hydrate formation could be observed was always well above the equilibrium pressure (P . P0) for that temperature, consistent with the need for a “driving force” for nucleation, as noted elsewhere.6,7 The gas uptake experiment goes to completion well within the lifetime of the HP state of the xenon, the only manifestation of spin-lattice relaxation being a steady decline in the signal level as time passes. We can conclude that the reaction stops when the thickness of the hydrate layer is enough to become a serious barrier to easy gas (or water) transport. Of course, complete conversion to hydrate can be expected to take place on a sufficiently long time scale as long as enough gas is available at sufficiently high pressure. Since we have estimates of the surface area of the ice and know the total gas uptake from the pressure drop, the thickness of the hydrate layer can be assessed. Again, this depends on the exact conditions of the experiment, and it varied between 20 and 100 unit cells (at 12 Å/unit cell, this amounts to a layer thickness between 240 and 1200 Å; see Table 1).

129Xe



TABLE 1: Ratio of Occupancies for Large and Small Cages θL/θS, Thickness of the Surface Hydrate Layer NL and Parameters l, k, and T1* Determined from Fitting of the Experimental Data with the Modified Avrami-Erofeyev Equation (Eq 10) small cage T, K

notes

193

a a

fresh a a

198 203

fresh fresh a

fresh a a a

213

fresh a

fresh a

223

a

fresh a

fresh a a a a a a

233

fresh a a a

fresh 243

a

fresh fresh a a

Pstart, mbar 745 705 724 775 585 560 755 675 574 567 835 570 720 538 856 770 580 758 760 760 681 810 340 333 190 942 1013 920 582 583 555 579 706 622 615

I

k, s-l

0.57 0.82 0.25 0.40 0.71 1.00 0.40 0.23 0.62 0.09 0.10 0.52 0.68 0.92 0.37 0.26 1.09 0.20 0.38 0.57 0.88 0.16 1.01 0.73 1.03 0.29 0.17 0.17 0.70 0.72 1.11 0.56 0.68 1.36 0.71

2.97 × 5.00 × 10-5 9.51 × 10-10 4.00 × 10-5 4.00 × 10-5 1.63 × 10-10 1.68 × 10-9 1.50 × 10-4 2.30 × 10-11 4.28 × 10-7 5.59 × 10-6 1.10 × 10-4 6.77 × 10-8 2.92 × 10-6 2.04 × 10-7 3.12 × 10-3 2.00 × 10-5 5.21 × 10-8 8.10 × 10-4 4.61 × 10-8 1.30 × 10-4 6.48 × 10-3 2.10 × 10-4 2.40 × 10-4 1.81 × 10-7 4.26 × 10-3 7.56 × 10-7 2.20 × 10-3 9.00 × 10-5 2.80 × 10-4 3.52 × 10-9 1.35 × 10-7 4.74 × 10-10 1.74 × 10-9 3.31 × 10-3 10-3

large cage l 1.96 2.65 3.17 2.73 2.59 4.35 3.83 3.01 3.85 2.33 2.71 2.91 3.29 2.49 3.24 2.60 3.31 3.82 2.44 3.78 2.96 2.51 2.05 1.89 2.95 2.10 2.87 1.94 2.17 2.21 3.53 2.80 3.07 2.82 1.18

T1,* s 5000 4149 1341 1471 1877 788 2111 1553 2757 5000 2661 1743 1334 3383 2253 1671 2258 4902 3444 3614 1806 1767 3250 2286 2367 1986 1881 1691 1897 3039 1914 3074 5000 2500 5225

I

k, s-l

l

T1,* s

NL, layers

θL/θS

1.68 2.42 0.83 1.20 2.05 2.97 1.19 0.69 1.89 0.41 0.31 1.52 2.21 2.93 1.23 0.82 3.65 0.59 1.40 1.96 2.92 0.50 3.74 2.68 4.03 0.79 0.55 0.53 2.34 2.85 3.87 2.26 2.21 5.63 2.92

2.72 × 5.78 × 10-6 5.91 × 10-10 3.01 × 10-6 4.00 × 10-5 6.04 × 10-11 1.37 × 10-9 1.10 × 10-4 1.32 × 10-11 1.00 × 10-5 4.23 × 10-6 1.00 × 10-4 3.45 × 10-8 5.11 × 10-6 3.17 × 10-8 1.96 × 10-3 2.00 × 10-5 2.10 × 10-8 1.33 × 10-3 2.01 × 10-8 1.30 × 10-4 3.32 × 10-3 1.60 × 10-4 1.80 × 10-4 7.14 × 10-8 2.16 × 10-3 4.71 × 10-7 2.00 × 10-3 7.00 × 10-5 2.70 × 10-4 3.52 × 10-9 8.58 × 10-8 1.75 × 10-10 6.58 × 10-10 2.73 × 10-3

2.00 3.22 3.28 3.42 2.53 4.53 3.85 3.08 3.93 1.80 2.79 2.93 3.41 2.38 3.62 2.76 3.32 3.59 2.33 3.51 2.93 2.81 2.10 1.94 3.11 2.31 2.96 1.98 2.22 2.20 3.87 2.87 3.60 2.94 1.20

5000 3321 1790 1444 1645 768 1889 1447 2463 4481 2492 1533 1212 2555 2030 1503 1975 4537 3213 3225 1601 1489 2771 2022 2031 1595 1708 1573 1749 2629 2011 2737 5000 2500 4624

32 29 37 35 45 35 27 23 43 39 16 42 30 36 45 35 53 63 50 46 42 13 41 43 25 17 15 22 29 101 80 76 58 31 64

2.96 2.96 3.40 3.01 2.87 2.97 2.95 2.96 3.07 4.29 3.17 2.94 3.25 3.20 3.37 3.11 3.33 2.91 3.73 3.45 3.31 3.11 3.71 3.68 3.92 2.74 3.32 3.15 3.36 3.97 3.49 4.07 3.25 4.13 4.09

10-3

Pre-exposed to xenon, then evacuated.

two distinct cages, the expression for the chemical potential difference is

∆µ )

Figure 5. Combined graph of induction time vs pressure and temperature for “fresh” samples of D2O ice.

B. Occupancy Ratio and the Solid Solution Model. The chemical shifts in all of the experimental runs are always consistent with the formation of structure I hydrate. For clathrates in general, the van der Waals and Platteeuw “solidsolution” model relates the composition to the chemical potential difference ∆µ between the dense, nonclathrate phase and the hypothetical empty clathrate lattice.24 The model also assumes that guest-guest interactions can be neglected and that the ∆µ value is independent of guest type. For structure I hydrate with 129Xe

RT [n ln(1 - θL) + nS ln(1 - θS)] N L

with N being the number of water molecules in the unit cell and n and θ the number of cages per unit cell and the cage occupancies, respectively. Therefore, within the framework of the solid-solution theory, once ∆µ is known, the occupancy ratio determines the overall composition. Earlier experiments on xenon hydrate prepared under equilibrium conditions have shown that the occupancy ratio is close to 4, consistent with absolute occupancies of ∼1 for the large cage and ∼0.75 for the small cage (∆µ ) 1297 J/mole at 0 °C, 1 atm).25 These values are the minimum cage occupancies for which the hydrate is stable. The occupancy ratio therefore is a significant structural parameter that gives important clues as to the composition of the solid phase formed and the establishment of equilibrium (as discussed in more detail below, equilibrium here refers only to the time scale of the experiment, not absolute equilibrium). Figure 3B shows the occupancy ratio as a function of the time for the experimental run shown in Figure 2. During the initial period of exposure, the experimental ratio changes rapidly with time but levels off at the start of the period of rapid growth. The plot is characteristic of all of the experimental runs; that is, the occupancy ratio becomes constant approximately at the time that the hydrate formation rate increases, i.e., at the end of the induction period. At shorter times, the occupancy ratio


Figure 6. (A) Occupancy ratio vs T for Structure I xenon hydrate formation on the surface of ice. (B) Occupancy ratio vs P for xenon hydrate on the surface of ice at 223 K.

drops sharply in favor of the small cage and is close to a value of 1 at the earliest times during which data could be obtained (we should note, however, that the error of integration is the largest at the shortest times, which limits the accuracy of the reported data). From these results, it is clear that during the induction period, the average composition of the hydrate phase is a rapidly varying function of time and that rapid growth commences approximately when the average composition becomes constant. Systematic measurements revealed that the occupancy ratio depends on pressure and temperature, its value increasing with temperature and decreasing with pressure. The temperature and pressure variations are shown in Figure 6, with the actual value of the occupancy ratio usually in the range from ∼3 to 4. It should be pointed out that the hydration number is a doublevalued function of the occupancy ratio, with equal occupancy of the cages at the minimum hydration number. The occupancy ratio less than 3 observed at the lowest temperature suggests that the smaller cages are now more fully occupied than the large cages. C. Surface Memory Effect. The surface hydrate can be decomposed again by exposing the sample to a vacuum, and this process can also be followed by NMR spectroscopy (Figure 7). The kinetics of decomposition can be fitted by a singleexponential decay (i.e., as a first-order process). The decomposition rate decreases with temperature; however, even at the lowest temperatures studied, τ1/2 did not exceed more than a few minutes. From the temperature dependence of the decomposition constant, the activation energy of 37 ( 7 kJ/mol is estimated (Figure 7A). The activation energy for decomposition is

Moudrakovski et al. somewhat less than the known activation energy for water selfdiffusion in ice Ih and bulk hydrates.26,27 An important question for further discussion of the nature and kinetics of the surface hydrate formation is how much hydrate has been left on the surface after evacuation. We tested this by measuring the amount of xenon released after melting the samples after evacuation. At 223 K, where most experiments were done, the amount of residual xenon did not exceed 2-3% of the initial amount after 20 min of evacuation and was always below 8% for an “interrupted” pumping experiment. If the surface of the ice after evacuation of xenon is reexposed to HP xenon, the reformation of hydrate is observed. In this case, however, the induction period is practically absent, and rapid hydrate growth commences as soon as xenon reaches the surface (see Figure 8). This suggests that the ice surface is now preorganized for hydrate growth and for the crystalline hydrate. This preorganization could be as simple as the retention of a significant number of hydrogen-bonded 5-rings of water molecules on the surface. Of course, if the surface “remembers” its previous structure, one also should be able to measure the rate at which the surface “forgets” its preorganized state. This was studied by lengthening the time between desorption and reabsorption of xenon. The variation of induction time with the evacuation time between consecutive admissions of xenon at 223 K is shown in Figure 9. After 2 h of continuous pumping, about 50% of the induction time has returned. The reintroduction of the induction time is slower for the less severe condition of interrupted pumping, nevertheless it is also unmistakable. We also observed that a higher temperature accelerates the loss of the preorganized state. Thus, if the sample temperature is increased from 223 to 243 K, the induction time will be almost completely restored after only 1 h. It appears that the history of the ice surface affects only the induction period of hydrate formation. The growth kinetics of surface hydrate formation after the induction period remains essentially the same for the same pressure and temperature. Similar so-called “memory” effects in connection with hydrate formation, but for liquid water, have been observed previously in experimental and modeling studies.7,28 D. Kinetic Model. The mechanism of any solid-state reaction is generally considered to include two or more of the following four types of elementary steps: (i) adsorption phenomena; (ii) reaction at the atomic or molecular scale (homogeneous or interface reactions); (iii) nucleation of new phase; (iv) growth of nuclei; (v) transport phenomena (diffusion of reactant and products). Usually, one of these steps is rate-determining and thus influences the overall reactivity of the system. Which step is rate-determining is often reflected by the experimentally observed dependence of the extent of the reaction as a function of time. Usually, a sigmoidal shape to the curve serves as an indication that nucleation is the most important step.29 The net equation for hydrate formation can be written as

Xe(g) + Ice(s) ) Str.I(s) In a purely formal sense, this is an analogue of the tarnishing reaction of metals. For the acceleratory period of reaction, the observed time dependence can be interpreted as arising from the formation and growth of nuclei of the product phase, the overall rate being the product of the rate of nucleus formation and of nuclear growth. Beyond the inflection point of the kinetic curves, the nuclei are thought to start overlapping so that the

129Xe



Figure 7. Decomposition of the surface hydrate in a vacuum at 223 K. Observations begin prior to exposing the sample to vacuum, which occurs at trace 8 (time 0). The delay time is increased each eight consecutive scans and is 0.2, 0.5, 1.0, 2.0, 5.0, 10.0, 20.0, and 30.0 s. Inset A shows the Arrhenius plot for the hydrate decomposition. Activation energy of the process is found to be 37 ( 7 kJ/mol.

interfacial area between reactant and products decreases and, hence, so does the reaction rate. Also if the reaction product (Str. I hydrate in our case) forms a very compact coating on the outer surface of the reactant, then the gas cannot penetrate into the bulk, and reaction will eventually cease on the time scale of our experiments. This type of kinetics can be generally accounted for by the Avrami-Erofeyev model,30-32 where random nucleation followed by three-dimensional growth of isolated nuclei is assumed. The model supposes that nucleation of a product phase is equally probable on all inner and outer surfaces of the reactant crystal and that the rate of the phase boundary reaction for the growth of the product phase is constant. Then the rate law for the reaction can be calculated, provided that the nucleation probability is known as a function of time. In further discussion of the kinetic model, we will follow closely the ideas developed in reports30-33 that describe the kinetics of nucleation and chemical reactions in the solid state. First, we need to make a few preliminary remarks about nucleation and its role in the chemical transformation of solids. It is natural to expect that nucleation occurs only when fluctuations in the local energy are sufficient to provide the necessary activation energy for nucleus formation. Speaking of adsorption of xenon and hydrate nucleation on the surface of ice, we generally should expect some kind of structure sensitivity, such that the nuclei are formed at definite localized spots where the activation energy is least. In this work, we do not need to make any assumptions about the existence of a so-called quasi-liquid layer on the surface of ice, for which there is good evidence, although a detailed description still is lacking. The number of nuclei formed in a given time will depend on the number of these potential nucleus-forming sites as well as on the mean activation energy for nucleus formation. Potential nucleus-forming sites are usually associated with some type of lattice imperfection such as cracks, lattice defects, dislocations, etc.

The rate law of nucleus formation will depend on how many steps are involved in the formation of a nucleus. Nucleation that involves only a single step is commonly associated either with an exponential or a linear law.29,30 For such a situation, if there are N0 potential nucleus forming sites, the rate of nucleus formation is

dN ) k1(N0 - N) dt

or

dN ) k1N0 exp(-k1t) dt

(1)

where k1 is a constant for nuclei formation (or in another terms, 1/k1 is the probability of this event). The number of nuclei increases with time exponentially. A limiting case of equations (eq 1) for early stages of the reaction (small t) or for very small k1 (weak interaction) is

dN ) k 1N 0 dt

(2)

so that the number of nuclei increases linearly with time. Another possible case is when k1 is very large, so N ) N0; i.e., the nucleation proceeds instantaneously. Nucleation involving more than one step is commonly described by a power law. For the general case of solid-state reaction with nucleation, Bagdassarian has shown33 that if m successive events are necessary to form a stable nucleus and if the probability of each is k1, then the number of nuclei formed in time t is N ) N0(k1t)m/m! ) Dtm, and

dN ) Dmtm-1 dt

(3)

The case with m ) 2 is a bimolecular combination of one active intermediary with one of a constant number of sites, m ) 3 corresponds to a bimolecular combination of two active intermediates, etc. Considering the case of hydrate formation on the surface of ice, it appears to be quite unlikely that the formation of hydrate


Moudrakovski et al.

Figure 8. Effect of sample history: kinetics of adsorption of HP xenon on fresh ice and ice from decomposed hydrate at 243 K: (A) Adsorption of HP xenon on a sample of “fresh” ice. Pressure of xenon at the start was 622 mbar. (B) Adsorption of HP Xe on a sample of ice previously “exposed” to xenon after continuous evacuation for 30 min (same sample of ice as used in experiment A). Initial Xe pressure was 615 mbar. Amount of adsorbed xenon for these two samples corresponds to 76 and 58 hydrates layers, respectively. Time between consecutive scans in the plot progressively increases from 1 s to 1 min, as shown on the right side of the spectra.

nuclei involves any combination of intermediates. Such intermediates should be quite bulky structures, and their mobility thus is expected to be very low even for the case of the quasiliquid layer, not mentioning the case of a “normal” hard surface. An exponential law for nucleus formation (or, a linear law as its special case) seems to be more reasonable for this situation. Of course, we cannot exclude that in such a complicated system there might be several parallel mechanisms of nucleation with comparable contributions. The latter situation would make the analysis particularly difficult.

If we consider that random (exponential) nucleation is followed by isotropic growth in n dimensions at a linear rate k2, then the volume V(t,y) at time t of a nucleus formed at time t ) y is

V(t,y) ) σ[k2(t - y)]n

(4)

where σ is a geometrical factor and n ) 3 for expansion into a volume, 2 for spreading on a surface, and 1 for unidimensional growth.

129Xe



Figure 9. Induction time vs evacuation time between consecutive admissions of xenon at 223 K. Filled points: continuous pumping of the samples between the xenon loadings. Open points: “interrupted” pumping between the loadings.

For further discussion, it is useful to introduce ω0sthe “active” part of the ice that can participate in the formation of hydrate (or “ready” to be converted to hydrate). Generally, ω0 could represent all of the ice, but most often, it is only part of it, since larger, dense chunks of ice will be converted only partly, without mechanical activation. A detailed analysis of random nucleation at potential nucleus-forming sites followed by threedimensional growth30-32 leads to the kinetic equation of transformation of this “active” ice

(

)

V(t) -ln(1 - R(t)) ) -ln 1 V0 )

)

σk1N0k23 V0

∫0t exp(-k1y)(t - y)3 dy

{

6σN0k23 V0k13

exp(- k1t) - 1 + k1t -

}

(k1t)2 (k1t)3 + 2! 3!

(5)

V0 is the final volume of product obtained from the complete decomposition of ω0 g of original “active” ice. For k1t small, this approximates to

-ln(1 - R) )

σN0k23k1 4 t 4V0

or

{

}

σN0k23k1 4 R ) 1 - exp t 4V0

(6)

Generally, depending on the shape of the nucleus and character of the processes defining growth/disappearance of nuclei, one can get the general equation of the form30-32

R ) 1 - exp(- kt ) l

(7)

which is known as the Avrami-Erofeyev equation.29 The parameter l in this equation is closely related to the type of

Figure 10. Examples of fitting experimental data with the modified Avrami-Erofeyev equation (eq 10). (A) Fresh sample of ice, T ) 223 K, P ) 760 mbar. (B) Second adsorption of Xenon on a sample of ice that was loaded with Xe and evacuated; T ) 223 K, P ) 580 mbar.

nucleation. For spherical nuclei expanding homogeneously in all dimensions, parameter l as it is shown above is equal to 4. For cylindrical nuclei (the centers of formation of the nuclei are edges or surfaces cracks), one can find an expression similar to the previous one: R ) 1 - exp(-kt3) and, correspondingly, for flat nuclei, R ) 1 - exp(-kt2). A detailed description of numerous different cases can be found in ref 30. In the case of multistep nucleation, parameter l can be even larger than 4, though this is not common. The composite parameter k in the equations is a function of the number of nucleation sites N0, the rate constant for nucleation k1, and the rate of the phase boundary reaction. Since k is a combination of several other constants, its interpretation is not straightforward and normally would require some assumptions about the mechanism of nucleation and the number of sites. Before application to our experimental data, the original Avrami-Erofeyev equation needs to be modified to account for the 129Xe NMR signal intensity decay due to the NMR RF pulses and spin-lattice relaxation. Since not all the xenon is in the detection area of the probe, it is difficult to separate the contributions of each effect to the total decay. To characterize this combined effect, we will use some effective constant of decay T1*. The combination of eq 7 and an exponential decay function gives us the fitting function

( )

I(t) ) I(1 - exp(-ktl)) exp -

t T1*

(8)

where I(t) is the integrated intensity of the signals, k is the composite nucleation constant as discussed above, and l is effectively the parameter of the nucleation dimensionality. Examples of the fits are shown in Figure 10, and parameters for the fits are summarized in Table 1.

12346 J. Phys. Chem. B, Vol. 105, No. 49, 2001 The most important observation is the large difference in l and k for fresh and “pre-exposed” samples of ice. For all fresh samples, l was consistently above 3, whereas for “pre-exposed” samples, in most cases, it is well below 3, suggesting a very different nucleation dimensionality. It is interesting to note that the corresponding thickness of the hydrate layer (NL) from Table 1 for the pre-exposed samples is smaller than that for the fresh ice samples. This suggest that the “preorganized” surface provides faster hydrate formation, but at the same time this dense hydrate layer becomes an obstacle to Xe penetration to continue hydrate formation. Changes in k are as large as several orders of magnitude. k also increases with increasing pressure and temperature, though the change is less pronounced with temperature. We might expect such behavior, since k is a combination of the concentration of nucleation sites and the probability of interaction of the sites with xenon. The constants of effective decay T1* can vary by as much as a factor of 2.5 for different samples, and this can be the result of different concentrations of paramagnetic impurities (O2). Within the same sample, however, the effective constants of decay T1* are essentially the same for large and small cages; i.e., spin-lattice relaxation times for xenon in the different cages are very similar. E. Some Observations on Nucleation. Considering the effort that has been devoted to studying the nucleation of hydrates and the current state of understanding, it would be presumptuous to suggest that we can do little more than offer some initial insight. Hydrate formation involves the formation of a new phase on the surface of ice. If we assume that standard nucleation theory applies, then successful nucleation depends on crystal embryo’s attaining a critical size before growth can occur. The free energy barrier between the embryo and stable crystal depends on the interfacial energy and the change in bulk molar free energy. The induction period then reflects the time during which nucleation takes place on the ice surfaces. At each nucleation site, the nucleation process can be assumed to be stochastic, but since there are numerous sites, the induction time is a measure of the time by which stochastic nucleation has occurred at all sites. Such a picture is consistent with the measurements on ice nucleation where for many individual nucleation events the induction time could be fit to an exponential function with a time constant characteristic of the driving force for nucleation, which for ice is the degree of sub-cooling. For hydrate nucleation on ice, the induction time is a function of temperature and pressure, and these two parameters can be identified as being the driving force for nucleation, although we have not determined a detailed functional relationship. At the beginning of the induction period, the occupancy ratio must reflect primarily the composition of precursor clusters. The occupancy ratio drops to a value less than or near 1, suggesting the presence of large numbers of small cages. However, this should not necessarily be seen as a cage with a definite geometry, as it is not confined to a crystal, but as a hydration sphere of 20 water molecules that may be quite fluxional (at least three distinct symmetries are known for the 512 cages in the three major families of hydrate structures and there is also the 435663 cagesall with similar chemical shifts). The end of the induction period signals the completion of nucleation of the crystal phase from precursor states, and during the induction period, the occupancy ratio is the weighted average of the occupancy ratios of the precursor state and the growing crystal hydrate.

Moudrakovski et al. F. Some Observations on Kinetics. We should examine our results in light of previous work, as there are some profound differences. First of all, we must distinguish between kinetic studies that take place at surfaces (e.g., on the ice surface, upon grinding of ice, in liquids) and hydrate formation that converts ice in bulk. In the latter instance, diffusion processes limit hydrate formation, and the activation energies are similar to those for diffusion and reorientation processes in ice and hydrates, with the rate increasing with temperature as for any activated process. For the processes that involve surface reactions, it is generally observed that the rate of hydrate formation increases with decreasing temperatures. This is especially so if simple first order rate laws are applied to the kinetic data. This observation is consistent with the view of the initial step of hydrate formation as an adsorption process rather than as a reaction involving an activated transition state. Our results show that for the experiments of xenon hydrate formation on ice surfaces, the kinetics fall strictly within the domain of the adsorption model. Diffusion-limited conversion can be expected to become important when the hydrate layer becomes thicker than ∼1000 Å, with the process taking place on a much longer time scale. Conclusions Using HP 129Xe NMR, the formation of Xe clathrate hydrate on the surface of ice was monitored. It was found that the kinetics of hydrate formation depends on the temperature, xenon pressure, and the history of the surface treatment. The relative concentration of occupied small cages was found to be higher at the initial moments after adsorption, which could indicate a special role for the 512 cage in the formation of Xe hydrate precursor. Induction periods of 10-100 s were observed preceding the formation of the surface hydrate, dependent on the temperature and pressure of the gas. The induction period is dramatically reduced for surfaces where the Xe hydrate was previously formed and then decomposed by evacuation. We explain these observations in terms that the decomposition of the surface hydrate leaves the surface “preorganized”, therefore eliminating the induction period for the consecutive hydrate formation. Overall, xenon hydrate formation on the surface of ice can be described by Avrami-Erofeyev kinetics, which emphasizes the importance of the nucleation step. References and Notes (1) Sloan, E. D., Jr. Clathrate Hydrates of Natural Gases, 2nd ed.; Marcel Dekker: New York, 1996. (2) (a) Ripmeester, J. A.; Tse, J. S.; Ratcliffe, C. I.; Powell, B. Nature 1987, 325, 135. (b) Ripmeester, J. A.; Ratcliffe, C. I.; Tse, J. S. J. Chem. Soc., Faraday Trans. 1 1988, 84, 3731. (3) Hammerschmidt, E. G. Ind. Eng. Chem. Res. 1934, 26, 851. (4) (a) Long, J. P.; Lederhos, J. P.; Sum, A.; Christiansen, R. L.; Sloan, E. D., Jr. Proceedings of the 73rd Gas Processors Association Annual Convention, New Orleans, LA, March 7-9, 1994. (b) Larsen, R.; Makogon, T.; Knight, C.; Sloan, E. D., Jr. Proceedings of the 2nd International Conference on Natural Gas Hydrates, Toulouse, France, June 2-6, 1996. (5) Lederhos, J. P.; Long, J. P.; Sum, A.; Christiansen, R. L.; Sloan, E. D., Jr. Chem. Eng. Sci. 1996, 51, 1221. (6) Englezos, P. ReV. Inst. Fr. Petr. 1996, 51, 789. (7) Vysniauskas, A.; Bishnoi, P. R. Chem. Eng. Sci. 1983, 38, 1061. (8) Lekvam, K.; Ruoff, P. J. Am. Chem. Soc. 1993, 115, 8565. (9) Lekvam, K.; Ruoff, P. J. Cryst. Growth 1997, 179, 618. (10) Malegaonkar, M. B.; Dholabhai, P. D.; Bishnoi, P. R. Can. J. Chem. Eng. 1997, 75, 1090. (11) Falabella, B. J. A Study of Gas Hydrates. Ph.D. Thesis, University of Massachusetts, Amherst, MA, 1975. (12) Sloan, E.; Fleyfel, F. AIChE J. 1991, 37, 1281. (13) Barrer, R. M.; Ruzicka, D. J. Trans. Faraday Soc. 1962, 58, 2262. (14) Barrer, R. M.; Edge, A. V. J. Proc. R. Soc. London 1967, A300, 1.

129Xe


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