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Modeling nanoporosity development in polymer films for low-k applications.

INTRODUCTION

The drive for advanced integrated circuits with submicron feature size has created the need for new dielectric materials [1, 2]. The 1997 Semiconductor Industry Association National Technology Roadmap for semiconductors mandates the need for materials with a dielectric constant (k) of 1.5 along with copper metallization for 0.1 mm feature size to be implemented by the year 2007. The goal of obtaining a material with a dielectric constant below 2.0 can be reached by introducing sufficient porosity into the inter-level dielectric layer, since air has a dielectric constant close to that of vacuum (i.e., 1). Porosity should be closed-cell to minimize the rate of water absorption and to enhance mechanical integrity; in addition, pore sizes should be on the order of 10 nm or less. In this work, we investigate the ability to form nanoporous materials by modeling bubble formation in a film saturated with C[O.sub.2] after a pressure quench.

The work is organized as follows. First, we review the literature on nanoporous low-k material development and that on bubble growth in supersaturated polymers. We then present the model we used to describe bubble growth in thin films, which is an extension of the model of Arefmanesh et al. [3-5] We follow with results, discussion, and conclusions.

Generating Nanofoams

Nanopores have been generated in low-k dielectric material by several research groups [6-21]. At IBM, researchers have used thermally labile porogens that are later removed by thermolysis to produce nanoporous materials [6-17]. For organic matrices, the porogen is a thermally labile component of a nanophase-separated block copolymer. Alternatively, thermally labile star or branched macromolecules have been used as templating structures for inorganic matrices. Pore sizes on the order of 10 nm have been achieved with a reduction in dielectric constant that is proportional to the amount of porogen initially contained in the material. In a collaborative effort between researchers at IBM and Texas Tech, C[O.sub.2]-soluble porogens have also recently been removed by supercritical C[O.sub.2] extraction [18, 19]. Another route to generating nanofoams has been reported by Xu et al. [20]. In that work, nanophase separation of low molecular weight organic material occurs during cure of a spin-coated monomer, followed by evaporation to produce nanofoams. In yet another approach, Cruden et al. [21] and Winder and Gleason [22] used a pulsed plasma deposition process to produce a film in which nonbonded species could be subsequently removed leaving a nanoporous matrix.

A different approach to reducing the dielectric constant based on introducing porosity in the dielectric material is modeled in this work in which an organic dielectric material is exposed to a high-pressure gas in the supercritical region. The gas has a finite solubility in the material and as it dissolves in it, the material's glass transition temperature ([T.sub.g]) is lowered. The pressure is then suddenly quenched down to atmospheric pressure, resulting in a supersaturated solution in which bubbles can nucleate and grow if the temperature is above the plasticized [T.sub.g]. As the gas diffuses out of the material into the bubbles or out of the film, the material's [T.sub.g] increases and vitrification may occur, thereby freezing the nanocellular foam structure. This process is similar to that of polymer foaming, which generates bubbles in the micron range (microcellular foam), often by a temperature ramp through the glass transition. Here, however, as in the work of Goel and Beckman [23, 24], bubbles are nucleated by a pressure quench as opposed to a temperature ramp.

The objective of this article is to model the growth of the gas bubbles nucleated by the pressure quench process in micron-size films for low-k applications. The model is an extension of the shell model of Arefmanesh et al. [3-5], which involves associating each bubble nucleated subsequent to the pressure quench process to a finite amount of material. We assume a model matrix with properties similar to those of poly(methyl methacrylate) (PMMA), and we assume supercritical C[O.sub.2] for the processing fluid.

Supercritical fluids have certain properties that make them especially suited for the foaming process. Specifically, their liquid-like densities permit solvent dissolution to much higher concentrations than gases, and their low gas-like viscosities lead to high rates of mass transfer [23]. This leads to the ability to readily swell most polymers. The solubility limits are easily tuned by changes in the pressure and temperature. Supercritical C[O.sub.2], in particular, is a preferred solvent because it is inflammable, has a low toxicity, is inexpensive, and is the only solvent not regulated by the EPA [23].

Modeling Bubble Growth

Since the early work of Street [25], modeling the growth and/or collapse of bubbles in Newtonian and viscoelastic fluids has been a subject of considerable interest. Street [25] modeled the growth of a bubble by using a non-Newtonian power law fluid to relate the viscous stresses to the strain rates for the fluid phase. Subsequently, the growth of bubbles in an infinite sea of fluid was modeled using a Zaremba-DeWitt viscoelastic model by Yoo and Han [26] in which oscillatory growth or collapse of bubbles at very short time scales was reported. The elasticity of the medium was found by Yoo and Han to enhance the oscillations, whereas the viscosity had an opposite effect.

Amon and Denson [27] modified the equations governing bubble growth to account for the simultaneous growth of a large number of bubbles in close proximity to each other to better describe macroscopic foaming processes. A cell model was assumed in which each bubble is surrounded by a finite shell of fluid. The diffusion of gas into the shell is restricted to the dissolved gas available in the shell, and there is no loss of dissolved gas to the surroundings through the outer shell boundary. This allows the model to predict a finite value for the bubble radius. Arefmanesh et al. [3-5] extended the model from the growth of gas bubbles in Newtonian fluids [27] to that in both viscous and viscoelastic fluids. Ramesh et al. [28] showed that the viscoelastic model of Arefmannesh et al. described foaming of polystyrene better than the Newtonian model of Patel [29] and a power law model, the latter two of which were developed for a bubble in an infinite sea of fluid.

Goel and Beckman [23, 24, 30] modified the Arefmanesh et al. [3-5] finite shell model by using a single element Maxwell equation to describe the viscoelastic behavior of the fluid and most importantly, accounting for the concentration and temperature dependence of the relaxation time. The thermodynamic equilibrium properties were modeled by using a mean field lattice gas model to obtain the equilibrium absorption of dissolved gas. A classical homogenous nucleation theory was used to model nucleation subsequent to the pressure quench. Their prediction of the average cell size dependence on temperature and pressure qualitatively followed the experimental observations as a function of temperature and pressure but did not quantitatively describe the data.

A further modification of the Arefmanesh et al. model was made by Shafi et al. [31] and Joshi et al. [32], who considered the simultaneous effects of diffusion and nucleation on bubble growth. In that work, the size of the influence volume surrounding each bubble was related to the size and the time of bubble nucleation. The factors enhancing bubble growth, such as high diffusion rates and high concentration of dissolved gas, were found to also yield a wider range of bubble size distributions with a predominance of larger-sized bubbles. Ramesh [33] also extended the cell model to account for diffusion out of the cell in a viscoelastic fluid and showed that the result is smaller bubble sizes with good agreement with the experimental data.

More recently, Venerus et al. [34] modeled the growth of a single gas bubble surrounded by an infinite sea of a viscoelastic fluid and found that nonlinear viscoelastic effects were of minor importance in determining bubble growth dynamics. In a subsequent paper, Venerus [35] compared model solutions for both finite and infinite extents of a viscous incompressible Newtonian fluid. The key conclusion was that the predictions of the two models coincide at the early stages of growth but deviate at later stages, because the infinite shell model predicts a bubble growing indefinitely while the finite shell model caps the bubble growth at a finite steady-state value.

The present work is based on the Arefmanesh et al. [3-5] finite shell model, which we also modified to allow for diffusion of gas out of the shell in addition to diffusion into the bubble. This modification more accurately simulates the bubble growth process in thin films where diffusion out of the film is important. It differs from the modification made by Ramesh [33] in that we choose to solve the diffusion out problem and the bubble growth problem separately and overlay the solutions as described in the next section. Similar to Goel and Beckman's extension [23, 24, 30] of Arefmanesh et al., we assume that the viscoelastic and transport properties of the fluid depend on temperature as well as on dissolved gas concentration, although the form of our equations is slightly different. In other respects our model is simpler than those of Goel and Beckman [23, 24, 30], Shafi et al. [31], and Joshi et al. [32]; in particular, we do not model nucleation; we assume that Henry's law defines the equilibrium at the gas bubble and polymer shell interface, and we assume the gas in the bubble is ideal. These simplifications are made because our primary goal is to examine the effect of film thickness on bubble growth in thin films exposed to high-pressure C[O.sub.2] after a pressure quench, and although these assumptions will affect the size of the bubbles at equilibrium, they will not affect our general conclusions.

[FIGURE 1 OMITTED]

MODEL DESCRIPTION

We model bubble growth in a thin film that is initially supersaturated with C[O.sub.2]. The film has one free surface, with the other surface constrained by a substrate (e.g., silicon wafer in low-k applications). As already mentioned, the finite shell model of Arefmanesh et al. [3-5] is adopted in this work. This cell model, which assumes a hypothetical shell of fluid of finite thickness concentrically surrounding the gas bubble, is modified to account for the simultaneous effects of diffusion of gas into the bubble as well as out of the film. A schematic of the dissolved gas in the shell diffusing into the bubble and out of the film through the outer shell surface is shown in Fig. 1. We account for both of these processes, that of diffusion into the bubble and out of the film, in a decoupled manner. To do this we solve the governing equations for a simple diffusion of gas out of the film problem first, and the concentration profiles of dissolved gas so obtained are then imposed as a boundary condition on the outer shell. We proceed by first describing the shell model formulation and then the formulation of the diffusion-out problem.

Shell Model Formulation

Here, we follow the development in Arefmanesh et al. [3-5]. The system is assumed to be isothermal and the continuum approximation valid. Bubble growth is purely radial, with no component of the velocity in the [theta] or [phi] directions, and the bubbles are assumed not to impinge. The fluid in the shell is assumed to be incompressible. The equation of continuity is applied over the material in the shell for spherical coordinates and assuming constant density:

[[partial derivative]/[partial derivative]r]([r.sup.2][v.sub.r]) = 0 (1)

where r, [theta], and [phi] are the spherical coordinate axes and [v.sub.r] = [v.sub.r](r,t) is the velocity of the shell in the radial direction. Ignoring gravity effects, the Cauchy momentum equation applied to the material in the shell can be reduced to the following:

[rho][[[[partial derivative][v.sub.r]]/[[partial derivative]t]] + [v.sub.r][[[partial derivative][v.sub.r]]/[[partial derivative]r]]] = - [[[partial derivative]P]/[[partial derivative]r]] + [[[partial derivative][[tau].sub.rr]]/[[partial derivative]r]] + [2/r]([[tau].sub.rr] - [[tau].sub.[theta][theta]]) (2)

where [rho] is density of the shell, t is the time coordinate, P = P(r,t) is the pressure in the shell, [[tau].sub.rr] = [[tau].sub.rr] (r,t) is the deviatoric stress tensor in the shell fluid in the r direction that is due to applied stress in the r direction, and [[tau].sub.[theta][theta]] = [[tau].sub.[theta][theta]] (r,t) is the deviatoric stress tensor in the shell fluid in the [theta] direction that is due to applied stress in the [theta] direction. The factor of 2 comes in the last term because of simplification of the term [1/[r.sup.2]][[[partial derivative]([r.sup.2][[tau].sub.rr])]/[[partial derivative]r]] in the Cauchy momentum equation and because [[tau].sub.[theta][theta]] = [[tau].sub.[phi][phi]] both depend on the radial velocity, where [[tau].sub.[phi][phi]] = [[tau].sub.[phi][phi]] (r,t) is the deviatoric stress tensor in the shell fluid in the [theta] direction that is due to applied stress in the [theta] direction. It is also important to realize that only normal deviatoric stress components get included in the equation; the off-diagonal stress components are all identically zero, since they are functions of the velocity components [v.sub.[theta]] and [v.sub.[phi]], which are zero. In addition, only the radial component of the momentum equation is required, since the velocity fields in the bubble and the shell are assumed to be purely radial.

The continuity equation for the dissolved gas can be written as:

[[[partial derivative][w.sub.A]]/[[partial derivative]t]] + [v.sub.r][[[partial derivative][w.sub.A]]/[[partial derivative]r]] = [1/[r.sup.2]][[partial derivative]/[[partial derivative]r]](D[r.sup.2][[[partial derivative][W.sub.A]]/[[partial derivative]r]]) (3)

where subscript A represents the dissolved gas species, [w.sub.A] = [w.sub.A](r,t) is the concentration of dissolved gas in the shell (weight fraction units), and D is the diffusivity of the gas in the polymer/gas mixture. Fick's law of diffusion is assumed to be valid but with a nonconstant diffusivity.

The momentum and continuity equations developed require appropriate initial and boundary conditions. The value of the velocity at time t = 0 is:

[v.sub.r](r,0) = 0 (4)

implying a stationary bubble at initial time. The boundary conditions at the two boundaries R = R(t) and S = S(t), with R being the radius of the bubble and S being the radius of the shell, are:

[v.sub.r](R, t) = [[dR]/[dt]] (5)

[v.sub.r](S, t) = [[dS]/[dt]] (6)

The initial bubble nucleated size and the outer shell radius are assumed to be [R.sub.o] and [S.sub.0], respectively:

R (t = 0) = [R.sub.o] (7)

S (t = 0) = [S.sub.o] (8)

The gas in the bubble is assumed to be ideal:

[[rho].sub.g](t) = [[[M.sub.A][P.sub.g]]/[[R.sub.g]T]]. (9)

where [[rho].sub.g](t) is the density of gas in the bubble, [P.sub.g] = [P.sub.g](t) is the pressure in the gas bubble, [M.sub.A] is the molecular weight of gas, [R.sub.g] is the universal gas constant, and T is the temperature of the system (assumed constant). The pressure in the gas bubble is related to the applied pressure [P.sub.[infinity]], which is initially [P.sub.[infinity]i] at t = 0 and is [P.sub.[infinity]f] after the instantaneous pressure quench:

[P.sub.[infinity]](t) = [P.sub.[infinity]i] t = 0 (10)

[P.sub.[infinity]](t) = [P.sub.[infinity]f] t > 0. (11)

The pressure in the gas at time zero can be written as:

[P.sub.g] = [P.sub.[infinity]i] + [[2[sigma]]/[R.sub.0]] (12)

where [sigma] is the surface tension (assumed constant in most of our simulations). The deviatoric components of the stress tensor are dropped from this equation, since at time zero the velocities, and therefore the deviatoric stresses, are zero. A similar equation for the stress equality at the bubble surface is:

P(R, t) - [P.sub.g](t) - [[tau].sub.rr](R, t) + [[2[sigma]]/R] = 0 (13)

which includes the radial deviatoric component of the stress tensor in the shell fluid. Because the gas is assumed to be ideal (inviscid), the radial deviatoric stress component for the gas is zero. The boundary condition at the outer boundary is set to the applied pressure:

P(S, t) = [P.sub.[infinity]](t) (14)

The above equation is approximate because the external applied pressure acts on the film surface and not on the outer shell of fluid. The outer film surface is exposed to the atmosphere, while the inner is constrained by substrate. Since the film is thin, the pressure is assumed to be transmitted without significant change to the outer shell.

To model the polymeric viscoelastic behavior, a single element Maxwell fluid is assumed. The Maxwell equation in terms of the radial and angular stresses is:

[[tau].sub.rr] + [lambda][[[partial derivative][[tau].sub.rr]]/[[partial derivative]t]] = 2[eta][[[partial derivative][v.sub.r]]/[[partial derivative]r]] (15)

[[tau].sub.[theta][theta]] + [lambda][[[partial derivative][[tau].sub.[theta][theta]]]/[[partial derivative]t]] = 2[eta][[[partial derivative][v.sub.r]]/[[partial derivative]r]] (16)

where [lambda] is the characteristic relaxation time for the shell fluid and [eta] is the viscosity of the shell fluid. The [phi] component is not required in the solution, since it is equal to the [theta] component of the stress tensor. The initial conditions for the stresses are zero, since initially there is no fluid motion:

[[tau].sub.rr](r, 0) = 0 (17)

[[tau].sub.[theta][theta]](r, 0) = 0. (18)

The boundary condition for the mass flux at the bubble interface is given by:

[d/[dt]]([4/3][pi][[rho].sub.g][R.sup.3]) = 4[pi][rho]D[R.sup.2]([[partial derivative][w.sub.A]]/[[partial derivative]r])[.sub.r=R]. (19)

This is the mass continuity equation applied at the inner boundary. It equates the rate of increase of mass in the bubble with the amount of dissolved gas diffusing through it from the shell at the bubble interface. Note that [[rho].sub.g] is the density of the gas in the bubble and [rho] is the density of the fluid in the shell. The weight fraction of gas in the fluid at the bubble/fluid interface is given by Henry's law:

[w.sub.A](R, t) = K[P.sub.g](t) (20)

where K is the Henry's law constant. The initial condition for the concentration is then the following, since we assume that initially the situation is at equilibrium with the concentration of gas being a constant in the shell:

[w.sub.A](r, 0) = K[P.sub.g](0). (21)

The concentration boundary condition at the outer shell boundary is the imposed concentration profile obtained by solving the diffusion out of film problem (formulated in the next section):

[w.sub.A](S, t) = [c.sub.A](x, t) (22)

where is the concentration of gas in the film (weight fraction units), and x is the coordinate axis for film thickness.

The dissolved gas in the shell will affect the physical, viscoelastic, and transport properties of the fluid. The dependence of the glass transition temperature ([T.sub.g]) on the concentration of dissolved gas can be expressed by the Fox equation:

[1/[T.sub.g]] = [[w.sub.A]/[T.sub.gA]] + [[1 - [w.sub.A]]/[T.sub.gB]] (23)

where subscript A represents the gas species, in our case [CO.sub.2], and subscript B represents polymer species. The value of [T.sub.gA]--that is, the glass transition temperature for pure [CO.sub.2]--is taken to be 127 K, a value estimated from the data of Chiou et al. [36] by fitting it to the Fox equation. Although the value is a long extrapolation of the data, we have used this value in our model calculations because of a lack of explicit data on the glass transition temperature of [CO.sub.2]. The material properties of the fluid will be a function of temperature and concentration, or more simply, temperature and [T.sub.g] according to the principles of time-temperature and time-concentration (or time-[T.sub.g]) superposition. Simon and coworkers [37] have similarly applied the principle of time-[T.sub.g] superposition to model the cure kinetics of an epoxy resin. The Williams-Landel-Ferry [38] (WLF) equation, or the equivalent Vogel-Fulcher-Tammann-Hess (VFTH) equation [39-41], describes the dependence of the shift factor on temperature and [T.sub.g]:

log [a.sub.T,[T.sub.g]] = [[-[C.sub.1](T - [T.sub.g])]/[[C.sub.2] + T - [T.sub.g]]] (24)

where [C.sub.1] and [C.sub.2] are constants and equal 17.44 and 51.6 K, respectively, for the "universal" case. The shift factor [a.sub.T,[T.sub.g]] is the shift factor for either a change in temperature or a change in [T.sub.g] and can be related to the viscosity [eta], the diffusion coefficient D, and the relaxation time in the Maxwell model [lambda]. We assume for this work that all of these parameters have the same temperature and [T.sub.g] dependences (i.e., that [C.sub.1] and [C.sub.2] are the same constants for [eta], D, and [lambda]). Although this assumption is valid for [eta] and [lambda], it is not as reasonable for the diffusion coefficient, since the diffusion of small molecules often has a weaker temperature dependence than the viscosity. However, we make this assumption for D because it makes the set of nonlinear equations to be solved more tractable. Hence, we have:

log([[eta](T, [T.sub.g])]/[[eta]([T.sub.ref], [T.sub.gref])]) = log([D(T, [T.sub.g])]/[D([T.sub.ref], [T.sub.gref])]) = log([[lambda](T, [T.sub.g])]/[[lambda]([T.sub.ref], [T.sub.gref])]) = [[-[C.sub.1](T - [T.sub.g])]/[[C.sub.2] + T - [T.sub.g]]]. (25)

We remark that Eq. 25 assumes that [T.sub.ref] = [T.sub.gref]. In our work, we take the reference temperature ([T.sub.ref]) and the reference [T.sub.g]([T.sub.gref]) to equal the glass temperature of the neat polymer-that is, [T.sub.ref] = [T.sub.gref] = [T.sub.g]([w.sub.A] = 0).

Formulation of Planar Diffusion-out Model

The diffusion of gas out of the thin film is modeled independently from the diffusion of gas into the bubble. The planar diffusion-out model essentially constitutes gas diffusing out of a planar film through the outer surface. The inner surface is assumed to be impermeable because of the substrate (i.e., silicon wafer for low-k dielectric films). In the diffusion-out model, the species continuity equation for the dissolved gas is:

[[[partial derivative][c.sub.A]]/[[partial derivative]t]] = [[partial derivative]/[[partial derivative]x]](D[[[partial derivative][c.sub.A]]/[[partial derivative]x]]) (26)

where [c.sub.A] is the concentration of gas in the film (in weight fraction units), D is the diffusion coefficient, and x is the coordinate axis for film thickness. The boundary condition for no diffusion through the inner boundary closest to the substrate can be written as:

([[partial derivative][c.sub.A]]/[[partial derivative]x])[.sub.x=0] = 0. (27)

The equilibrium concentration of gas at the outer surface of the film is given by Henry's law:

[c.sub.A]([[delta].sub.o], t) = K[P.sub.[infinity]f] (28)

where [[delta].sub.o] is the film thickness that is assumed to be constant. The assumption of a constant film thickness in spite of diffusion of gas/[CO.sub.2] out of the film is a first approximation. Because of the difference in the size of the gas and polymer species and because of the dissolution of gas from the gas to the fluid state, non-ideal mixing will occur, and the volume changes on mixing are expected to be considerably smaller than additive; hence the volume changes can reasonably be neglected in this attempt at solving the bubble growth problem with diffusion out of the film. We also note that since the mass diffusion out problem is solved independently of the bubble growth problem, the effect of the radial velocity field emanating from each growing bubble is not accounted for in this approximation. At time zero, the initial concentration of gas in the film is a constant and is also given by Henry's law:

[c.sub.A](x, 0) = K[P.sub.[infinity]f]. (29)

METHODOLOGY

The set of equations involved in this model is highly nonlinear and coupled. A numerical solution is achieved by using finite difference formulas to approximate the partial derivatives. The region between the inner boundary and the outer boundary is divided into eight sections, and the difference formulas are appropriately evaluated at each point. A nonlinear equation solver, MINPACKTM, was applied to obtain the converged solution. The equations were then reevaluated for the next time step. The initial guesses required for the new solution were the solution values obtained from the previous time step. The problem is made more complex by the fact that it involves moving boundaries. To assure an even difference between mesh points, the spatial domain is remeshed at each time step before a new solution is attempted.

A total of 50 time steps were involved in each simulation. The longest time step used was 0.04 for the thickest films and considerably shorter for the thinner films. These can be compared with the relevant time scales in the problem. One pertinent time scale is that associated with the characteristic relaxation of the polymer; at [T.sub.g],

[[tau].sub.relaxation] = [[lambda].sub.o] = 100 s. (30)

As gas diffuses out of the film or into the bubble, the polymer will become more glassy and this time scale will increase; at high [CO.sub.2] concentrations when T > [T.sub.g], the time scale will be shorter: [lambda] = 5.2 s at T = 105 [degrees]C for K = 5 X [10.sup.-9] [m.sup.2]/N. In all cases, this time scale was considerably longer than the time steps used. The other important time scale is that pertaining to diffusion out of the film; when the temperature equals the glass temperature and for a film 10,000 nm thick, the time scale for diffusion out of the film is given by:

[[tau].sub.diffout] = [[[delta].sub.o.sup.2]/[D.sub.o]] = [[(10000 X [10.sup.-9]m)[.sup.2]]/[1 X [10.sup.-10] [m.sup.2][s.sup.-1]]] = 1 s (31)

where [[delta].sub.o] is the film thickness and [D.sub.o] is the value of the diffusion coefficient at [T.sub.g]. For films of 1000 nm, the time scale for diffusion out is 0.01 s at T = [T.sub.g], but in this case the time steps were much smaller than 0.01 s.

As already mentioned, the planar diffusion-out problem was solved independently form the bubble growth model by using a similar finite difference approach. The concentration profile for the dissolved gas at specified depths in the film was evaluated by using the planar diffusion model. The diffusion-out profile was fit to a ninth order polynomial equation obtained by a least squares fit to the actual modeled profile. The profile was then imposed as an outer boundary condition at for the concentration of gas in the shell, and the bubble growth model was solved.

Model Comparison to the Model Developed by Venerus et al. [34]

Before presenting the results obtained from the model calculations for the problem of interest, the results are compared with a simplifying case that was solved by Venerus et al. [34]. To make the comparison, Eqs. 22 and 25 are changed:

([[partial derivative][w.sub.A]]/[[partial derivative]r])[.sub.s] = 0 (22a)

log([[eta](T, [T.sub.g])]/[[eta]([T.sub.ref], [T.sub.gref])]) = log([D(T, [T.sub.g])]/[D([T.sub.ref], [T.sub.gref])]) = log([[lambda](T, [T.sub.g])]/[[lambda]([T.sub.ref], [T.sub.gref])]) = 0. (25a)

Our set of equations then reduces to a case solved by Venerus et al. [34], except that instead of using a single element Maxwell fluid. Venerus et al. used the nonlinear Phan-Thien Tanner fluid. However, since they concluded that the effects of nonlinear viscoelasticity are of minor importance, a good comparison between the two models is expected. To compare the two models, we convert the values of the dimensionless groups as given in Venerus et al. [34] (in fig. 1 of the referenced work) to corresponding values of dimensionless groups as used in this model. The numerical values used in our model (converted) are given below:

[[[rho][D.sub.o.sup.2]]/[[R.sub.o.sup.2]([P.sub.[infinity]i] - [P.sub.[infinity]f])]] = 10.183 X [10.sup.-9] (32)

[[[[eta].sub.o][D.sub.o.sup.2]]/[[R.sub.o.sup.2]([P.sub.[infinity]i] - [P.sub.[infinity]f])]] = 10.183 X [10.sup.-9] (33)

[[sigma]/[[R.sub.o]([P.sub.[infinity]i] - [P.sub.[infinity]f])]] = 4.073 X [10.sup.-3] (34)

[[S.sub.o]/[R.sub.o]] = [infinity] (35)

[[P.sub.[infinity]f]/[[P.sub.[infinity]i] - [P.sub.[infinity]f]]] = 10.183 X [10.sup.-3] (36)

[[[rho]kRT]/[M.sub.A]] = 10.183 X [10.sup.-3] (37)

[[[rho][D.sub.o.sup.2]]/[R.sub.o.sup.2]] = 1 (38)

[[[delta].sub.o]/[R.sub.o]] = [infinity] (39)

[1/[k([P.sub.[infinity]i] - [P.sub.[infinity]f])]] = 1.008 X [10.sup.3]. (40)

[FIGURE 2 OMITTED]

The growth of the bubble radius with the square root time obtained from our formulation is shown in Fig. 2, as well as the solution of Venerus et al. [34], with the diamonds representing the Venerus solution and the circle points representing our solution. The two solutions agree quite well, indicating that our numerical methodology is correct.

Values of Parameters Used

The main objective of this work is to conduct a parametric analysis to elucidate the effects of key model parameters on bubble growth in thin films following a pressure quench after exposure to high-pressure [CO.sub.2]. The purpose of the process is to induce nanoporosity in the films, thereby lowering the dielectric constant k. Table 1 lists the values and/or ranges of the model parameters used in our simulations. The nominal values listed in the table, which are based reports in the literature [26, 36, 42-44], are the values used in all simulations except when a particular parameter is varied in the parametric analysis. It is noted that only one parameter was varied at a time. In addition, the reasoning behind the choice of the nominal values is given in Table 1. The values of process parameters are presented in Table 2. The parametric analysis involved varying the film thickness ([[delta].sub.o]), the diffusion coefficient at [T.sub.g]([D.sub.o]), the temperature (T), and Henry's law constant (K).

RESULTS

The planar diffusion-out model was first solved, and the evolution of concentration profiles for [CO.sub.2] as it diffuses out of a film is shown in Fig. 3. The results show that the concentration of gas decreases with time and is a function of the fractional depth x/[[delta].sub.o]. Note that x/[[delta].sub.o] = 1.000 is the free surface of the film and that x/[[delta].sub.o] = 0.000 is the surface against the underlying substrate.

The concentration profile is imposed as an outer boundary condition on the shell to get the bubble growth dynamics. The effects of incorporating the diffusion out of the film are shown in Fig. 4, where the reduced bubble radius R/[R.sub.o] is plotted as a function of the reduced time t[D.sub.o]/[R.sub.o.sup.2] for the case where diffusion out of the film occurs and for the case where there is no diffusion out. The results are shown for a bubble growing at the midpoint of a film that is 10,000 nm thick. The nominal values of [D.sub.o] and K were used in the simulation. As expected, when diffusion out of the film occurs, the final steady-state radius is significantly lower than when diffusion out does not occur.

[FIGURE 3 OMITTED]

As the [CO.sub.2] gas diffuses either into the bubble or out of the film, the concentration of [CO.sub.2] in the shell decreases with time, and this in turn causes [T.sub.g] to increase with time, as shown in Fig. 5 for the same cases as shown in Fig. 4. The initial value of [T.sub.g] at time zero is around 315 K, corresponding to the [T.sub.g] of the polymer containing the equilibrium concentration of [CO.sub.2] at the applied high pressure (before the quench). As the gas diffuses into the bubble or out of the shell after the pressure quench. [T.sub.g] increases. The matrix vitrifies when [T.sub.g] reaches the processing temperature of 358 K (85[degrees]C) for this case. The limiting value approached at long times in the diffusion-out case is the [T.sub.g] of the pure polymer (378 K), since in this case all of the dissolved gas has either diffused into the bubble or diffused out of the film except for the very small fraction remaining at equilibrium at the lower quenched pressure. The [T.sub.g] for the no diffusion-out case only gradually increases in the time scale shown in Fig. 5. In both cases, the [T.sub.g] of the material closest to the bubble (r = R) increases faster than that at the outer shell radius (r = S); however, this difference is not discernible in the figure.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

The growth of bubbles will introduce porosity into the film, and this in turn is expected to decrease the dielectric constant of the film. Because of the [CO.sub.2] diffusion-out process, however, the amount of [CO.sub.2] available for bubble growth varies with depth, as was shown in Fig. 1. The result is a gradation in bubble sizes with smaller sized bubbles near the film surface, which was observed experimentally by Goel and Beckman [23]. The porosity and density of the film is expected to follow a similar gradation. From the plot of the radii with film depth, the density of the "system" (defined as the shell plus the bubble) can be determined as a function of the film depth:

[[rho].sub.sys](x, t) = [rho] + (R/S)[.sup.3] ([[rho].sub.g] - [rho]) (41)

[FIGURE 6 OMITTED]

where [[rho].sub.sys] is density of the system (shell + bubble) at a particular fractional depth x and time t. The bubble radius and shell radius, R and S, respectively, are also functions of x and t. The net density of the film can be calculated by integrating the density over film thickness. To determine the effective reduction in the dielectric constant associated with the reduction in film density, we assumed that the polymer matrix and dispersed gas bubbles act as two individual capacitors in series. For this case, the effective dielectric constant is given by [45]:

[1/[k.sub.eff]] = [z/[k.sub.air]] + [[1 - z]/[k.sub.poly]] (42)

where [k.sub.eff] is the effective dielectric constant of the nanoporous material, [k.sub.air] is the dielectric constant of air ([approximately equal to] 1.0), [k.sub.poly] is the dielectric constant of the polymer and z is the volume fraction of air and is equal to (R/S) (3). Calculating the volumes of gas and the polymer for the simulation shown in Figs. 4 and 5 for the diffusion-out case and assuming [k.sub.poly] to be 2.7 (a typical value for current low-k dielectrics), the dielectric constant decreases to 2.09, a 22.6% reduction.

The effect of temperature on the steady-state bubble radius at the midpoint of the film is shown in Fig. 6 for a 10,000 nm film and nominal values of [D.sub.o] and K. As the temperature increases to [T.sub.g] (105 [degrees]C), the bubble radius also increases because of the temperature dependence of the transport properties--that is, the decrease in viscosity and shear modulus and increase in the diffusivity with increasing T. The lower viscosity and modulus offer less resistance to bubble growth, and high diffusion rates accelerate the growth process. It should be noted that high diffusion rates accelerate both the diffusion into the bubble and diffusion out of the film. However, since the length scale leading to diffusion out of the film is much longer than the length scale leading to diffusion into the bubble (since the initial shell radius is only two times the initial bubble radius, while the film is at least a thousand times bigger), diffusion into the bubble is the preferred path, and the equilibrium radius increases with higher [D.sub.o]. Increasing bubble size with increasing temperature is observed experimentally for temperatures below [T.sub.g] [23, 28, 46, 47]. At temperatures above [T.sub.g], however, the steady-state radius that we calculated appears to level off in contrast to experimental observations that bubbles formed will collapse because of the low viscosity of the matrix at temperatures sufficiently above [T.sub.g]. This effect is not captured by our model, since we neglect gravity effects in the momentum equation and since we only assume radial flow.

The dependence of the bubble growth dynamics at 85[degrees]C on film thickness is shown in Fig. 7 for nominal values of [D.sub.o] and K. Vitrification (i.e., when [T.sub.g] rises to T) is marked for the cases in which it occurred within the time scale plotted. Growth is observed to be faster and the steady-state radius is larger for thicker films because of the longer length scale for diffusion out of the film as compared with that for diffusion into the bubble. Although the results show that the change in the equilibrium radius from a 1000 nm film to a 10,000 nm film is only several percent in absolute terms (i.e., from 1.002 for the 1000 nm film in Fig. 7 to 1.10 nm for the 10,000 nm film as shown in Fig. 4), it is considerably greater if one compares the relative growth (R/[R.sub.o] - 1).

Two distinct types of growth behavior are observed, as indicated by the results shown in Figs. 4 and 7: bubble growth profiles for films thinner than 5000 nm in which the radius increases, passes through a maximum, and then decreases before leveling off to its steady-state value; and profiles for films greater than or equal to 5000 nm in which the radius grows monotonically to its steady-state value. The maximum in the bubble radius for the thinner films is attributed to the elastic expansion subsequent to the applied pressure quench. This effect of the elastic response on the velocity profiles is reflected in the variation of the velocity profiles shown for a 1000 nm thick film in Fig. 8 for various values of [lambda], the characteristic relaxation time that is related to the elastic force. A zero value for [lambda] implies a viscous fluid, while an infinite value implies a perfectly elastic solid. The simulations shown in Fig. 8 demonstrate that lower values of [lambda] dampen the oscillations and that no undershoot (and hence, no maximum in bubble radius) is expected for a purely viscous material.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

The effect of the film thickness and temperature on the steady-state bubble radius is further investigated by plotting the steady-state radius versus film thickness for two processing temperatures in Fig. 9. The results show that the steady-state radius increases with temperature for all film thicknesses. Similarly, Figs. 10 and 11 show the steady-state bubble radius as a function of film thickness for various values of the diffusivity at [T.sub.g] ([D.sub.0]) and for various values of Henry's law constant (K), respectively. The results demonstrate that the steady-state bubble radius increases with increasing [D.sub.o] and with increasing K for all film thicknesses, as might be expected. Again, although the absolute changes in steady-state bubble radius are not large, the relative values (R/[R.sub.o] - 1) change significantly. It is also important to note that although [D.sub.o] and K were varied over orders of magnitude, the effect of changing these parameters is not as great as the effect of film thickness.

[FIGURE 9 OMITTED]

DISCUSSION

The bubble growth dynamics in micron-size films subjected to high pressure C[O.sub.2] and then a subsequent pressure quench have been investigated by a parametric analysis. The key results obtained are summarized below.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

* Modeling the diffusion of gas out of thin films is important, since the diffusion of gas out has a substantial impact on bubble growth and steady-state bubble radius.

* The bubble growth rate and the steady-state bubble radius decrease with decreasing film thickness because of increasing importance of diffusion of gas out of the film as the film thickness decreases.

* The steady-state bubble radius increases with the temperature up to the glass transition temperature of the pure polymer. Above [T.sub.g], the steady-state radius is observed to level off; however, the model does not incorporate gravity and nonradial flow effects that would result in collapse of bubbles at the highest temperatures.

* The steady-state bubble radius increases with increasing diffusivity of the gas ([D.sub.o]) and with increasing Henry's law constant (K), although the effects of these variables are small over wide ranges of the parameter space relative to the effects of film thickness.

* Two distinct types of growth behavior observed. For the nominal set of parameters studied, bubble growth profiles for films thinner than 5000 nm showed that the radius increased, passed through a maximum, and then decreased before leveling off to its steady-state value, whereas for films thicker than or equal to 5000 nm, the radius grew monotonically to its steady-state value.

For the range of parameters investigated in this work, the simulations predict a limiting value for the final steady-state bubble radius below 1.2 nm. This steady-state bubble radius is only slightly larger than the nucleation size of 1 nm. The lack of substantial bubble growth in our simulations is due to the incorporation of diffusion out of the film in the modeling. Note that in the simulations where diffusion out did not occur (Figs. 2 and 4), substantial bubble growth relative to the assumed nucleation size was observed.

For low-k nanoporous films, approximately 30% porosity with pore sizes less than 10 nm is desirable. Although this appears to be achievable based on our simulations, the result depends on extremely high nucleation rates as well as on the size of the nucleated radius and the stability of the small bubbles formed. We have assumed [R.sub.o] = 1 nm and [S.sub.o] = 2 nm in our simulations; this is akin to assuming a nucleation density of 3 X [10.sup.22] nuclei/[cm.sup.3]. Goel and Beckman predicted homogeneous nucleation densities ranging from approximately 1 X [10.sup.7] to 1 X [10.sup.12] for a PMMA/C[O.sub.2] system with the density of nucleation increasing with increasing initial pressure of C[O.sub.2] and then leveling off at approximately 30 MPa [24]; their predictions agreed with the experimentally observed number of cells (bubbles). To obtain the high nucleation rates needed to obtain porosities on the order of 30%, heterogeneous nucleation/seeding would be necessary. The size of the of the critical nucleus radius, which we took to be 1 nm, is reasonable for homogenous nucleation based on the calculations of Goel and Beckman at moderate initial C[O.sub.2] pressures ([approximately equal to] 20 MPa). However, the lack of growth subsequent to nucleation does raise questions on the stability of the bubbles formed. The initial bubble nucleus might be expected to be thermodynamically stable to minor perturbations. Larger perturbations, however, could cause the bubbles to collapse [48]. It is concluded that the ability to obtain nanoporous micron-size films of substantial porosity contents by using a supercritical C[O.sub.2] foaming process is not possible unless the path for gas diffusion out of the film is blocked--for example, by putting a sacrificial thick layer over the film to be foamed. This is in agreement with experimental observations in which PMMA films less than 10,000 nm thick did not show any significant porosity after exposure to supercritical C[O.sub.2] and a subsequent pressure quench [J. Sun and S.L. Simon, unpublished data].

SUMMARY

The bubble growth dynamics subsequent to nucleation resulting from supersaturation of a polymer-dissolved gas system have been modeled by using the shell model of Arefmanesh et al. [3-5] modified to allow for diffusion of dissolved gas out of the shell in addition to diffusion into the bubble. The model equations include mass and momentum conservation equations along with appropriate initial and boundary conditions, equations for thermodynamic equilibrium criteria, equations of state, equations describing the evolution of transport properties with concentration of dissolved gas and temperature, mechanical constitutive equations for the polymer-dissolved gas phase, and an equation of state for the supercritical gas phase. The model equations were solved by using a finite difference approach. A parametric analysis was performed to examine the effects of film thickness, temperature, diffusivity, and Henry's law constant on bubble growth dynamics. The results show that the process of diffusion out of thin films will substantially reduce the final steady-state radius of the bubbles. Hence, the foaming process is not suitable for producing nanoporous micron-size films unless the diffusion path out of the film can be blocked and sufficient heterogeneous nucleation can be accomplished.
TABLE 1. Material parameters for simulations.

Parameter Nominal value

[rho] 1169 kg/[m.sup.3]
[sigma] 2.8 X [10.sup.-2] N/m
[[eta].sub.o] [10.sup.12] Pa s
[[lambda].sub.o] 100 s
[D.sub.o] [10.sup.-10] [m.sup.2]/s
K [10.sup.-9] [m.sup.2]/N
[T.sub.gB] 378 K
[T.sub.gA] 127 K

Parameter Range

[rho] --
[sigma] --
[[eta].sub.o] --
[[lambda].sub.o] --
[D.sub.o] [10.sup.-11] [m.sup.2]/s to [10.sup.-9] [m.sup.2]/s
K 5 X [10.sup.-10] [m.sup.2]/N to
 5 X [10.sup.-9] [m.sup.2]/N
[T.sub.gB] --
[T.sub.gA] --

Parameter Comment

[rho] Value adopted for PMMA at 25[degrees]C [42]
[sigma] Value adopted for a PS-[CO.sub.2] system [26]
[[eta].sub.o] Reference value of the viscosity at [T.sub.g] [43]
[[lambda].sub.o] Reference value of the relaxation time at [T.sub.g]
 [44]
[D.sub.o] Value adopted for the diffusivity at [T.sub.g]
K Value adopted for the Henry's law constant
[T.sub.gB] [T.sub.g] for pure PMMA [42]
[T.sub.gA] [T.sub.g] for pure [CO*.sub.2]

*Calculated as by extrapolation of data of Chiou et al. [36] by fitting
to the Fox equation.

TABLE 2. Process parameters for simulations.

Parameter Nominal value

[P.sub.[infinity]i] 6.895 X [10.sup.7] N/[m.sup.2]
 (10,000 psi)
[P.sub.[infinity]f] 1.013 X [10.sup.5] N/[m.sup.2]
 (14.7 psi)
[R.sub.o] 1 X [10.sup.-9] m
 (1 nm)
[S.sub.o] 1 X [10.sup.-9] m
 (1 nm)
T 358 K
 (85[degrees]C)
[[delta].sub.o] [10.sup.-5] m
 (10,000 nm)

Parameter Range

[P.sub.[infinity]i] --
[P.sub.[infinity]f] --
[R.sub.o] --
[S.sub.o] --
T 298 K to 413 K
 (25[degrees]C to 140[degrees]C)
[[delta].sub.o] [10.sup.-6] m to [10.sup.-5] m
 (1,000 to 10,000 nm)


Contract grant sponsor: National Science Foundation (NSF-CMS); contract grant number: 0210230.

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Prathamesh Doshi, Sindee Simon

Department of Chemical Engineering, Texas Tech University, Lubbock, Texas 79409

Correspondence to: S. Simon; e-mail: sindee.simon@coe.ttu.edu
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