Kristofer Paso,† M. Senra,† Y. Yi,‡ A. M. Sastry,‡ and H. Scott Fogler*,†
Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, andDepartment of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109
Incipient wax-oil gel deposits form in crude oil transport pipelines when long-chain n-paraffinsprecipitate at the cold interior surface of the pipe wall. The kinetics of paraffin gel formationwas studied using model fluids consisting of monodisperse and polydisperse n-paraffincomponents dissolved in petroleum mineral oil. Classical homogeneous nucleation theory wasapplied to investigate the supersaturation conditions necessary for crystal formation. Differentialscanning calorimetry was used to monitor paraffin crystallization rates and to provide solid-phase fraction measurements. Gelation occurs when growing n-paraffin crystals interlock andform a volume-spanning crystal network which entrains the remaining liquid oil among thecrystals. Paraffin wax-oil gels exhibit a mechanical response to an imposed oscillatory stress,which is characterized by the elastic storage modulus G′ being greater in magnitude than theviscous loss modulus, G′′. Low-temperature rheological gels can form from model fluids withn-paraffin contents as low as 0.5 wt %. Images of wax-oil gel morphologies were obtained usinga cross-polarized microscope fitted with a z-drive and indicated crystal lengths of ∼10-20 µm.A microstructural gelation model based on percolation theory was introduced to providepredictions of gel formation conditions among randomly oriented paraffin crystals. The structuralmodel provides correlations of crystal morphologies and solid fractions at the percolation thresholdcondition. Comparison of the initial wax contents required for gelation of monodisperse andpolydisperse n-paraffin wax indicates that sharp crystal edges and ordered crystal faces hinderthe paraffin crystal-crystal “anchoring” interactions which result in mechanical gelation.
1. Introduction
Paraffin deposition and wax plugging are ubiquitousproblems faced in cold-environment petroleum produc-tion, such as from arctic climates or subsea producingfields. When the temperature of a crude oil or gascondensate fluid drops below the paraffin solubilitylimit, known as the cloud point, the heaviest n-paraffinfractions precipitate out of solution as solid wax crystals.Gelation occurs when paraffin crystals interact to formvolume-spanning networks which entrain the remainingliquid oil and impart solidlike mechanical properties tothe fluid. In petroleum transport pipelines, paraffin geldeposits form at the cold interior surface of the pipe walland increase in thickness and hardness with time until,gradually, a complete blockage is formed, at an enor-mous economic loss to the producer. Successful riskabatement strategies typically incorporate a compre-hensive planning and implementation program, utiliz-ing mechanical, thermal, and chemical remediationmethods. Proper design of paraffin management sys-tems is dependent upon a priori knowledge of paraffindeposition rates and gel deposit properties. To ac-curately assess paraffin deposition tendencies, knowl-edge of pipeline flow characteristics and temperaturegradient conditions are required, which are often dif-ficult to reproduce in laboratory settings. A fundamentalunderstanding of the paraffin gelation and depositionprocess is necessary in order to appropriately scalelaboratory deposition rate measurements to field condi-tions and obtain useful forecasts.
A plethora of investigations have focused on charac-terizing low-temperature phases and solid-solid transi-tions of n-alkane waxes using X-ray diffraction,1-5
Raman and IR spectroscopy,6 and thermal analysis.7,8
Nucleation studies performed with undiluted singlecomponent n-paraffins have characterized metastablezone-widths9 and solid-liquid interfacial energies10,11
in relation to solid structural phases and surface freez-ing effects. Polydisperse distributions of n-paraffincomponents dissolved in crude oil1 or other organicsolvents12,13 assume orthorhombic equilibrium struc-tural phases. Thermodynamic models have been devel-oped which provide accurate predictions of the solid-liquid equilibrium of multicomponent paraffin fluidswhile accounting for the formation of multiple solidphases due to chain-length variations.14,15 Solid-liquid-phase equilibria models are useful in establishing thedeposition potential of a petroleum fluid on a composi-tional basis.
Paraffin deposition models have been developed bySingh et al.16,17 which provide accurate predictions ofdeposit buildup and aging rates based on pipeline heat-and mass-transfer characteristics. The aging processinvolves the molecular diffusion of n-paraffins from thebulk fluid into the occluded liquid within the deposit,where the paraffin components subsequently precipi-tate, causing an increase in the solid fraction (andhardness) of the deposit with time.18 In addition, theinfluence of shear reduction on the deposition rate hasbeen quantified and applied to predictive depositionmodels.19 Lab-scale wax deposition tests using coldfin-gers20 or flow loop systems16 are commonly used toprovide forecasts of field deposition and aging rates. Toappropriately scale the deposition process and accountfor effects such as shear reduction and kinetic limita-
* To whom correspondence should be addressed. Tel.: (734)763-1361. Fax: (734) 763-0459. E-mail: [email protected].
† Department of Chemical Engineering.‡ Department of Mechanical Engineering.
tions, flow conditions must be properly mapped betweenthe laboratory and field systems. Because of the dif-ficulty in establishing pipeline flow patterns in labora-tory-scale equipment, predictive deposition models mustbe sufficiently robust to account for varying shear andtemperature-gradient conditions and adhere to funda-mental scaling laws.
Singh et al.21 established the applicability of usingsuperimposed oscillatory rheometric measurements todetermine the gelation point of waxy fluids under shearconditions. Paraffin contents of gelled samples in therheometer provided excellent correlation to measuredincipient solid fractions of analogous flow loop depositsfor a variety of shear conditions. Venkatesan et al.22
demonstrated that the rheometric gelation temperatureprovides a more useful definition of the onset of gelationthan the commonly used pour point temperature. Theimportance of gelation in the wax deposition process hasbeen established by Singh et al.16,17,21 as well as byVenkatesan et al.22 Gelation is responsible for theformation of incipient wax deposits in petroleum trans-port flowlines, as well as for the occlusion of liquid oilat the deposit-fluid interface during deposit growth.
In the work presented here, the role of the kineticsof formation of incipient wax-oil gels is assessed byapplying the fundamentals of homogeneous nucleation,crystallization, and gelation theory. Model fluids con-sisting of n-paraffin components dissolved in petroleummineral oils are formulated in order to provide insightinto the crystal-crystal interactions which ultimatelylead to gel formation. The experimental methods ofdifferential scanning calorimetry, rheometry, and cross-polarized microscopy provide measures of the crystal-lization rate, mechanical properties, and crystal mor-phologies, respectively. One of the primary limitationsof current predictive deposition models is the assump-tion of the complete absence of kinetic limitations in therates of incipient gel formation and deposit buildup,which allows for the direct application of scaled heat-and mass-transfer correlations based on flow charac-teristics. Neglecting gelation kinetics considerationsmay lead to predictions of wax deposition in cases wherea stable gel cannot form, resulting in unnecessarycapital expenditures related to paraffin remediation andcontrol systems. The current research lays the ground-work for assessing kinetic limitations in paraffin crys-tallization and gelation rates, which may be applied toexisting deposition models and may ultimately resultin less conservative wax deposition predictions. Theanalytical methods developed here using simple modelfluids consisting of wax dissolved in mineral oil can bereadily extended to the real waxy petroleum fluidsencountered in production systems.
2. Experimental Section
2.A. Model Fluids. Wax-oil gels were prepared frommodel fluids consisting of n-paraffin waxes dissolved inmineral oil. The waxes were obtained from Sigma-Aldrich and include a polydisperse wax (mp 65 °C), n-C36(mp 75 °C), and n-C35 (mp 74 °C). The carbon numberdistribution of the polydisperse wax, analyzed using anAgilent 6890N gas chromatograph with a 0.25 µm fusedsilica stationary phase and an FID detector, is shownin Figure 1. The mineral oils include Coray-15 lubricat-ing oil obtained from ExxonMobil, as well as a heavymineral oil obtained from Sigma-Aldrich. Coray-15exhibits an average molecular weight of 290 and a
viscosity at 25 °C of 29 cP, while the heavy mineral oilexhibits an average molecular weight of 424 and aviscosity at 25 °C of 143 cP. Model oils were formulatedwith either monodisperse or polydisperse n-paraffinconcentrations ranging from 0.5% to 8% in the mineraloils. Cloud point solubility measurements were per-formed by heating the samples until a transparentliquid was obtained and subsequently placing thesamples in a temperature-controlled water bath for atime period of at least 1 h. Paraffin crystal formationwas detected by visual turbidity observation. For eachmodel fluid, the procedure was repeated until the cloudpoint temperature was determined within an accuracyof (0.1 °C.
2.B. Differential Scanning Calorimetry. Heat flowmeasurements during cooling were performed using aPerkin-Elmer DSC-7 instrument to monitor crystalliza-tion kinetics and provide accurate measures of solid-phase fractions under quiescent conditions. Model fluidsamples of at least 25 mg were carefully prepared inaluminum sample pans and weighed. Sample chambercooling was provided by pumping water through an icebath, which limits the operating temperature range ofthe heat flow measurements in accordance with thetemperature difference necessary to sustain samplecooling. At cooling rates of 5, 1, and 0.5 °C/min, thelower operating temperature limits are 20, 5.5, and 3.6°C, respectively. To obtain accurate heat flow measure-ments, a linear heat capacity baseline was first estab-lished by performing a cooling scan with mineral oil inthe sample pan. Subsequently, heat flow measurementswere performed with the formulated model oils uponsample cooling from a temperature of 70 °C to the lowertemperature limit of heat flow detection at rates of 5,1, and 0.5 °C/min. Nucleation temperatures were ob-tained by identifying the temperature at which themeasured heat flow diverges from the sample heatcapacity baseline. The heat flow attributed to crystal-lization was obtained by subtracting a linear heatcapacity interpolation from the measured heat flow. Theintegrated heat of crystallization was normalized to theappropriate crystallized fraction at the lower tempera-ture limit.
2.C. Rheometry. A controlled stress rheometer (TAInstruments AR1000) was used for the rheologicalexperiments. The instrument was configured with a 4cm, 1.59° steel cone and equipped with a Peltier platetemperature-control device. During each run, a modelfluid sample was contained between the cone and the
Figure 1. n-Paraffin carbon number distribution of the mp 65°C wax.
Ind. Eng. Chem. Res., Vol. 44, No. 18, 2005 7243
Peltier plate, and a single-frequency oscillatory stresswas imposed upon the sample while the temperaturewas cooled at a constant rate starting from a temper-ature of ∼10 °C above the respective fluid cloud point.Each rheometric experiment was performed using anoscillatory frequency of 0.1 or 1 Hz and a constantoscillatory stress amplitude within the range of 0.01 to1 Pa. The mechanical response was characterized by G′,the elastic storage modulus; G′′, the dissipative lossmodulus; and the oscillatory strain amplitude.
2.D. Cross-Polarized Microscopy. A Nikon EclipseE600 microscope fitted with a 50X objective lens andequipped with a “z-drive” automatic focusing controllerwas used to obtain 3-D images of the wax-oil gels.Digital images were captured with a Hamamatsu ORCAER digital camera. The 2-D domain captured in eachimage represents a rectangular area of 131 × 172 µm.Model oil samples were first cooled at 0.5 °C/min from∼10 °C above the respective fluid cloud point to 10 °C.For the 0.5% polydisperse wax model fluid, the finaltemperature was reduced to 7.6 °C in order to providea sufficiently high solid-phase fraction to establishcrystal sizes. During the imaging process, the autofo-cusing controller was synchronized with the digitalcamera to rapidly obtain 2-D cross-polarized images at80 focal planes, spaced 1 µm apart, for a total distancein the z-direction of 80 µm. Reconstruction software(SimplePCI from Compix, Inc.) was used to create 3-Dimages from the 2-D z-slices by calculating imageprojections viewed from angles of 0°-360° at 5° inter-vals. Reconstructed 3-D images can be viewed in con-tinuous rotational mode with the software.
3. Results and Discussion
3.A. Paraffin Solubilities. Paraffin solubilities ob-tained from cloud point measurements are correlatedin Figure 2. The enthalpy and entropy of dissolution areobtained by applying experimental cloud points to thevan’t Hoff relation23
where xsol is the soluble n-paraffin mole fraction,∆Hdissolution is the dissolution enthalpy (kJ/mol), and∆Sdissolution is the dissolution entropy (kJ/mol/K).23 Thevan’t Hoff relation assumes ideal solubility behavior anda negligible heat capacity change upon crystallization.
Table 1 shows dissolution enthalpy and entropy valuesfor the paraffin waxes in mineral oil. The van’t Hoffrelation provides a robust thermodynamic solubilitymodel for single-component n-paraffins dissolved inorganic solvents at low mass fractions. Dissolutionenthalpy and entropy values associated with the n-C36component are nearly independent of the mineral oilviscosity. When the van’t Hoff correlation is applied tosolubility measurements of polydisperse distributionsof n-paraffin components in solution, significant uncer-tainty exists in the xsolubility values because only thehighest carbon numbers contribute to the precipitatedphase at the initial nucleation point; midrange n-paraffin components remain in the liquid phase at thenucleation point. However, if the upper n-paraffinfraction which contributes to the first precipitated phaseis considered a pseudocomponent in constant proportionto the total n-paraffin distribution, the left-hand sideof eq 1 can be expressed as ln(1/xsolubility) + ln(1/c) forpolydisperse n-paraffin model fluids. The constant crepresents the molar fraction of the wax solute whichconstitutes the upper pseudocomponent; xsolubility repre-sents solubility measurements based on the initial waxfraction of the model fluid on a molar basis. Applicationof the pseudocomponent assumption provides an esti-mate of the dissolution enthalpy of the heaviest paraffinfraction of 107.1 kJ/mol, indicating an average chainlength of ∼36-38 in the first precipitated phase of thepolydisperse wax fluid.
3.B. Classical Homogeneous Nucleation. The su-persaturation conditions necessary to induce nucleationare investigated by application of the homogeneousnucleation theory. For simple model fluids consistingof refined paraffins dissolved in mineral oil, the nucle-ation mechanism is homogeneous. The supersaturationratio can be expressed as
where C represents the n-paraffin concentration insolution and Ceq represents the equilibrium n-paraffinconcentration, provided by the van’t Hoff solubilitytheory. Experimental supersaturation values at thenucleation point are obtained by establishing equilib-rium paraffin concentrations based on DSC-measurednucleating temperatures. A generalized form of thenucleation rate, applicable to solutions, can be expressedas24
where σ represents the surface energy of the criticalnucleus, ν represents the molecular volume of thenucleating species and is a weak function of tempera-ture, and k represents the Boltzman constant. Thepreexponential factor A is commonly known as thecollision factor and is theoretically related to an equi-
Figure 2. n-Paraffin solubilities of monodisperse and polydispersemodel fluids, determined from visual cloud point observations.
ln( 1xsol
) )∆Hdissolution
RT-
∆Sdissolution
R(1)
Table 1. Paraffin Dissolution Enthalpy (kJ/mol) andEntropy (kJ/mol/K) Values Derived from the Van’t HoffSolubility Relation,23 Based on Experimental CloudPoints
librium precritical cluster size distribution. When amodel paraffin fluid is supercooled under a constantcooling rate, the nucleation rate will increase with timebecause of the changing temperature and supersatura-tion conditions. Consequently, to establish the nucle-ation point dependence upon cooling rate, the nucleationrate expression must be integrated over the nucleationinduction time. The integrated nucleation expressionyields a nuclei number density.24
tcp represents the time at which the sample temperaturereaches the cloud point; tn represents the time at whichthe sample nucleates. The concept of a critical nucleinumber density, Fn*, is established to define the nucle-ation point within the framework of homogeneousnucleation theory and to facilitate the comparison ofexperimental nucleation point measurements. Nucle-ation is assumed to occur when the critical nucleinumber density is attained. Integration of the nucle-ation rate expression is performed by utilizing the van’tHoff solubility relation in the supersaturation term ofthe exponential denominator. Details of the homoge-neous nucleation theory development are provided inthe Appendix. Application of the nucleation theoryframework to measured nucleation points providescorrelated values of σ and A/Fn*.
Figure 3 shows experimental and fitted theoreticalsupersaturation ratios at the nucleation point for themodel fluids consisting of the single component n-C36in mineral oil. Each individual line representing nucle-ation theory corresponds to specific values of A/Fn* andσ. Hence, A/Fn* and σ depend on the fluid composition.The results indicate that slightly larger supersaturationratios are necessary to induce nucleation at highercooling rates. Low initial wax fractions also result inhigher supersaturation ratios at the nucleation point.Comparison of supersaturation ratios in the high- andmedium-viscosity mineral oils (Figure 3) indicates anabsence of transport limitations in the nucleation kinet-ics. To provide nucleation correlations for the polydis-perse model fluids, supersaturation calculations arebased on only the pseudocomponent which contributesto the first precipitated fraction, as defined by applica-tion of the van’t Hoff solubility correlation to measuredcloud points. Figure 4 shows experimental and theoreti-cal supersaturation ratios at the nucleation point for
the polydisperse wax in Coray-15. Similar to the caseof single-paraffin-component fluids, low initial waxfractions and high cooling rates result in increasedsupersaturation ratios at nucleation. Optimized surfaceenergy values obtained by fitting the homogeneousnucleation theory expressions to the supersaturationratios (Figures 3 and 4) measured by DSC (see Figures6-8) are shown in Table 2. In all cases, surface energiesassociated with polydisperse paraffin critical nuclei are
Figure 3. Supersaturation conditions at nucleation for monodis-perse model fluids. Solid lines represent homogeneous nucleationtheory.
Fn ) ∫tcp
tnJ dt (4)
Figure 4. Supersaturation conditions at nucleation for polydis-perse wax in Coray-15. Solid lines represent homogeneous nucle-ation theory.
Figure 5. Linearized nucleation rate expression. Solid data pointsrepresent theoretical ln(F*/J) values plotted versus experimentalsupersaturation ratios.
Figure 6. DSC-measured crystallization rate for 4.0% n-C36 inCoray-15 at a cooling rate of 1 °C/min. The thick solid curverepresents the equilibrium crystallization rate, and the solidvertical line represents the cloud point temperature.
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lower than surface energies of the single-componentn-C36 critical nuclei. Linearized nucleation rates areshown in Figure 5 for the 4% initial wax fraction fluids.
3.C. Crystallization Kinetics. Paraffin crystalliza-tion was investigated at constant cooling rate conditionsby DSC heat flow measurements. The enthalpy of fusionrelates heat flow to the paraffin crystallization rate.Differences in liquid- and solid-phase heat capacitiescause the enthalpy of fusion to vary with temperatureaccording to the following relation
where ∆Hf,i represents the enthalpy of fusion of com-ponent i, Tm,i represents the melting temperature of thepure component i, and Cp,L,i and Cp,S,i represent theliquid- and solid-phase paraffin heat capacities of the
ith component, respectively. The correlations for n-paraffin melting enthalpy (cal/mol) and heat capacity(cal/mol/K) are obtained from Petersen et al.25 as follows:
In these correlations, MWi represents the n-paraffinmolecular weight and ∆Cp,i
L-S represents the heat ca-pacity difference between the liquid and solid phase ofthe pure-component paraffin. We will let X representthe mass fraction of paraffin in solution. The experi-mental crystallization rate, expressed as -dX/dt, isobtained by division of the crystallizing heat flow by theenthalpy of fusion and is expressed in terms of massfraction (of the total wax and oil present) crystallizedper minute. Figure 6 shows the experimental crystal-lization rate for the case of 4% n-C36 in Coray-15. Alsoplotted is the theoretical equilibrium crystallizationrate, which is established assuming the complete ab-sence of kinetic limitations in the crystallization rate.The equilibrium crystallization rate, -dXs/dt, is calcu-lated by the following equation using the chain rule ofdifferentiation
The term dXs/dT is obtained by differentiating the massfractional form of the van’t Hoff equilibrium solubilityrelation with respect to temperature. The cooling rateis represented by dT/dt. Comparison of experimentaland equilibrium crystallization rates indicates threeregimes in the crystallization process at low coolingrates. The first regime, starting from high-temperatureconditions, is a nucleation lag period, characterized bythe onset of supersaturation conditions but without theformation of a solid phase. The second regime, asupersaturation growth period, is driven by the super-saturation established during the nucleation lag periodas well as by continuing decreasing solubility conditions.The third regime, an equilibrium growth period, ensueswhen the supersaturation ratio is diminished and thecrystallization rate converges with the equilibriumpredictions. The accurate match between experimentaland equilibrium crystallization rates in the equilibriumgrowth regime attests to the robustness of the van’t Hoffequilibrium solubility model. Figure 7 shows crystal-lization rates associated with monodisperse n-C36 modelfluids of varying paraffin fractions during cooling at 1°C/min. Because the equilibrium crystallization rate (eq9) is independent of the initial paraffin fraction, a singlemaster equilibrium crystallization curve applies to allinitial paraffin fractions at a particular cooling rate. Ineach case, the nucleation lag period results in a super-
a Surface energies and A/Fn* derived from the nucleation relation expressed in eq 4 applied to experimental nucleation points obtainedfrom DSC measurements.
Figure 7. Crystallization rates obtained by DSC for n-C36 inCoray-15 at a cooling rate of 1.0 °C/min. The thick solid linerepresents the equilibrium crystallization rate, and the dashedvertical lines represent the respective cloud point solubility limits.
Figure 8. Crystallization rate and solid fraction obtained by DSCfor 4% polydisperse wax in Coray-15 at a cooling rate of 1 °C/min.The dashed vertical line represents the cloud point solubility limit.
Tm,i ) 374.5 + 0.02617MWi - 20 171/MWi (6)
∆Hf,i ) 0.1426MWiTm,i (7)
∆Cp,iL-S(T) ) 0.3033MWi - (4.635 × 10-4)MWiT (8)
-dXs
dt) -
dXs
dTdTdt
(9)
∆Hf,i(T) ) ∫T
Tm,iCp,L,i(T) dT + ∆Hf,i(Tm,i) +
∫Tm,i
TCp,S,i(T) dT (5)
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saturation growth period in which the crystallizationrate temporarily spikes well above the equilibriumcrystallization rate. Subsequently, the crystallizationrate converges with the equilibrium crystallization ratepredicted by the van’t Hoff relation.
Figure 8 shows the crystallization rate of the 4%polydisperse wax in Coray-15 at a cooling rate of 1 °C/min. A nucleation lag period and a supersaturationgrowth regime are clearly evident in the high-temper-ature range. Because of the broad distribution of n-paraffin components in the polydisperse wax modelfluid, the equilibrium crystallization rate does notexhibit an exponential profile as in the monodispersecase. Instead, the total crystallization rate reflects a sumof crystallization rates associated with a range ofn-paraffin components and exhibits a nearly linearprofile in the low-temperature range. The abundantpresence of n-paraffin carbon number components in themid-20s shifts a large fraction of the crystallizationactivity to low single digit temperatures, below theoperating limits of the DSC instrument. Therefore, forthe polydisperse wax model fluids, the low-temperaturecrystallization rates are linearly extrapolated to zero inorder to provide a closed form integral for solid fractiondetermination. Calculated solid-phase fractions of mon-odisperse and polydisperse wax model fluids are appliedin gelation correlations.
When comparing measured crystallization profileswith the master equilibrium crystallization rate curveshown in Figure 7, it is evident that the crystallizationpeaks are a result of the supersaturation generatedduring the nucleation lag regime. Therefore, at lowcooling rate conditions, homogeneous nucleation limita-tions represent the primary deviation of real crystal-lization rates from equilibrium crystallization ratepredictions. Within the supersaturation growth regime,the crystallization rate is limited by the rate of paraffinchain addition to the growing crystal faces. The durationof the supersaturation growth regime is established bydefining the temperature span of the supersaturationgrowth regime as ∆T ) Tnuc - Teq.cryst, where Teq.crystdenotes the temperature at which the crystallizationrate converges with the equilibrium crystallization rateprediction. Figure 9 shows supersaturation growthregime ∆T widths for the 4% n-C36 and 4% polydispersewax model fluids. The ∆T temperature span of thesupersaturation growth regime follows a linear depen-dence upon cooling rate, reflecting a constant super-
saturation growth time of ∼4.2 min before equilibriumcrystallization is achieved. The ∆T temperature spanof the supersaturation growth regime is independent ofthe model fluid viscosity, providing evidence of theabsence of transport limitations in the crystallizationrate.
3.D. Gelation Kinetics. Changes in the mechanicalproperties of waxy model fluids were investigated atconstant cooling rate conditions by using controlled-stress rheometric measurements. To characterize themechanical properties of model fluids during the gela-tion process, an oscillatory stress is imposed upon thefluid sample. In general, mechanically weak gels, formedfrom low wax-content fluids, require a smaller imposedoscillatory stress amplitude in order to obtain a quan-tifiable mechanical response. An imposed oscillatorystress amplitude of 0.1 Pa at a frequency of 0.1 Hz wassufficient to monitor the mechanical response duringcooling of all Coray-15-based model fluids with theexception of the 4% polydisperse wax model fluid. Atthe midrange cooling rates of 1 and 2 °C/min for the4% polydisperse wax fluid, the characteristic timeassociated with the initial strain amplitude reductioninterferes with the oscillatory periodicity of the imposedstress, resulting in unquantifiable G′ and G′′ values nearthe gel point. Therefore, to obtain a quantifiable me-chanical response, an imposed oscillatory stress ampli-tude of 1 Pa at a frequency of 1 Hz was utilized at themidrange cooling rates for the 4% polydisperse waxfluid.
When the fluid temperature is maintained above thecloud point, a Newtonian mechanical response is ob-tained, characterized by a finite value of the viscous lossmodulus, G′′. After the initial nucleation event, thepresence of solid paraffin crystals causes an increasein the viscosity of the model fluid, resulting in increasedvalues of the measured loss modulus, G′′. The formationof random interaction networks among the growingparaffin crystals imparts a solidlike mechanical re-sponse to the fluid, which is characterized by finitevalues of the elastic storage modulus, G′. Figure 10shows changes in the mechanical moduli and oscillatorystrain amplitude associated with paraffin crystallizationin the 0.5% n-C36/Coray-15 model fluid at a cooling rateof 0.5 °C/min. At temperatures below the fluid cloudpoint of 26.1 °C, an increase is observed in the viscousloss modulus, G′′, which correlates to a decrease in theoscillatory strain amplitude. The formation of crystal-
Figure 9. Temperature widths of the supersaturation coolingregime for polydisperse and monodisperse model fluids of 4%initial wax fraction. The fitted line corresponds to a supersatura-tion regime time span of 4.2 min.
Figure 10. Mechanical response of 0.5% n-C36 in Coray-15resulting from an imposed oscillatory shear stress of 0.1 Pa at afrequency of 0.1 Hz (cooling rate ) 0.5 °C/min).
Ind. Eng. Chem. Res., Vol. 44, No. 18, 2005 7247
crystal interactions is detected at temperatures below22.3 °C by finite values of the elastic storage modulus,G′. As the temperature is further lowered, the degreeof interaction among the wax crystals increases, whichis reflected in an increase in the storage modulus G′.However, the value of the elastic storage modulus doesnot exceed the value of the loss modulus, indicating theabsence of a volume-spanning crystal network. Thesample behaves as a viscous fluid with a small degreeof elasticity at low temperatures.
Figure 11 shows rheometric data of the 0.5% poly-disperse wax in Coray-15 sample obtained at a coolingrate of 0.5 °C/min. A crossover in the values of G′ andG′′ occurs at an interpolated temperature of 14.73 °C,defining the experimental gelation temperature of thefluid and confirming the formation of a continuousparaffin crystal interaction network. The reduction inoscillatory strain amplitude associated with cooling thepolydisperse wax fluid from above the cloud point to ∼0°C is ∼3 orders of magnitude, confirming the gellikemechanical properties of the polydisperse paraffin modelfluid at low temperatures. The strain amplitude reduc-tion of the analogous monodisperse n-C36 model fluid(Figure 10) is ∼1 order of magnitude and is primarily aresult of the viscosity increase associated with crystal-lization.
Rheometric gelation temperatures were establishedat cooling rates ranging from 0.5 to 5 °C/min for themodel fluids consisting of n-C35, n-C36, and the polydis-perse n-paraffin wax in Coray-15. The gel point tem-perature exhibits a dependency upon the imposedoscillatory stress amplitude, as shown in Figure 12 forthe 0.5% polydisperse wax fluid. An increase in theoscillatory stress amplitude from 0.01 to 0.1 Pa at afrequency of 0.1 Hz causes a 1.4 °C depression in theexperimental gel point temperature, demonstrating thatgel point temperatures obtained at dissimilar imposedstress conditions cannot be directly compared. The geltemperature reduction indicates a shear degradationeffect on the gel structure, in which the imposed stresshinders the formation of a volume-spanning crystalnetwork. When a paraffin crystal network begins toform under an imposed stress, the occluded liquid isunable to bear a mechanical load, and the applied stressis localized to the low-density crystal linkages. Asample-spanning network of crystal interactions cannotform unless the density of crystal linkages is sufficientto sustain the maximum mechanical load associated
with the applied oscillatory stress. When a higher stressis imposed upon the sample during the gelation process,a higher density of crystal linkages is necessary tosustain the applied stress and provide a gellike me-chanical response. Hence, when the imposed oscillatorystress amplitude increases, the gel point is depressedto lower temperatures corresponding to larger solidfractions. Figure 12 shows experimental gel point solidfractions for the 0.5% polydisperse wax fluid, confirmingthat larger solid fractions are necessary to gel modelwaxy fluids under higher imposed oscillatory stressconditions.
To facilitate comparison of gel point temperaturesobtained under the dissimilar frequency and stressamplitude conditions for the 4% polydisperse wax modelfluid, a correction is established to account for the geltemperature depression effect of changing the rheo-metric conditions from 0.1 Pa and 0.1 Hz to 1 Pa and 1Hz. At the higher stress conditions, experimental gelpoint temperatures are reduced by 0.5 and 0.67 °C atcooling rates of 0.5 and 5 °C/min, respectively. Applyinga linear interpolation to the gel temperature reductionsbetween 0.5 °C and 5 °C, gel temperature reductions of0.52 and 0.56 °C are predicted for cooling rate conditionsof 1 and 2 °C/min, respectively. The corrections for thegel temperature depression effect are applied to the gelpoint temperatures obtained at the higher stress andfrequency conditions, such that gel temperature cor-relations remain consistent with the imposed 0.1 Pa and0.1 Hz rheometric condition.
Figure 13 shows gelation temperature dependenciesupon cooling rate for the polydisperse model fluidsconsisting of 0.5, 1, and 4 wt % of paraffin in Coray-15.Experimental gel point temperatures decrease withincreasing cooling rate; a stronger cooling rate depen-dence is exhibited at lower initial wax fractions. Theoffset in gel point temperatures of the various paraffinweight fraction fluids correlates to solubility differences.Figure 14 shows gel point temperatures of the mono-disperse paraffin fluids consisting of 1% and 4% weightfractions of n-C35 and n-C36 in Coray-15 mineral oil.Stronger cooling rate dependencies are observed for then-C36 fluids than for the n-C35 fluids, indicating astronger time-dependency in the formation of the crystalinteraction network of the n-C36 paraffin fluid. For the0.5% n-C36 fluid, no crossover in G′ and G′′ values wasobserved at any cooling rate. For the 0.5% n-C35 fluid,no crossover in G′ and G′′ values was observed at 0.5°C/min, although poorly reproducible gel point temper-
Figure 11. Mechanical response of 0.5% polydisperse wax inCoray-15 resulting from an imposed oscillatory shear stress of 0.1Pa at a frequency of 0.1 Hz (cooling rate ) 0.5 °C/min).
Figure 12. Gel point temperatures and correlated solid fractionsfor 0.5% polydisperse wax in Coray-15 at a frequency of 0.1 Hzfor varying shear stress amplitudes (cooling rate ) 0.5 °C/min).
7248 Ind. Eng. Chem. Res., Vol. 44, No. 18, 2005
atures were obtained at higher cooling rates. The poorreproducibility is a result of the fragile gel structure aswell as the finite sample volume of the rheometer.
Gelation temperatures obtained from the controlled-stress rheometric measurements are correlated withsolid fractions obtained by integration of DSC-measuredcrystallization rates, providing qualitative correlationsof the solid-phase fraction at the gelation temperature.Table 3 provides incipient solid-fraction estimates ofmonodisperse n-C36 and n-C35 as well as polydispersemodel fluids. The large dependence of incipient gel solidfractions upon cooling rate for the 4% monodispersen-C36 and n-C35 cases is an indication of time-depen-dency in the crystal aggregation and network formation.The even carbon number n-C36 paraffin exhibits astronger time-dependency than that of the odd carbonnumber n-C35. In the case of the polydisperse wax fluids,the observed reductions in incipient gel solid fractionwith increasing cooling rate result from the sheardegradation effect. At the lower cooling rate, 0.5 °C/min,
the imposed shear degrades the crystal network struc-ture for a longer time duration during the gelationprocess, requiring an increased solid fraction at theincipient gel point in order to obtain a volume-spanningcrystal network. The results suggest that fundamentaldifferences exist in the morphology and/or structuraldynamics of monodisperse and polydisperse n-paraffinwax-oil gels. To investigate the microstructural mech-anism of gelation, cross-polarized images are obtainedof the wax-oil gels.
3.E. Cross-Polarized Imaging. The 3-D imagesseen on the computer by rotating the sample are difficultto see in 2-D. Consequently, the description that followswill contain information that is difficult to deduce fromFigure 15. Figure 15a shows a composite image of 0.5%n-C36 in Coray-15 at a temperature of 10 °C, corre-sponding to a DSC-measured solid fraction of 0.47%,indicating nearly complete crystallization of the addedn-paraffin. The image is obtained by reconstruction ofthe 80 single z-slice images spaced at 1 µm intervals,calculated about a rotational angle of 0°. The crystalswhich are visible in the 0° composite image are posi-tioned at varying z-depths in the 3-D sample domain.The monodisperse crystals depicted in Figure 15a arerandomly oriented and exhibit minimal crystal-crystalinteraction, which is confirmed by rheometric measure-ments indicating the elastic storage modulus is of lowermagnitude than the viscous loss modulus throughoutthe entire temperature range. A small curvature in thecrystal faces is evident for the monodisperse n-C36crystals, which is the result of dislocations in thelamellar structure of the solid phase.
Figure 15b shows a 0° rotation composite image ofthe 0.5% polydisperse wax in Coray-15 at a temperatureof 7.6 °C, corresponding to a DSC-determined solid-phase fraction of 0.44%, indicating nearly completecrystallization. Correlated rheometric measurementsindicate a gellike mechanical response at temperaturesbelow 14.58 °C. The polydisperse crystals appear toexhibit less separation than the monodisperse crystals.Image analysis results shown in Table 4 indicate asmaller average crystal length and width of the poly-disperse crystals compared to the monodisperse crystalsof 0.5% initial wax fraction. Figure 15c shows a 0°rotation composite image of the 4% polydisperse waxin Coray-15 at a temperature of 10 °C, correspondingto a DSC-determined solid-phase fraction of 3.66% anda gellike mechanical response. The formation of avolume-spanning crystal network which results in me-chanical gelation is clearly evident from the compositeimage. Because of the difficulties in representing re-constructed 3-D wax crystal images, a 2-D schematic ofthe crystal orientation concepts is provided based onobservations of the crystal images in continuous rota-tional mode. Figure 16a represents the condition inwhich paraffin crystals have formed in solution, butmaintain minimal crystal-crystal interaction, such asexhibited by the monodisperse crystals. Conversely,
Table 3. Solid Fraction Estimates at the Incipient Gel Point Temperaturea
5 N/A N/A N/A 0.0095 0.031 0.0098 0.0058a Gel point temperatures which are out of range of DSC solid fraction measurements are denoted by N/A.
Figure 13. Experimental gelation temperatures for polydispersemodel fluids. The dashed horizontal lines represent the respectivecloud point solubility limits.
Figure 14. Experimental gelation temperatures for monodispersemodel fluids. Cloud point temperatures are indicated by the dashedhorizontal lines.
Ind. Eng. Chem. Res., Vol. 44, No. 18, 2005 7249
Figure 16b represents a volume-spanning crystal inter-action network which forms a strong gel and entrains
the remaining liquid oil within the crystal structure, asexhibited by the polydisperse crystals.
Several limitations are inherent to the use of cross-polarized imaging in determining n-paraffin crystaldimensions. Although 2-D z-slice images are obtainedat single focal planes spaced at 1 µm intervals, therotated polarized light intensity originating from acrystalline particle is projected across several focalplanes. Higher intensity images are projected across anincreased number of focal planes, producing diffuseshadow images in the z-direction of the originatingparticle. Hence, the sharpest 2-D composite crystalimages are obtained at 0° rotation angles where thediffuse shadow images are concealed by genuine crystalimages. Light intensity projection in the z-directionresults in distortion of crystal dimensions. For crystalorientations in which the primary crystal face normalvector is parallel to the focal plane, a high intensity ofrotated polarized light is produced because of crystalalignment with the source illumination. The high polar-ized light intensity correlates to an apparent elongationof the crystal in the z-dimension. Paraffin crystalsoriented such that the primary crystal face normalvector forms an angle with the focal plane exhibitdistortions in apparent crystal width as well as inz-dimension length. Light intensity projected acrossadjacent focal planes increases the illumination widthassociated with the in-focal-plane crystal position. Hence,
Figure 15. A composite image obtained using a cross-polarizedmicroscope fitted with a 50X objective lens of (a) 0.5% n-C36 inCoray-15, (b) 0.5% polydisperse wax in Coray-15, (c) 4% poly-disperse wax in Coray-15.
Figure 16. (a) An artist’s rendition from 3-D images of freecrystals in solution. The crystals do not form a volume-spanningcrystal interaction network. (b) An artist’s rendition from 3-Dimages of physical gelation by paraffin crystals. The crystals forma volume-spanning crystal interaction network.
7250 Ind. Eng. Chem. Res., Vol. 44, No. 18, 2005
the image analysis crystal aspect ratios provided inTable 4 do not reflect accurate crystal dimensions.
Paraffin birefringence properties favor illuminationof crystals oriented such that the in-plane componentof the crystal face normal vector forms a 45° angle witheither the x-axis or y-axis of the focal plane. Paraffincrystals aligned vertically or horizontally in the x-yplane are not captured by cross-polarized imagingtechniques. Hence, exact measures of crystal lengths arenot provided by standard image analysis of luminescentdomains in 2-D cross-polarized images. Instead, theupper bound to the range of crystal lengths observed ina cross-polarized image may provide a more accurateestimate of the primary crystal dimension. Orientationalimaging effects must be considered when correlatingexperimental gelation conditions with percolation theorypredictions.
3.F. Percolation Theory Application. To providefundamental insight into the structures and morphol-ogies of wax-oil gels formed from randomly nucleatedwax crystals, an extension is made to an establishedthree-dimensional analytical percolation approximationand is applied to wax-oil gel systems. The percolationthreshold is considered to be the fractional volume ofthe solid crystalline phase at which it forms a continu-ous, domain-spanning path connected by crystal-crystalinteractions. Robust three-dimensional percolationthreshold points have been established for only a fewsimple particle geometries.26 Geometric assumptionsmust be made concerning paraffin crystal morphologiesin order to utilize percolation predictions for wax-oilgels. In this work, ellipsoidal geometries with primaryaxis R1, R2, and R3 are used to represent paraffincrystals, as shown in Figure 17. If the two longestprimary axes are equal in length, the geometry is anoblate spheroid. Conceptually, the ellipsoid geometriesare rotated about their origin in order to map out thespherical rotational volume of interaction.27 Thus, the
paraffin crystals are treated as spherical particle do-mains, which allow the use of spherical percolationthreshold models to predict gelation conditions. Figure18 shows a visual representation of the “crystal interac-tion volume” concept used in applying the percolationthresholds. Theoretical gelation is equated to the condi-tion of an unbounded spherical interaction network. Ananalytical percolation threshold for randomly orienteduniform overlapping spheres26 of 0.295 is used to obtaingelation correlations. Implementation of the crystalinteraction volume for ellipsoid geometries provides thefollowing prediction of the solid-phase fraction necessaryfor theoretical gelation:
In this relation, φg represents the solid-phase fractionat the percolation threshold, θp ) 0.295 represents thespherical percolation threshold, and R1 and R2 representthe primary and secondary ellipsoidal aspect ratios.
Morphological and orientational assumptions must bemade concerning individual crystals in order to provideinsightful correlations between percolation theory andcrystal morphologies in the gel. Application of thespheroid (i.e., R2 ) 1) approximation to paraffin crystalsallows the crystal length to be established as themaximum individual 2-D crystal lengths observed in thecross-polarized images, circumventing distortions as-sociated with shadow image projection in the z-direction.In addition, it is assumed that paraffin crystals orientedat 45° in the x-y focal plane are representative of theparaffin crystals which contribute to the percolationstructure of the wax-oil gel.
The image in Figure 15b represents a 0.44% solid-fraction polydisperse paraffin gel with maximum single-crystal lengths (at 45° orientation) of ∼16 µm. Appli-cation of percolation theory yields a spheroid primaryaspect ratio, R1, of 67, corresponding to a mean crystalthickness of ∼0.24 µm and a crystal number density of
Table 4. Paraffin Crystal Dimensions Obtained by Averaging the 2-D Image Analysis Results of 80 z-Slices (1 µmIntervals) of Each Sample
Figure 17. A pictorial representation of the “crystal interactionvolume” concept. Ellipsoid geometries are utilized to representparaffin crystals.
Figure 18. A visual representation of the “crystal interactionvolume” concept for randomly oriented paraffin crystals. A 3-Dspherical percolation threshold of 0.295 is utilized in the gelationmodel.
φg ) θp1R1
1R2
(10)
Ind. Eng. Chem. Res., Vol. 44, No. 18, 2005 7251
1.11 × 104 mm-3. By comparison, maximum monodis-perse n-C36 single-crystal lengths are ∼25 µm, corre-sponding to a spheroid primary aspect ratio of 29, amean crystal thickness of ∼0.86 µm, and a crystalnumber density of 0.27 × 104 mm-3.
Percolation prediction correlations are ideal theoreti-cal constructs that are independent of crystal-crystalinteraction dynamics. As such, percolation is a necessarybut not sufficient condition for gelation. The formationof a crystal percolation network will lead to gelation onlyif the number density and strength of the crystal-crystal interactions are sufficient to impart solidlikeproperties to the fluid. Evidence exists that crystalsurface properties influence wax-oil gel strengths. Imiaet al.28 have demonstrated that smooth crystal surfacesassociated with monodisperse n-C32 crystals form me-chanically weak gels; increased gel strengths wereobtained by mixing two paraffin chain lengths (n-C30and n-C32) to induce lamellar surface structure disorder.It is mechanistically consistent that sharp crystal edgesand smooth crystal faces associated with monodispersen-paraffin crystals provide a smaller area of interactionbetween crystal faces and edges, resulting in weakercrystal-crystal “anchoring” interactions.
Percolation threshold theory provides direct gelationcondition predictions only in the limit of ideal crystal-crystal anchoring interactions, in which rigid solid-solidinteractions dominate the fluid mechanical response.For gelation to occur in fluids with weak crystal-crystalinteractions, a larger number density of interaction sitesis necessary to compensate for the weaker interactions,necessitating an increased solid fraction at the gelationpoint. Because strong crystal-crystal interactions aredependent upon the polydispersity of the solid-phasen-paraffin, a larger crystal number density (and solidfraction) is required for gelation to occur in monodis-perse paraffin fluids with weak crystal-crystal interac-tions. Hence, a gel is able to form from the 0.5%polydisperse wax fluid, while the monodisperse n-C35and n-C36 fluids require a 1% paraffin content in orderto form coherent gels, indicating that the interactionvolume for the monodisperse fluid must be significantlygreater than the percolation threshold fraction in orderto induce gelation. Therefore, the theoretically cor-related monodisperse crystal aspect ratio of 29 under-predicts the real aspect ratio of the crystals, which mayexhibit values closer to those of the polydisperse paraffincrystals.
Beyond the morphological ambiguities associated withcrystal imaging and percolation threshold application,it is clear that polydispersity in the n-paraffin composi-tion of a model petroleum fluid facilitates gelation by areduction in the surface energy of the critical nuclei,resulting in higher nucleation rates and increasedcrystal number densities, as well as providing surfaceroughness to paraffin crystals on a nanometer scale,facilitating strong crystal-crystal interactions of Lon-don van der Waals forces.
4. Conclusions
Application of the van’t Hoff solubility model withinthe framework of classical homogeneous nucleationtheory demonstrates conclusively that nucleation rep-resents the primary kinetic limitation associated withcrystallization of n-paraffins in organic solution at lowcooling rate conditions. Crystallization rate limitationsbecome significant at high cooling rates. The introduc-
tion of chain-length variations effects a reduction in thecritical nucleus surface energy via cocrystallization ofdissimilar chain-length paraffins. The initial nucleationevent is dependent upon the solubility behavior of thehighest fraction of n-paraffin components in a fluid,which can be readily established using a pseudocompo-nent analysis based on cloud point measurements. In areal petroleum fluid with a broad carbon numberdistribution, the majority of n-paraffin components willcrystallize in the equilibrium cooling regime at lowcooling rate conditions.
London van der Waals interactions between paraffincrystals in model waxy petroleum fluids result in drasticchanges in mechanical properties at low temperatures.Model fluids consisting of n-paraffin components dis-solved in mineral oil exhibit a low-temperature gellikemechanical response to an imposed low-frequency oscil-latory stress. The gel point of a waxy model petroleumfluid is dependent on the morphologies and surfacecharacteristics of the randomly oriented paraffin crys-tals. Paraffin crystals composed of a single-chain-lengthcomponent exhibit ordered surfaces and sharp edges,providing minimal crystal-crystal contact and weakinteractions. Polydisperse n-paraffin crystals exhibitnanoscale surface roughness which provides contact“mesh” points for strong crystal-crystal interactions,allowing mechanical gelation at smaller wax contents.Percolation threshold models provide accurate gel pointpredictions for physical gelation systems which exhibitstrong particle-particle interactions, while underpre-dicting the solid fraction necessary to induce gelationin weakly interacting particles systems. The inherentn-paraffin polydispersity of real petroleum fluids en-sures that a mechanically strong gel may form fromnearly any paraffinic fluid at a sufficiently low temper-ature.
This work demonstrates the importance of consideringnucleation effects in the rate of incipient gel formation.In pipeline systems where the bulk fluid temperatureis maintained at a temperature above the fluid cloudpoint, nucleation may be a significant factor in thecrystallization kinetics and may occur via a homoge-neous or heterogeneous mechanism depending upon thecontent of the fluid. Advancements in computationalfluid dynamics of wax deposition should incorporatenucleation theory to accurately predict rates of incipientparaffin gel formation.
Acknowledgment
The authors wish to acknowledge financial supportfrom the following members of the University of Michi-gan Industrial Affiliates Program: Baker Petrolite,ChevronTexaco, ConocoPhillips, Shell Oil, Schlumberg-er, and Total.
Appendix
As a thermodynamic framework for the calculationof the critical nucleus surface energy, we use the van’tHoff solubility theory, assuming ideal mixing in theliquid phase and a negligible heat capacity change uponcrystallization. Application of the supersaturation defi-nition in mole fraction yields the following functionalityfor the logarithm of the supersaturation ratio
ln(S) ) ∆HR (1
T- 1
Tcloud)
7252 Ind. Eng. Chem. Res., Vol. 44, No. 18, 2005
which can be rewritten as
where R ) kNA. Insertion into the nucleation rateexpression, eq 3, yields
The temperature dependencies exhibited in the denomi-nator are T(Tc - T)2. A Taylor expansion about Tc yieldsthe following exact relationship for the temperaturedependencies in the denominator:
A comparison of terms with the incorporated experi-mental nucleation point temperatures confirms that thesecond term represents <1.5% of the total value for allnucleation conditions. Application of the first-termTaylor approximation, as well as constant cooling rateconditions, such that ∆T ) (Tc - T) ) Rct (where Rcrepresents the cooling rate and t represents time),results in the following expression for the integratednucleation rate:
An analytical solution to the integral is obtained where
using the following definite integral
such that
Applying the definition of ∆T, the relationship can beexpressed as
which constitutes the working equation for correlatingthe classical homogeneous nucleation theory with ex-perimentally measured nucleating points expressed as∆Tnuc. The values of B and (A/F*) are varied as fittingparameters, and a least-squares minimization procedureis used to fit nucleation theory ∆Tnuc predictions toexperimental ∆Tnuc points across the cooling rate range0.5 to 5 °C/min. Correlated values of the surface energyσ derived from the optimized parameter B are shownin Table 2. With correlated values for the variables σ
and (A/Fn*), the inverse of the nucleation rate canreadily be expressed as a function of the supersaturationratio in the following linearized form:
At constant cooling rate conditions, the maximumnucleation rate, Jnuc, is attained at the nucleation point,where the supersaturation ratio is denoted Snuc and thetemperature is denoted Tnuc. Hence,
Figure 5 shows the linearized form of the maximumnucleation rate (for 4% initial paraffin contents) as afunction of 1/(ln S)2 and facilitates the appropriatecomparison of theoretical and experimental nucleationrates.
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ln(S) ) 1kT(∆H
NA)(Tc - T
Tc)
J ) A exp[ -16πσ3ν2
3kT( ∆HNATc
)2(Tc - T)2]
T(Tc - T)2 ) Tc(Tc - T)2 - (Tc - T)3
∫t)0
t)tnucA exp( -16πσ3ν2
3kTc-1(∆H
NA)2
Rc2t2) dt ) Fn*
B ) 16πσ3ν2
3kTe-1(∆H
NA)2
∫x)0
x)Xexp(-R
x2 ) dx ) exp(-RX2 )X - xRπ erfc(xR
X )
A exp(-B/R2
tnuc2 )tnuc - AxBπ
R2erfc(xB/R2
tnuc) ) F*
( AF*)∆Tnuc[exp( -B
∆Tnuc2) -
xBπ∆Tnuc
erfc( xB∆Tnuc
)] ) R
ln(Fn*J ) ) ln(Fn*
A ) + 16πσ3ν2
3k3T3(ln S)2
ln( Fn*Jnuc
) ) ln(Fn*A ) + 16πσ3ν2
3k3Tnuc3(ln Snuc)
2
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Received for review March 8, 2005Revised manuscript received June 10, 2005
