Formation of Perfluoropolyether Coatings by the Rapid Expansion of

Ind. Eng. Chem. Res. 2001, 40, 6127-6139

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Formation of Perfluoropolyether Coatings by the Rapid Expansion of Supercritical Solutions (RESS) Process. Part 2: Numerical Modeling Randall K. Franklin,† Jack R. Edwards,† Yury Chernyak,‡ Richard D. Gould,† Florence Henon,‡ and Ruben G. Carbonell*,‡ Department of Mechanical and Aerospace Engineering and Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27695

The rapid expansion of supercritical solutions (RESS) process is a promising method for the production of ultrafine powders and aerosols of narrow size distribution for coatings and other applications. In this article, part 2 of a two-part study, the nucleation and subsequent growth of 2500 Mw perfluoropolyether diamide (PFD) from supercritical carbon dioxide (CO2) by expansion through a small-diameter nozzle is modeled in a three-stage, multidimensional fashion. The stages include a hydrodynamic solution, solvent-solute phase equilibria analyses, and an aerosol transport model. The hydrodynamics model successfully captures the vapor-liquid transition that occurs as carbon dioxide is expanded to ambient conditions. Cloud-point pressures and equilibrium compositions of the separated solvent-solute system are determined and are used in a multidimensional aerosol transport model. This model incorporates various mechanisms influencing droplet growth. Parametric studies are conducted to investigate the influences of the interfacial tension, the equilibrium addition of carbon dioxide, and the diffusion coefficient on the predicted droplet diameter. Turbulent coagulation in the ambient region downstream of the expansion nozzle is found to be the dominant mechanism responsible for the production of micron-sized droplets observed in companion experiments. 1. Introduction The rapid expansion of supercritical solutions (RESS) process is a promising method for the production of ultrafine powders and aerosols. The process can be distinguished by its ability to produce particles of uniform size and by the high degree of supersaturation that can be obtained within the carrier fluid. RESS has been used to produce a wide variety of substances,1,2 including polymeric materials.3,4 Part 1 of this work5 describes an experimental effort designed to gain an understanding of the effects of RESS process conditions on the formation of perfluoropolyether coating materials, specifically a low-molecularweight perfluoropolyether diamide (PFD) dissolved in carbon dioxide. Phase equilibria measurements and polymer droplet size characterization results are reported, and the effects of temperature, solute concentration, and nozzle configuration on product characterization and transfer efficiency are discussed. This paper, part 2 of the two-part study, presents a multidimensional computational fluid dynamics model of the RESS expansion process and describes the use of the model in further analyzing and interpreting experimental data. Although RESS has been used for other applications, the focus of this work is on the deposition of fluorinated polymer coating materials, dissolved initially in supercritical carbon dioxide (CO2). An isothermal solution consisting of CO2 and polymer can separate into polymer* Author to whom correspondence should be addressed. E-mail: [email protected], fax: (919)-515-5831. † Department of Mechanical and Aerospace Engineering. ‡ Department of Chemical Engineering.

rich and polymer-lean phases as the pressure is lowered below a certain threshold pressure, termed the cloudpoint pressure. Beyond this point, nucleation and growth of polymer-rich droplets occurs, obeying rate expressions that primarily depend on the equilibrium compositions of the two phases, the interfacial tension between the phases, the properties of the pure components, and the local pressure and temperature of the solution. The prediction of the nucleation and growth of polymer droplets as a function of spatial location within (and outside) the spraying nozzle is the key output of the computational fluid dynamics (CFD) model described in this paper. Only a few efforts toward the simulation and modeling of the RESS process have appeared in the literature, with none directly focusing on the precipitation of polymeric compounds. Lele and Shine3 presented a onedimensional adiabatic flow model for the internal RESS expansion process, during which the fluid remains in a supercritical state. They employed the Peng-Robinson equation of state to model the real-fluid behavior of the solvent. Perhaps the most complete one-dimensional flow model is that of Kwauk and Debenedetti,9 who considered a partial (strictly subsonic) expansion of a supercritical solution of CO2 and phenanthrene through a small-orifice, converging nozzle. Their work coupled an aerosol transport model for solute nucleation and diffusional growth with fluid dynamics. The model was thus able to predict such properties as the nucleation rate, number density distribution, and average solute particle size as functions of the axial distance along the tube. Kwauk and Debenedetti’s model did not consider either the entrance region to the capillary tube or the free-jet expansion, and the possible effects of coagulation on particle growth were neglected. Furthermore, the

10.1021/ie010268e CCC: $20.00 © 2001 American Chemical Society Published on Web 11/27/2001

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Ind. Eng. Chem. Res., Vol. 40, No. 26, 2001

underlying equilibrium thermodynamics are relatively simple, as the solute precipitates as a single-component solid substance with a known intrinsic density. Nevertheless, their work remains perhaps the most comprehensive effort to date, and their procedures are closely followed (but modified) in the present work. To the authors’ knowledge, the only prior multidimensional RESS simulation reported in the literature is that of Ksibi and Subra,10 who considered the flow of supercritical/gaseous CO2 in a two-zone domain consisting of a small-orifice nozzle and an external expansion region. They used a Roe-type discretization, modified for real-fluid effects, and thus were able to capture the Mach disk that terminates the free-jet expansion. They then employed Kwauk and Debenedetti’s aerosol transport model to determine growth characteristics of the dissolved precipitate (salicylic acid). This decoupled approach for modeling particulate precipitation is also followed in the present work, both because of its comparative simplicity and because of the diluteness of the polymer solutions (less than 5.1 wt % of dissolved polymer). This work presents a three-step approach for modeling PFD precipitation in a RESS apparatus such as that described in part 1 of this work.5 Details of each stage of the approachsCFD modeling of the carbon dioxide flow in the apparatus, liquid-liquid equilibria (LLE) and vapor-liquid equilibria (VLE) calculations, and aerosol transport modeling of polymer precipitations are provided in the sections that follow. The paper concludes with discussions of the flow structure and predicted droplet growth trends, along with comparisons with experimental data.

Table 1. Reference Values Used in Equations 1 and 2 CO2 PFD

T*

P*

328 538

464.2 × 273.0 × 106 106

M

1426 2039

44 2500

The system is closed by application of appropriate thermodynamic state equations. The ideal-gas law is incorporated for air, as are gas-phase mixing rules between CO2 and air. This is warranted as the CO2 and air do not mix until they reach the region downstream of the expansion nozzle, where the CO2 has been expanded to vaporlike densities. The lattice-fluid state equation of Sanchez and Lacombe14 is used for both single-phase CO2 and PFD, which is a liquid at the conditions of interest

[

(

F˜ 2 + P ˜ +T ˜ ln(1 - F˜ ) + 1 -

1 F˜ ) 0 r

)]

(1)

where

T ˜ )

P F P*M T , P ) , F˜ ) , r ) T* P* F* F*RT*

(2)

are the reduced temperature, pressure, and density and number of lattice sites occupied by a r-mer, respectively. Other definitions are found in the Nomenclature section. T*, P*, and F* are determined by least-squares fitting of pressure-volume-temperature data.15,16 The values used in this work are listed in Table 1. An enthalpy departure function is included to complete the thermodynamic state description of singlephase CO2. The enthalpy per unit mass for the SanchezLacombe equation of state is written as

2. Governing Equations 2.1. Hydrodynamics. In the system of interest, PFD is dissolved in supercritical CO2 and is then precipitated by expansion through the RESS apparatus.5 As PFD never exceeds 5.1% of the system mass for the experiments to be modeled, its effects on the hydrodynamics are neglected. This assumption might be questionable for the higher weight percentages, but it is retained for reasons of tractability and simplicity. The hydrodynamics calculations therefore model the expansion of supercritical CO2 to ambient conditions, followed by mixing with quiescent air. The axisymmetric Favre-averaged compressible Navier-Stokes equations,27 expanded to include separate continuity equations for air and CO2, are solved to simulate the hydrodynamics of the RESS process. The Navier-Stokes system also includes an equation describing conservation of the energy of the fluid mixture. The equation set is formulated in a cell-vertex finitevolume manner, with the inviscid components discretized using a low-diffusion flux-splitting scheme valid for real-fluid flows at all speeds.12 Viscous and diffusive terms are discretized using central differences. Timederivative preconditioning is used to alleviate the timestep restriction associated with solving nearly incompressible flows (such as that encountered in the reservoir region) with a compressible formulation. The system is solved using an implicit technique based on incomplete lower-upper (ILU) factorization. The algorithm operates on a set of simply connected multiblock domains and uses grid sequencing to provide a good starting solution for the calculation on the finest mesh. Further information can be found in refs 12 and 13.

F*

h(F, T) ) hI(T) - 2F˜ P* +

[

T ˜ P* ln(1 - F˜ ) +1 F* F˜

]

(3)

where hI(T) is the enthalpy per unit mass based on an ideal-gas description at the corresponding temperature.17 The viscous and conductive components of the NavierStokes set are closed by providing expressions for the molecular viscosity and thermal conductivity. These expressions are supplied by ref 18 for CO2 and ref 17 for gaseous air. Wilke’s law19 is used to determine mixture values. Fick’s law is used to determine molecular diffusion velocities. One diffusion coefficient, related to the mixture viscosity and density through the assumption of a constant Schmidt number, is used for both CO2 and air. Standard Boussinesq/gradient diffusion approaches are adopted for turbulence closure, reducing the problem to the specification of an eddy viscosity and turbulent Prandtl and Schmidt numbers, both set to 0.9 in this work.27 The eddy viscosity is provided by the Spalart-Allmaras one-equation turbulence model.20 One difficulty encountered in modeling the RESS process is the fact that, because of the high reservoir pressure and low reservoir temperature, pure-component CO2 undergoes a phase transition during expansion to ambient conditions. The model presented by Ksibi and Subra10 was applied to RESS process paths that existed above the vapor dome entirely, thus avoiding this difficulty. Kwauk and Debenedetti9 noted that their process path did penetrate the two-phase region within the nozzle, but they did not explicitly model the two-phase fluid that resulted. This phase transition

Ind. Eng. Chem. Res., Vol. 40, No. 26, 2001 6129

Figure 1. Pressure-density process path (pure CO2).

results in localized regions of two-phase vapor-liquid CO2 flow, which are modeled by a homogeneous equilibrium state description.12 Figure 1 shows a typical pressure-density process path as extracted from computational analysis of the nozzle flow (discussed later). Also shown is the vapor dome for pure CO2 as determined from the Sanchez-Lacombe state equation. This figure indicates that, before rapid expansion, the carbon dioxide is essentially in a liquid state, indicating that the phase transition most probably occurs through the formation and growth of gaseous CO2 bubbles in the CO2 liquid medium. Whether this situation holds in the presence of dissolved polymer is unclear; the decoupled analysis used in this work does not rigorously account for the possibility of three coexisting phases or the associated influences on particle growth. 2.2. Solvent-Solute Phase Equilibria. According to Lele and Shine,3 the precipitation of high-molecularweight polymers from supercritical solutions occurs by liquid-liquid phase separation, which produces a polymer-rich phase and a polymer-lean phase, each containing both polymer and solvent. In dilute polymer solutions, the polymer-rich phase can nucleate as globules21 in the initial stages of liquid-liquid phase separation. Phase equilibrium for the PFD-CO2 binary system is assumed to be described by a mixture extension of the Sanchez-Lacombe equation of state,23 as detailed in the Appendix. For liquid-liquid phase equilibria, the following system of chemical potential equations is solved, with subscripts 1 and 2 referring to CO2 and PFD, respectively, and superscripts I and II referring to the polymer-lean and polymer-rich phases, respectively I II µI1(yl,eq , P, T)L ) µII 1 (yl,eq, P, T)L

(4)

I II , P, T)L ) µII µI2(yl,eq 2 (yl,eq, P, T)L

(5)

If the temperature and pressure are specified, the I II and yl,eq , the equilibsolution of eq 4 and 5 yields yl,eq rium mass fractions of PFD (l ) 2) and CO2 (l ) 1) in each phase. If the temperature and the polymer-lean I are specified, the system yields phase composition yl,eq the cloud-point pressure and the polymer-rich phase composition. The polymer-lean phase composition is

assumed to be the overall system composition prior to expansion through the RESS apparatus. If the pressure at a point in the flow field is just below the cloud-point pressure for the corresponding temperature, the solution enters a metastable state, in which random thermodynamic fluctuations can induce a separation into polymer-lean and polymer-rich phases.21 This separation is described by the aerosol transport model presented later. The degree of penetration into the metastable region is quantified by the supersatuI , which evolves according to the aerosol ration, yI2/y2,eq transport model along the process path. For the model presented later, the local supersaturation remains greater than 1 until all polymer is exhausted from the solution, implying that the solution remains in a metastable state after the onset of precipitation. As discussed in part 1,5 lower reservoir temperatures do not result in liquid-liquid phase separation within the nozzle, so the system presented above will not provide a solution. In this case, the system represented by eq 4 and 5 is replaced by the liquid-vapor equilibrium constraint I µI1(yl,eq , P, T)L ) µ1(P, T)V

(6)

The polymer-rich droplet phase at equilibrium is assumed to consist of pure polymer. 2.3. Aerosol Transport. The behavior of the nucleated polymer-rich phase is assumed to be described by classical aerosol transport theory, modified to account for real-fluid effects following Kwauk and Debenedetti.9 The general dynamic equation (GDE) of aerosol transport theory24 describes the evolution of a droplet number density distribution function under the influences of nucleation, diffusional growth, and coagulation, among other effects. The GDE is posed in a four-dimensional space, with the fourth dimension representing the droplet size. This additional dimension renders the GDE difficult to solve for multidimensional analyses. To reduce this complexity, one can average the GDE over the space of the droplet size dimension to yield an average droplet number density at every point in space and time, which can be associated with an average droplet size if the volume fraction of the droplet phase is also determined. This approach, however, requires knowledge of the shape of the local droplet size distribution. In this work, it is assumed that the local droplet size distribution is monodisperse, which greatly simplifies the modeling of coagulative effects in particular. This assumption is somewhat inconsistent with the results of ref 5, which indicate that the RESS apparatus produces a polydisperse, albeit narrow, droplet size distribution under most conditions. However, it should be recognized that the volume of fluid sampled by the particle analyzer is much larger than that within a particular mesh cell in the hydrodynamics model, so that the model could return an effective polydisperse distribution if local results for droplet size and number density are sampled over the experimental volume. With these assumptions, the aerosol transport model used in this work consists of transport equations governing droplet number density, mass density, and volume fraction. These equations are each a particular moment of the GDE24 and are written as follows

|

∂ ∂N ∂ (N) + (NUj) ) I + ∂t ∂Xj ∂t coag

(7)

6130


M2 ∂ ∂ (Fby˜ II [g*I + NF2] (Fby˜ II 2) + 2 Uj) ) ∂t ∂Xj NA

(

(8)

)

II y1,eq FII ∂ II 2 ∂ II (R ) + (R Uj) ) η2 g*I + NF2 II II ∂t ∂Xj y F 2,eq

1

(9)

The primary outputs from these equations include N, the droplet number density; y˜ II 2 , the mass fraction of PFD referenced to the system mass; and RII, the volume fraction of the polymer-rich phase. The hydrodynamics model supplies the velocity, temperature, and pressure fields. The aerosol transport model is advanced in time by use of a first-order semi-implicit Euler time integration scheme on the same simply connected multiblock domain used for the hydrodynamic solution. The coagulation term in eq 7 is treated in an implicit manner. An examination of the right-hand side of eq 7 reveals that the droplet number density is governed by nucleation and coagulation effects. The nucleation rate can be written as9

I ) 2N

Px2

x2πM2NA

-1

exp

{

kbT

x

σ (η )2 × kbT 2

[

3 2 kbT -16π σ η2 3 I I 3 (k T) µ (y ) - µI (yI) b 2 l,eq 2 l

]}

|

|

|

|

|

(10)

(11)

|

()

(14)

where the kinematic viscosity ν is determined from the CO2-air hydrodynamic solution. The dissipation rate is determined by solution of the turbulence kinetic energy equation apart from the hydrodynamic solution, assuming that the eddy viscosity νt as extracted from the Spalart-Allmaras model20 can be written as

νt )

0.09k2

(15)

For integral-scale coagulation, the characteristic time is assumed to exhibit a form similar to that of the Kolmogorov-scale turbulence, written as

(16)

Droplets smaller than the Kolmogorov length scale are affected by eddies of all sizes. As a droplet grows, the effects of smaller eddies diminish. To the authors’ knowledge, very few attempts have been made to model turbulent coagulation for droplets larger than the Kolmogorov length scale, as the isotropic turbulence assumption becomes invalid and turbulence history effects become important. An empirical model is thus used to account for changes in the turbulent coagulation rates as the droplets change in size. This model allows droplets with diameters less than the Kolmogorov length scale to be affected by both Kolmogorov-scale and integral-scale turbulence, with the effect of the Kolmogorov-scale turbulence being reduced as the droplet grows beyond that length scale. The empirical model is

1 τturb

(12)

Droplets in a laminar shear field might also collide because of their relative motion. The change in number density due to such collisions is

∂N 16 d 3 2 )- S N ∂t ms 3 2

xν

2

Coagulation due to Brownian motion affects droplets with diameters less than ∼1 µm. For droplets much larger than the mean free path of the carrier fluid, experimental evidence shows that this process is diffusion-limited.24 Kinetic theory gives

∂N 4 kbT 2 )N ∂t ms 3 µ1

1 ) τK

1 ) τint k

with particular quantities defined in the Nomenclature section. The local composition of the polymer-lean phase, yIl with l ) 1, 2, can be related to the transported quantity y˜ II 2 and the bulk density, as shown in ref 13. For highly miscible fluorinated polymer/supercritical CO2 systems, the interfacial tension σ is expected to be small, i.e., on the order of 0.001 N/m or less, if measurements obtained for nonfluorinated polymers in CO225 can be used as a rough guideline. Three mechanisms for coagulation are investigated in this work. These three mechanisms are Brownian, mean shear, and turbulence-driven coagulation24

∂N ∂N ∂N ∂N ) + + ∂t coag ∂t b ∂t ms ∂t turb

coagulation mechanism is a strong function of the droplet size with respect to the eddy size. Turbulent eddies vary in size from Kolmogorov (∼10-7-10-6 m) to integral (∼10-3 m) scales for the system of interest. The characteristic time for Kolmogorov-scale coagulation is24,27

(13)

where S is the magnitude of the local strain rate and d is the monodisperse droplet diameter. Coagulation is also affected by droplets residing in energy-dissipating turbulent eddies which increase the collision frequency of the droplets. The magnitude of this

(

)

(

)

λK 1 λint 1 + min 1.0, C (17) d τK d τint

) min 1.0, C

where λK and λint are the Kolmogorov and integral length scales, respectively, and C is a user-specified constant of O(1). The Kolmogorov and integral length scales are27

λK )

() ν3

1/4

, λint )

k3/2

(18)

This scaled characteristic time for turbulence is related to the turbulent coagulation mechanism by24,26

|

16 1 d 3 2 ∂N )N ∂t turb 3 τturb 2

()

(19)

It should be mentioned that, for most of the conditions considered in this work, the droplet sizes do not exceed the Kolmogorov length scale. In this case, the empirical model presented above reduces to the theoretical development of Saffman and Turner,26 as applied to a monodisperse collection of droplets.


The droplet mass density of PFD, given by eq 8, is influenced by both the nucleation rate and the condensation rate F2 of PFD droplets. The source term for eq 8 states that the droplet mass density is increased by the nucleation of droplets containing g* molecules of PFD and is further increased by condensation of other molecules onto this critical nucleus at a rate of F2 molecules per second. The number of PFD molecules in a critical nucleus is

g* )

[

32π σ (η )2/3 3 kb T 2

][

kbT

3

I µI2(yl,eq ) - µI2(yIl )

]

3

(20)

and the condensation rate is given by

F2 ) 2πdD(N - Neq)

(

)

1 + Kn 1 + 1.71Kn + 1.333(Kn)2

(21)

The diffusion coefficient D is given by9

TxM1 D ) 7.4 × 10-15 F1ν1(η2)0.6

Figure 2. RESS apparatus computational domain.

(22)

The Knudsen number is defined as

Kn )

x

2µ1 dF1

πM1 2NAkbT

(23)

and the equilibrium droplet number density is defined as

Neq )

I NAF2,eq RI M2

(24)

I is a function of the equilibrium composition where F2,eq of the polymer-rich phase at the local temperature and pressure. The volume fraction of the polymer-rich phase, given by eq 9, is likewise influenced by nucleation and condensation. The term multiplying F2 on the righthand side of eq 9 accounts for swelling of the droplets by addition of CO2 (see the Appendix. The composition of the droplets is assumed to remain in equilibrium throughout their growth. This assumption is warranted because of the short time required (10-12 s) for diffusion of CO2 in to and out of the nucleated droplet with respect to the typical time required for a droplet to traverse a mesh cell (10-7 s or longer). The key result from the aerosol transport model is the prediction of the average (locally monodisperse) droplet diameter at each location within the flow field

d)

( ) 6RII πN

1/3

(25)

3. Results 3.1. Hydrodynamics. The computational domain used to simulate the RESS apparatus is illustrated in Figure 2. The geometry in Figure 2 is shown reflected about the x axis, and axisymmetry is assumed in the calculations. Although part 1 of this article also presents data obtained from a pinhole nozzle and a nozzle with a 25-mm capillary tube, only the baseline geometry with an 11-mm capillary tube is considered in this paper.

The region prior to the expansion nozzle consists of a straight-walled reservoir region (block 1, 65 × 129 mesh points), followed by a converging nozzle (block 2, 81 × 81 mesh points) that blends the reservoir diameter smoothly into that of the 11 mm capillary tube. The tube has an inside diameter of 0.1524 mm (block 3, 177 × 81 mesh points). The ambient region downstream of the expansion nozzle (block 4, 201 × 137 mesh points, and block 5, 201 × 73 mesh points) is the region in which the gaseous carbon dioxide mixes with the quiescent air. For clarity, blocks 4 and 5 are shown merged in Figure 2. The “expansion nozzle” terminology used in this paper refers to the combination of the converging nozzle and the capillary tube. The inflow conditions for the reservoir region are taken from the experiment. The pressure is fixed at 2400 psi, and the temperature is set to either 300 K (27 °C) or to 330 K (57 °C). The higher temperature results in precipitation within the capillary tube, whereas the lower temperature results in precipitation at the nozzle exit (see Figure 8 in ref 5). The axial velocity at the inflow boundary of the reservoir is extrapolated from the interior. The pressure along the boundaries of the ambient region is set as standard atmospheric pressure. All other flow variables are extrapolated from the interior along the boundaries of the ambient region. Standard no-slip, adiabatic wall boundary conditions are applied for all solid surfaces. Figure 3 compares the shock wave structure for the 300 and 330 K cases. In both cases, the CO2 remains in a liquid or supercritical state until near the outlet of the expansion nozzle, at which point the fluid transitions into a vapor-liquid mixture. This transition results in much lower values for the fluid sound speed, leading to a sonic transition and a supersonic free-jet expansion. Because of the more liquidlike character of the fluid at 300 K, more of the pressure drop occurs within the nozzle (see also Figure 4), and the gaseous flow accelerates to about 400 m/s before the Mach disk, compared with 500 m/s for the 330 K case. The adverse pressure gradient induced by the Mach disk results in the formation of two recirculation regions near the center line for the 300 K case. These recirculation zones affect polymer droplet growth, as shown later. In contrast, the higher-momentum 330 K jet does not stagnate downstream of the Mach disk.

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Figure 3. Comparison of Mach disk structure in ambient region for different reservoir temperatures.

Figure 4. Pressure and temperature along the center line of the RESS apparatus for different reservoir temperatures.

Figure 4 shows the pressure and temperature along the center line of the RESS apparatus. As can be seen, the passage of the fluid through the nozzle (X < 0.0186 m) causes the pressure along the centerline to be reduced to ∼50 and ∼36% of the reservoir value for the

330 and 300 K cases, respectively. The temperature is reduced only slightly during this internal part of the expansion. The CO2 expands rapidly at the outlet of the expansion nozzle, reaching pressures as low as 6 kPa and temperatures as low as 137 K before being recompressed through a stationary Mach disk to nearly ambient pressures and temperatures of ∼174 K. The temperatures approached in the free-jet expansion are well below the triple-point temperature of CO2 (217 K), meaning that the formation of solid CO2 in the ambient region is a possibility. The current work does not account for sublimation of CO2, although, as mentioned in part 1, the presence of solid CO2 particles could help explain particle-sizing trends at lower preexpansion temperatures. For the higher-momentum 330 K jet, mixing with the entrained air causes the temperature of the mixture to increase over the remainder of the domain, with values of over 240 K attained at the maximum boundary of the ambient region. The 300 K jet does not display this behavior, indicating that the overall mixing process is much less intense. Figure 5 shows a comparison of computed axial velocity and CO2 mass fraction with experimental laser light-sheet imaging data for a case in which the reservoir pressure was set at 4000 psi.29 The spreading rate of the jet (defined as the rate of change of the jet half-


Figure 6. Center-line equilibrium mass fractions in polymer-lean and -rich phases for different reservoir temperatures.

Figure 7. Effect of interfacial tension on center-line droplet diameter (1.5 wt % PFD, 330 K reservoir temperature).

Figure 5. Comparison of computed CO2 jet structure with laser light-scattering data.

width with distance away from the nozzle) is captured reasonably well except for very near the expansion nozzle outlet, where the computation predicts a more rapid spreading rate of the jet. These deviations can be explained partially by the inexact rendering of the expansion nozzle outlet geometry and by possible inconsistencies in the specification of the reservoir inflow boundary conditions. 3.2. Solvent-Solute Phase Equilibria. Figure 6 presents the equilibrium mass fractions of CO2 in both the polymer-lean and polymer-rich phases along the center line of the RESS apparatus. The left axis corresponds to the polymer-lean phase, and the right axis corresponds to the polymer-rich phase. The cloud point is crossed near X ) 0.01 m for this case (330 K reservoir

temperature). The equilibrium composition of the polymer-lean phase is specified as containing no PFD in the ambient region, whereas the polymer-rich phase is assigned the values obtained from the cloud-point analysis at the outlet of the expansion nozzle. Also shown is the equilibrium mass fraction of CO2 in the polymer-lean phase for the 300 K reservoir temperature. Rapid phase separation occurs near the exit of the capillary tube, within the region of two-phase CO2 flow as predicted by the homogeneous equilibrium model. 3.3. Aerosol Transport. Results from the application of the aerosol transport model are discussed in this subsection. Several parametric studies illustrating the factors that govern droplet growth along the center line are presented first for the 330 K case (precipitation inside the capillary tube). Comparisons with experimental droplet size data described in part 1 are presented next for both the 330 and 300 K cases. The value chosen for the interfacial tension σ ultimately has an influence on the predicted droplet diameter. For this work, σ was chosen to be 0.001 N/m. As can be seen in Figure 7, lower values of σ decrease the predicted droplet diameter within the expansion nozzle

6134


Figure 8. Effect of diffusion coefficient on center-line droplet diameter (1.5 wt % PFD, 330 K reservoir temperature).

while increasing the predicted droplet diameter downstream of the expansion nozzle. Because g* ∼ σ3, a reduction in σ reduces the number of molecules that must come together to produce a stable nucleus. This affects the droplet diameter predictions in that a greater number of nanoscale droplets are nucleated in the expansion nozzle. Larger droplet diameters in the ambient region result, as the magnitudes of the coagulation mechanisms vary with the second power of the number density, i.e., N2. The nucleation delay observed within the expansion nozzle for higher σ values can be attributed to the fact that, as σ increases, the number of molecules that must come together to produce a stable nucleus increases, which requires a slightly longer period of time. The diffusion coefficient D also has a noticeable effect on the predicted droplet diameters. As shown in Figure 8, an increase in the diffusion coefficient results in an overall increase in the predicted droplet diameter. This is warranted as the condensation rate F2 of molecules onto a particular droplet (eq 21) varies linearly with D. Because of the very short diffusion times of CO2 in to and out of a droplet, the composition of the droplet is assumed to be in equilibrium throughout its growth. The effect on the predicted droplet diameter of the equilibrium addition of CO2 into the droplets, compared to droplets consisting entirely of PFD, is illustrated in Figure 9. Although the droplet sizes are predicted to be somewhat smaller within the expansion nozzle for the case without swelling, the most noticeable difference is the rate of increase of the droplet size upon rapid expansion (X ≈ 0.02 m). Taking the substantial derivative of eq 25 yields

(

)

Dd DN 1 DRII ) - Vp Dt NAp Dt Dt

(26)

where Ap and Vp are the surface area and volume of the average droplet, respectively. This equation indicates that the rate of droplet growth will increase with a decrease in the rate of change of the number density (through coagulation) and with an increase in the rate of change of the volume fraction. The latter is influenced strongly by accounting for droplet swelling as in eq 9, leading to the more rapid growth evidenced in Figure 9.

Figure 9. Effect of CO2 swelling on center-line droplet diameter (1.5 wt % PFD, 330 K reservoir temperature).

Figure 10. Volume fraction of polymer-rich phase and mass of PFD in polymer-rich phase (center line, 1.5 wt % PFD, 330 K reservoir temperature).

Figure 10 illustrates the transfer of PFD into the polymer-rich phase along the center line of the RESS apparatus. Downstream of the expansion nozzle outlet, further growth of the droplets is dominated by coagulation mechanisms, as no PFD is present in the polymerlean phase. As expected, this behavior is independent of whether droplet swelling by CO2 addition is modeled (curves of y˜ II 2 with and without swelling coincide in the figure). The volume fraction of the polymer-rich phase and the rate at which it increases are affected by swelling, as discussed above. Figure 11 illustrates the influence of the three coagulation mechanisms modeled in the present work. Coagulation due to Brownian motion of the droplets has only a slight effect on the predicted droplet diameter when compared with the other mechanisms. The addition of coagulation due to mean shear in the fluid has a minor influence on the predicted droplet diameter along the center line. This coagulation mechanism can affect droplet sizes farther from the center line of the jet, where the magnitude of the strain rate is greater. Coagulation due to turbulence effects is the most influential of the three coagulation mechanisms inves-


Figure 11. Effects of coagulation mechanisms on center-line droplet diameter (1.5 wt % PFD, 330 K reservoir temperature). Table 2. Measured and Predicted Droplet Diameters in the Ambient Region with a Reservoir Temperature of 330 K Sauter mean diameter (µm)

PFD wt %

maximum center-line diameter (µm)

experimental

predicted, 0.5 µm cutoff

predicted, no cutoff

0.7 1.5 2.0 3.4

0.18 1.35 2.93 4.61

NDa 1.35 1.56 2.24

NDa 0.56 0.87 2.75

0.099 0.35 0.72 2.75

a

ND ) not detected.

tigated. Turbulent coagulation is mostly responsible for droplet diameter predictions that agree reasonably well with experimental observations. The effect of the constant C in eq 17 is also illustrated in Figure 11. For a particular eddy size λ, the constant C restricts the full effect of the associated time scale to droplets whose diameters are less than or equal to Cλ. As the droplets become larger than Cλ, the effects of that eddy size diminish. The lower value of C (0.2) results in the lowering of the cutoff threshold for influences due to Kolmogorov eddies (the shortest time scale). Droplets that grow beyond this cutoff are influenced more by integral-scale eddies, which induce collisions less frequently. The coagulation rate drops as a result, and the average droplet diameter is reduced. Predictions for C ) 1.0 and C ) 5.0 are identical. This simply indicates that, for a polymer concentration of 1.5%, droplets never grow larger than the local Kolmogorov length scale in the mixing region. Coagulation then is primarily a result of collisions induced by Kolmogorovscale eddy motion. Predicted droplet diameters for the 330 K reservoir temperature are compared with averaged data produced by a Malvern particle analyzer in Table 2. Experimental Sauter mean diameters are shown in column 2, followed by the maximum predicted droplet diameter along the jet center line. Estimates of the Sauter mean diameter as obtained from the definition

D[32] )

∫VNd3 dV ∫VNd2 dV

(27)

Figure 12. Effect of solution composition on center-line droplet diameter (330 K reservoir temperature).

are shown in columns 4 and 5. These values were obtained by numerically evaluating the volume integrals over the latter half of the mixing zone. This region corresponds roughly to one-half of the volume traversed by the analyzer laser beam at its closest position to the nozzle exit. For the results tabulated in column 4, only droplets with diameters greater than the detection limit of 0.5 µm were included in the integration. All droplets were included in the integration for the results shown in column 5. As indicated, the particle analyzer could not detect droplets produced from the 0.7 wt % solution spray. For the 1.5 and 2.0 wt % solution compositions, the predicted Sauter droplet diameters are less than those measured experimentally but are of a similar order of magnitude. Agreement is quite good for the 3.4 wt % solution. The inclusion of the analyzer detection limit in the calculation of the Sauter diameter results in larger average droplet sizes, as expected. As indicated earlier, these predictions are most strongly influenced by the polymer-rich phase composition and the diameter of the droplets with respect to the characteristic sizes of the turbulent eddies. It is of note that, without the inclusion of turbulent coagulation mechanisms, the model underpredicts measured droplet sizes. It is also of note that the experimental measurements were taken at a distance of 15.2 cm downstream of the nozzle exit plane, well beyond the end of the computational domain (1.57 cm). If turbulent coagulation in the ambient region the dominant mechanism, it is certainly possible that further droplet growth could occur over this interval. Figure 12 illustrates the effect of the overall solution composition on the predicted droplet diameter profiles along the center line of the RESS apparatus for the 330 K case. A trend toward increasing droplet diameter with increasing concentration is evidenced, in accord with experimental results. Table 3 shows a comparison between predicted and measured droplet diameters for the 300 K reservoir temperature. Particle growth takes place entirely outside the nozzle for this case, and no swelling of the polymer-rich phase is assumed to occur. Although large droplets are predicted along the center line, the bulk of the flow field is dominated by submicron-sized droplets, particularly at the lower concentrations. Only at the highest concentration of 5.1 wt % are the predictions in reasonable accord with experimental measurements.

6136


Figure 13. Contour plots of droplet diameter in ambient region for different reservoir temperatures (3.4 wt % PFD). Table 3. Measured and Predicted Droplet Diameters in the Ambient Region with a Reservoir Temperature of 300 K Sauter mean diameter (µm)

PFD wt %

maximum center-line diameter (µm)

experimental

predicted, 0.5 µm cutoff

predicted, no cutoff

1.5 2.0 3.4 5.1

0.78 1.64 4.02 6.90

1.86 2.13 2.53 2.80

NDa 0.62 0.70 1.02

0.039 0.13 0.21 0.38

a

ND ) not detected.

Figure 13 illustrates the differences between the droplet diameter predictions at 330 and 300 K. The larger center-line droplet diameters for 300 K can be attributed to the presence of recirculating regions of fluid downstream of the stationary Mach disk, as illustrated previously in Figure 3. These are sources of high turbulence intensity and long residence times, both of which act to promote turbulent coagulation. Only small droplets initially entrained in these regions experience accelerated growth. The majority of droplets bypass the recirculation regions and grow at much slower rates. Figure 14 illustrates the effect of concentration on center-line droplet diameter predictions for the 300 K case. A small, sharp rise in droplet size is observed immediately downstream of the Mach disk, at X ) 0.0196 m. This increase corresponds to the first recirculation region and is followed by a second, more pronounced increase in droplet diameter at the second recirculation region, which begins at X ) 0.0206 m.

Figure 14. Effect of solution composition on center-line droplet diameter (300 K reservoir temperature).

Although the effects of measurement location cannot be discounted, predictions for the 300 K case do not agree qualitatively with the experimental results of part 1.5 In particular, the experimental results show increases in the mean droplet size, transfer efficiency, and spray volume concentration as the temperature is reduced. All of these observations imply that larger droplets are being formed. The computational results indicate that the delay in precipitation until the nozzle


exit and the absence of droplet swelling both act to reduce the predicted droplet sizes relative to the highertemperature case. Similar trends regarding the influences of the preexpansion temperature have been noted in other theoretical9 and experimental4 studies. As discussed in part 1,5 the presence of solid CO2 in the jet at lower preexpansion temperatures could both bias the particle-sizing instrument and provide a mechanism for the coagulative growth of polymer/solid CO2 aggregates, which could maintain more of their forward momentum. Higher transfer efficiencies could thus result, even if the actual polymer droplet sizes are smaller, as predicted. This conjecture remains to be explored in detail. 4. Conclusions A three-step approach for modeling perfluoropolyether diamide precipitation from supercritical CO2 during the rapid expansion of supercritical solutions (RESS) process has been described. The stages involve an initial calculation of the internal and external flow fields, followed by equilibrium analyses of the solvent-solute system and then aerosol transport modeling of droplet precipitation and growth. The key result from the aerosol transport model is a prediction of average droplet diameter at all locations within the flow field. The onset of phase separation and droplet formation is most influenced by the interfacial tension between polymer-lean (solvent) and polymerrich (solute) phases. Droplet sizes are most affected by the swelling of droplets through the equilibrium addition of CO2 and by modeled turbulent coagulation within small-scale (Kolmogorov) eddies. The latter effect is particularly important in the region downstream of the free-jet expansion/Mach disk, and the results are likely to be affected by the quality of the turbulence model in predicting the jet structure in this region. Comparisons with experimental droplet size data described in part 15 indicate that the model can predict micron-sized polymer droplets under certain conditions, in qualitative agreement with experimental observations. In general, though, predicted Sauter mean diameters are below experimental values. The model predicts an increase in the average droplet size with increasing polymer concentration, again in accord with experimental results. In contrast with the experimental measurements, the model predicts a decrease in average droplet size as the preexpansion temperature is lowered, in accord with other theoretical and experimental work. The experimental trend of part 1 remains to be explained but might result from the presence of solid CO2 particulates in the jet at low preexpansion temperatures. 5. Acknowledgment This work was supported by the U.S. Office of Naval Research under Grant NOOO14-98-1-0157; the DuPont company; the Kenan Institute for Engineering, Technology, and Science; and the National Science Foundation (STC Program, Agreement CHE-9876674). Cray T-90 computer time was provided by a grant from the North Carolina Supercomputing Center. 6. Nomenclature C ) model constant d ) average droplet diameter (m)

D ) diffusion coefficient (m2/s) F2 ) condensation rate [molecules/(s droplet)] gi ) number of molecules of component i g* ) number of molecules in critical nucleus h ) enthalpy (J/kg) I ) nucleation rate (droplets/s) kb ) Boltzmann’s constant [J/(molecule K)] k ) turbulence kinetic energy (m2/s2) Kn ) Knudsen number M ) molecular weight (kg/kmol) N ) droplet number density (droplets/m3) NA ) Avogadro’s number P ) pressure (N/m2) P ˜ ) reduced pressure in Sanchez-Lacombe equation P* ) reference pressure in Sanchez-Lacombe equation (N/ m2) r ) number of lattice sites occupied by an r-mer R ) universal gas constant [J/(kmol K)] S ) norm of strain rate tensor (1/s) t ) time (s) T ) temperature (K) T ˜ ) reduced temperature in Sanchez-Lacombe equation T* ) reference temperature in Sanchez-Lacombe equation (K) Uj ) Cartesian velocity component (m/s) Xj ) Cartesian coordinate (m) xi ) mole fraction of component i within a particular phase yji ) mass fraction of component i in phase j, referenced to mass of phase j y˜ ji ) mass fraction of component i in phase j, referenced to total system mass Rj ) volume fraction of phase j δij ) binary interaction parameter ) dissipation rate of turbulence kinetic energy (m2/s3) / i,j ) entries in the cohesive energy matrix (J/kmol) ζij ) binary interaction parameter ηi ) molecular volume of component i (m3/molecule) φi ) close-packed volume fraction of component i within a particular phase λ ) turbulence length scale µi ) molecular viscosity of component i [kg/(m s)] µji ) chemical potential of component i in phase j (J/ molecule) ν˜ ) reduced specific volume in Sanchez-Lacombe equation (m3/kmol) ν ) kinematic viscosity (m2/s) νt ) eddy viscosity (m2/s) F ) density (kg/m3) Fb ) bulk density (kg/m3) F˜ ) reduced density in Sanchez-Lacombe equation F* ) reference density in Sanchez-Lacombe equation (kg/ m3) Fji ) density of component i in phase j (kg/m3) σ ) interfacial tension (N/m) τ ) time scale (s) χij ) binary interaction parameter Subscripts 1 ) component 1 in mixture (CO2) 2 ) component 2 in mixture (PFD) eq ) equilibrium i, j, l, m ) general component index (equal tovaries from 1 orto 2) int ) integral K ) Kolmogorov L ) liquid V ) vapor

6138


M ) x1M1 + x2M2

Superscripts I ) polymer-lean phase II ) polymer-rich phase III ) entrained air “phase”

Pure-component values for P/i , T/i , Mi, and F/i as used in the above equations are specified in Table 1. The chemical potential of CO2 (component 1) is given by

7. Appendix 7.1. Mixing Rules and Chemical Potential Definition. The reference quantities in eq 1 must be adjusted to account for displacement in the fluid lattice by multiple components. The particular mixing rules used in this investigation are taken from ref 23 and are repeated here for completeness. The close-packed volume fraction φi of component i (i ) 1, 2) in the binary mixture is defined as

xiri x1r1 + x2r2

φi )

MiP/i RT/i F/i

(29)

From this equation, the mixture value of r is given by

r)

v* v/o

(30)

where v* is the mixture close-packed molecular volume

)

v* and

v/o

φi

2

1

∑ i)1

v/i

(31)

( )

µ1 r1 ) ln φ1 + 1 - φ2 + r1F˜ χ12φ22 + RT r2 ˜ 1ν˜ 1 -F˜ P + + (ν˜ - 1) ln(1 - F˜ ) + ln F˜ (38) r1 T ˜1 T ˜1 r1

[

]

with

T ˜1 )

(28)

where xi is the mole fraction of component i and ri is the number of lattice sites occupied by a molecule of component i

ri )

T P 1 , P ˜ 1 ) /, and ν˜ ) F˜ T/1 P1

2

)

v/o

φi ∑ i)1

P/i

RT/i

RTχ12 ) /11 + /22 - 2/12

P* )

where

/11 ) RT/1, /22 ) RT/2, /12 ) (/11 /22)1/2(1 - δ12) (41) and δ12 is an adjustable interaction parameter, set to zero in this work. The chemical potential of PFD can be determined by interchanging the indices in eq 38. 7.2. Swelling of Polymer-Rich Phase. The volume fraction of the polymer-rich phase can be written as

RII ) N(g1η1 + g2η2)

(

RII ) g2η2N 1 + (32)

(42)

)

g1η1 g2η2

(43)

The ratio term in the above equation can be multiplied II and divided by FII 1 F2 to give

2

φiφjP/ij ∑ ∑ i)1 j)1

(40)

This expression can be rearranged to read

The mixture reference pressure P* is defined as 2

(39)

The interaction factor χ12 is defined as

is the molecular volume of a “hole” in the lattice

1

(37)

(33)

(

RII ) g2η2N 1 +

)

II FII 1 g1η1F2 II FII 1 g2η2F2

(44)

By dimensional arguments, the following expression can be substituted into the previous equation, to give

where

P/ij ) (P/i P/j )1/2(1 - ζij)

(34) yII m

and ζij is an interaction parameter (set to zero in this work). The mixture reference temperature is defined as

T* )

P*v/o R

(35)

M v*

II

(36)

where M is the molecular weight of the mixture, written as

∑i

(45)

FII i giηi

which results in the following expression for the volume fraction

and the mixture reference density is given by

F* )

)

FII mgmηm

(

R ) g2η2N 1 +

)

II yII 1 F2 II yII 2 F1

(46)

By assuming the polymer-rich phase to be in equilibrium, eq 46 is rewritten using the equilibrium mass


fractions produced by the solvent-solute phase equilibria analysis

(

RII ) g2η2N 1 +

)

II y1,eq FII 2 II y2,eq FII 1

(47)

Taking the substantial derivative of eq 47 gives the source term in eq 9, noting that

Dg2 DN ) I and ) F2 Dt Dt

(48)

Literature Cited (1) Debenedetti, P. G.; Tom, J. W.; Kwauk, X.; Yeo, S.-D. Rapid Expansion of Supercritical Solutions (RESS): Fundamentals and Applications. Fluid Phase Equilib. 1993, 82, 311. (2) Jung, J.; Perrut, M. Particle Design Using Supercritical Fluids: Literature and Patent Survey. J. Supercrit. Fluids 2001, 20, 179. (3) Lele, A.; Shine, A. D. Morphology of Polymers Precipitated from a Supercritical Solvent. AIChE J. 1992, 38, 742. (4) Mawson, S.; Johnston, K. P.; Combes, J. R.; DeSimone, J. M. Formation of Poly(1,1,2,2-tetrahydroperfluorodecyl acrylate) Submicron Fibers and Particles from Supercritical Carbon Dioxide Solutions. Macromolecules 1995, 28, 3182. (5) Chernyak, Y.; Henon, F.; Harris, R. B.; Gould, R. D.; Franklin, R. K.; Edwards, J. R.; DeSimone, J. M.; Carbonell, R. G. Formation of Perfluoropolyether Coatings by the Rapid Expansion of Supercritical Solutions (RESS) Process. Part 1: Experimental Results. Ind. Eng. Chem. Res. 2002, 41, 6118. (6) Kumar, S. K.; Johnston, K. P. Modelling the Solubility of Solids in Supercritical Fluids with Density as the Independent Variable. J. Supercrit. Fluids 1988, 1, 15. (7) Krukonis, V. J. Supercritical Fluid Nucleation of Difficultto-Comminute Solids. Presented at the AIChE Fall Annual National Meeting, San Francisco, CA, 1984; Paper 140f. (8) Hannay, J. B.; Hogarth, J. On the Solubility of Solids in Gases. Proc. R. Soc. London 1879, 30, 178. (9) Kwauk, X.; Debenedetti, P. G. Mathematical Modelling of Aerosol Formation by Rapid Expansion of Supercritical Solutions in a Converging Nozzle. J. Aerosol Sci. 1993, 24, 445. (10) Ksibi, H.; Subra, P. Influence of Nozzle Design on the Nucleation Conditions in the RESS Process. Chem. Biochem. Eng. Q. 1996, 10, 69. (11) Matson, D. W.; Fulton, J. L.; Petersen, R. C.; Smith, R. D. Rapid Expansion of Supercritical Fluid Solutions: Solute Formation of Powders, Thin Films, and Fibers. Ind. Eng. Chem. Res. 1987, 26, 2298. (12) Edwards, J. R.; Franklin, R. K.; Liou, M.-S. Low-Diffusion Flux Splitting Methods for Real Fluid Flows with Phase Transitions. AIAA J. 2000, 38, 1624. (13) Franklin, R. K. Numerical Simulation of the Rapid Expansion of Supercritical Solutions Process for Deposition of Polymeric

Coating Materials. Master’s Thesis, North Carolina State University, Raleigh, NC, 2000. (14) Sanchez, I. C.; Lacombe, R. H. An Elementary Molecular Theory of Classical Fluids: Pure Fluids. J. Phys. Chem. 1976, 80, 2352. (15) Span, R.; Wagner, W. A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple-Point Temperature to 1100 K at Pressures up to 800 MPa. J. Phys. Chem. Ref. Data 1996, 25, 1511. (16) Zoller, P.; Walsh, D. Standard Pressure-Volume-Temperature Data for Polymers; Technomic Publishing Co., Inc.: Lancaster, PA, 1995; p 255. (17) McBride, B. J.; Gordon, S.; Reno, M. A. Coefficients for Calculating Thermodynamic and Transport Properties of Individual Species; NASA Report TM-4513; National Aeronautics and Space Administration: Wshington, D.C., 1993. (18) Olchowny, G. A.; Sengers, J. V. Crossover from Singular to Regular Behavior of the Transport Properties in the Critical Region. Phys. Rev. Lett. 1988, 61, 15. (19) Wilke, C. R. A Viscosity Equation for Gas Mixtures. J. Chem. Phys. 1950, 18, 517. (20) Spalart, P. R.; Allmaras, S. R. A One-Equation Turbulence Model for Aerodynamic Flows. Rech. Ae´ rosp. 1992, 1, 5. (21) Schaaf, P.; Lotz, B.; Wittman, C. Liquid-Liquid Phase Separation and Crystallization in Binary Polymer Solutions. Polymer 1987, 28, 1930. (22) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice Hall: Upper Saddle River, NJ, 1986. (23) Sanchez, I. C.; Panayiotou, C. G. Equation of State Thermodynamics of Polymer and Related Solutions. In Models for Thermodynamic and Phase Equilibria Calculations; Sandler, S. I., Ed.; Marcel Dekker: New York, 1994; p 187. (24) Friedlander, S. K. Smoke, Dust, and Haze: Fundamentals of Aerosol Behavior; Wiley-Interscience: New York, 1977. (25) Harrison, K. L.; da Rocha, S. R. P.; Yates, M. Z.; Johnston, K. P. Interfacial Activity of Polymeric Surfactants at the Polystyrene-Carbon Dioxide Interface. Langmuir 1998, 14, 6855. (26) Saffman, P.; Turner, J. On the Collision of Drops in Turbulent Clouds. J. Fluid Mech. 1956, 1, 16. (27) Wilcox, D. C. Turbulence Modeling for CFD, 2nd ed.; KNI Inc.: Anaheim, CA, 1998. (28) Chernyak, Y.; Franklin, R. K.; Edwards, J. R.; Gould, R. D.; DeSimone, J. M.; Carbonell, R. G. Delivery of Perfluoropolyether Coatings from Homogeneous Solution by the Rapid Expansion of Supercritical Solution (RESS) Process. In Proceedings of the 5th International Symposium on Supercritical Fluids; American Chemical Society: Washington, D.C., 2000. (29) Lindsay, J. An Experimental Study of the RESS Process for the Deposition of Polymeric Materials. M.S. Thesis, North Carolina State University, Raleigh, NC, 1999.

Received for review March 22, 2001 Revised manuscript received October 3, 2001 Accepted October 5, 2001 IE010268E

Formation of Perfluoropolyether Coatings by the Rapid Expansion of

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