Evaluation of Nanoparticle Effect on Bubble Nucleation in Polymer

Article pubs.acs.org/JPCC

Evaluation of Nanoparticle Effect on Bubble Nucleation in Polymer Foaming Linyan Wang,†,‡ Hongfu Zhou,† Xiangdong Wang,*,† and Jianguo Mi*,‡ †

School of Materials and Mechanical Engineering, Beijing Technology and Business University, Beijing 100048, China State Key Laboratory of Organic−Inorganic Composites, Beijing University of Chemical Technology, Beijing 100029, China

‡

ABSTRACT: We present a density functional approach to calculate the freeenergy barriers, critical radii, and nucleation rates of bubble nucleation in polystyrene and poly(methyl methacrylate) nanocomposites. In particular, the effects of surface geometry and chemistry of nanoparticles on bubble morphology and cell density have been evaluated with consideration of the local supersaturation of dissolved CO2 molecules and the local subsaturation of polymer chains. It is shown that addition of SiO2 or fluorinated SiO2 particles can improve the nucleation rates up to 4 or 5 orders of magnitude, and the critical radii shrink down to approximately half of the homogeneous nuclei, which are very helpful to fabricate low density foaming materials. The theoretical approach has been tested by the available experimental data and is expected to provide a reasonable explanation for the mechanism of inhomogeneous polymer foaming at the molecular level. underlying mechanism that is responsible for the final structure and properties of the material. Theoretical modeling of bubble formation, nucleation free-energy, and bubble growth rate are as crucial as experimental measurement for a comprehensive understanding of the process. The bubble growth mechanism in the polymer matrix has long been studied by a few of models, such as the sea−island model,12 cell−shell model,13,14 modified cavity model,15,16 and selfconsistent field theory model.17 These studies have proven that bubble growth is coherently related to the nucleation process, which decides the cell density, and in turn decides the bubble growth. Notable progress toward predicting bubble nucleation in thermoplastic polymers is the commonly used classical nucleation theory (CNT).18−21 The theory provides a basic reference with which to evaluate other theories and to interpret experimental measurements for many years. Since the macroscopic interfacial tension makes it difficult to predict the experimentally observed bubble nucleation density, the important contributions of recent investigations22−24 to the development of CNT concentrate on the correction of the interfacial tension, which is crucial to the free-energy barrier. Even though the preexponential factor has been correctly presented in the theoretical model, the final answer of freeenergy for the formation of a critical size bubble is fortuitously correct, and some errors could emerge in intermediate steps in deriving the latter quantity. The reason is that the free-energy barrier and critical radius are calculated as a function of the equilibrium gas−liquid interfacial tension, whereas actually, the

1. INTRODUCTION Polymer foams are important materials in a variety of applications due to their inherent advantages, such as low density, low thermal conductivity, and good sound isolation.1−3 The properties of polymer foams depend not only on the intrinsic properties of polymer matrix, but also on the foam morphology, such as cell density, cell size and pore size distribution. There are two types of bubble formation in polymer foaming processes: homogeneous vs heterogeneous nucleation. Homogeneous behavior refers to the nucleation in a supersaturated solution that takes place in the bulk polymer. It is not considered the primary mechanism by which most bubbles form in plastic foams, because too high activation energy is needed for bubble nucleation. In contrast, heterogeneous nucleation is supported by the existence of foreign surfaces, which promotes the growth of bubbles on the surfaces. In recent years, considerable effort has been made to optimize the foaming process to decrease the cell size and increase the cell number.4−8 Nanoparticles have received much attention as heterogeneous nucleation sites to increase cell density, reduce cell size, improve cell uniformity, and at the same time reinforce the polymeric matrix. Good control over porosity in homogeneous or heterogeneous foaming is possible in the reported experiments by manipulation of temperature, pressure, and dispersion of the selected nucleation agent.9−11 These experiments are invaluable in providing limitations and guidelines for the choice of processing conditions and properties of polymer and particles in making microcellular and nanocellular foams. However, due to the small time scales and rapid mass transport involved, it is quite difficult to achieve optimum cellular structures. One of the key reasons for the limited success is a lack of knowledge on the © 2016 American Chemical Society

Received: August 29, 2016 Revised: September 30, 2016 Published: November 1, 2016 26841

DOI: 10.1021/acs.jpcc.6b08723 J. Phys. Chem. C 2016, 120, 26841−26851

Article

The Journal of Physical Chemistry C nucleation process contains complicated interfacial phenomena. Nowadays a suitable theory that can predict the bubble nucleation and growth in polymer nanocomposites is still unavailable, owing to the deficiency in reliable description of the interfacial structure and properties in CO2−polymer−particle trinary mixtures at the molecular level. In this work, we apply a density functional theory (DFT) approach to study the thermodynamic behaviors of homogeneous and heterogeneous nucleation of CO2 bubbles in foaming of polystyrene (PS) and poly(methyl methacrylate) (PMMA). DFT is an inhomogeneous theory and has achieved good performance in polymer nanocomposites.25−27 Within the theoretical framework, the bubble nucleation occurs in the actual nanocomposite environment, involving the local supersaturation of dissolved CO2 and subsaturation of polymer matrix, as well as the interfacial tensions of bubble−melt−particle threephase contact. The morphology of the nucleated bubble and the density profiles inside and outside the bubble can be determined by minimizing the grand potential of the system. In addition, it goes beyond the classical approach by taking the free-energy to depend not on the bubble radius but on the actual spherical or sphere-cap morphology.28−30 The physical behaviors of bubble nucleation with or without addition of nanoparticles are compared to interpret different nucleation mechanisms. For instance, bare and fluorinated silica particles with different particle sizes are selected to evaluate the contribution of the chemical and geometric characteristics of particle surface on the local densities of CO2 and polymer, and thereby the free-energy barrier, nuclei radius, and growth rate of critical nuclei. In addition, the surface wettability is also considered, which is defined through the contact angle of a bubble on a particle surface in the selective polymer matrix. We note that the preexponential factor is related to the nucleation energy and thus related to the interfacial structure of bubble− melt−particle triphase contact. As a result, this work is expected to provide a solid theoretical basis to achieve better control of cell size and number in thermoplastic foams.

Figure 1. Schematics of homogeneous bubble nucleation in bulk polymer melt dissolved with CO2 molecules (a) and heterogeneous bubble nucleation on a nanoparticle surface surrounded by CO2− polymer mixture (b). The yellow spheres represent the nucleated bubbles, the gray chains denote polymer melt, the red spheres stand for CO2 molecules, the purple region indicates the nanoparticle surface, and θ is the contact angle.

Ω[ρ1(r), ρ2 (r)] = kBT ∑

∫ drρα (r)[ln(ρα (r)) − 1]

α

+ F ex[ρ1(r), ρ2 (r)]

2. THEORETICAL SECTION

+ kBT ∑

The schematics of homogeneous and heterogeneous nucleation are shown in Figure 1. For a CO2−polymer binary system without particle addition, the bubble is formed in the bulk phase surrounding by polymer and dissolved CO2 molecules (shown in Figure 1a). In homogeneous nucleation, the density profiles of CO2 and polymer melt only vary in the radial direction. While in a CO2−polymer−particle ternary system, particles provide the nucleation sites (shown in Figure 1b), and the bubbles formed on the particles show a sphere-cap morphology. To improve the computational efficiency, CO2 is modeled as spherical molecule, PS and PMMA are regarded as coarse-grained semiflexible chains, each chain contains 100 monomers, and each monomer contains two sites. The specific force field parameters for CO2, PS and PMMA molecules have been given elsewhere.31 The particle sizes vary from 15 to 150 nm, and the biggest one can be regarded as a planar wall. The grand potential, Ω[ρ1(r), ρ2(r)], can be expressed as

∫ ρα (r)(Vα(r) − μα ) dr

α

(1)

where the first term on the right-hand side indicates the intrinsic Helmholtz free-energy of ideal gas, the excess free-energy contribution of Fex[ρ1(r), ρ2(r)] has been given in the Appendix, α denotes a CO2 molecule or a site in polymer monomer, μα stands for the bulk chemical potential of component α, and the external field Vα(r) = Uαβ(r) is the overall potential exerted by the particle β. Since the particle concentration is very low, other particles have no influence on the bubble. In a homogeneous system, Vα(r) no longer exists. The pair potential Uαβ(r) is calculated through the Hamaker theory32,33 26842


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The Journal of Physical Chemistry C ⎡ 2R αRβ 2R αRβ A + 2 Uα , β(r) = − ⎢ 2 6 ⎢⎣ |r| − (R α + Rβ)2 |r| − (R α − Rβ)2

Epl =

AσLJ 6 ⎡ |r|2 − 7|r|(R α + Rβ) + 6(R α 2 + 7R αRβ + Rβ 2) ⎢ + 37800|r| ⎢⎣ (|r| − R α − Rβ)7

Epv =

−

2πr(r + R cos θ − t /2)[f [ρpl (r )] − ρpl (r )μ

∫R

7

(|r| + R α + Rβ) (|r| + R α − Rβ)

(2)

(9)

(10)

where ρv,α is the gas density of component α at 0.1 MPa, ρs,α is the density of component α around the bubble at given supersaturation (or subsaturation), and R is the nucleus radius. According to the density profiles, the nucleus boundary can be determined on the location of ρpolymer(r) = 0.5ρs,polymer. For the considered systems, all the nuclei are flat spherical caps, and the effect of particle surface plays a dominate role in the density distributions inside the bubbles. Therefore, we use the central height to approximately denote the geometry of nucleus for representation of the density profile. The calculated length in the radial direction is 5000 × 0.01σCO2, which is sufficiently large to ensure that the density profile is insensitive to the value. The final density profile and subsequent thermodynamic properties of the critical nucleus are calculated using a “pseudo” grand canonical ensemble method with ΩC[ρ(r)] = Ω[ρ(r)] + λ(N° − N). Here λ is the Lagrange multiplier, N° is the target number of fluid sites (in the given volume of the nuclei), and N is the actual value.40 Tolman length (δ) for fluid in contact with a spherical particle can be written as a function of interfacial tension41

(3)

(4)

where P is the bulk pressure and A is the interfacial area. In order to analyze the microstructure of the nucleated bubble in the heterogeneous state, the bubble morphology should be determined in advance with the curved interfacial tensions of bubble−melt, bubble−particle, and melt−particle, as well as the bubble−melt−particle contact line tension. The line tension, τ, correlates to the contact angle with the modified Young equation,36,37

⎛ 2δ ⎞⎟ γ (R N ) = γ ⎜1 − ⎝ R⎠

(5)

(11)

where γ and γ(RN) denote the interfacial tensions of fluid−plate and fluid−particle, respectively. The constrained free-energy for the bubble nucleation can be calculated with

where γpl, γpv, and γvl are the melt−particle, bubble−particle, and bubble−melt interfacial tensions, respectively. To solve the equation, an additional free-energy equilibrium equation needs to be set down. Following Marmur’s analysis,38,39 we define the line tension as the difference per unit length of the contact line lplv = 2πR sin θ between the actual interfacial energy and the independent interfacial energies: τlplv = Epl + Epv + Evl − (γplSpl + γpvSpv + γvlSvl)

[f (ρ) − ρμ + P] dρ] dr

⎧ ρ r < R(1 − cos θ ) ⎪ v, α ρα (r ) = ⎨ ⎪ ρ r ≥ R(1 − cos θ ) ⎩ s, α

An ordinary Picard iteration scheme is used in solving the equations. The procedure is repeated until the average fractional difference over any grid point between the old and the new ρα(r) is less than 1.0 × 10−3. According to the density profile, the interfacial tension γ can be estimated by

τ cos θ γpv − γpl − γvl cos θ + =0 R sin θ

ρpl (r )

∫R sin θ 2πr[∫ρ (r)

(8)

where Spl = 2πR2(1 + cos θ), Spv = 2πR2(1− cos θ), and Svl = ∫∞ Rsinθ2πr dr are the interfacial areas of melt−particle, bubble− particle, and bubble−melt, respectively, f(ρ) is the free-energy density with f(ρ) = ∂(Fid[ρ(r)]+Fex[ρ(r)])/∂ρ, and t is the thickness of the bubble−melt interface. After determination of bubble morphology, the density distributions in the region of bubble−melt−particle triphase contact can be calculated by minimizing of the global grand potential. Therefore, the nucleation free-energy barrier can be calculated by eq 1 with respect to the optimized density distributions. For bubble nucleation, the initial guess density distributions along the central height of the sphere-cap can be written as

where A = 4π2εLJρ1ρ2σLJ6 is the Hamaker constant, Rα is the radius of polymer site or CO2 molecule, Rβ is the particle radius, | r| is the distance between their centers, εLJ and σLJ are the Lennard−Jones (LJ) energy and distance cross-interaction parameters, and ρ is the density of interaction sites in the macroscopic body. According to refs 34 and 35, a silica particle contributes only the interactions of its oxygen atoms with polymer sites and CO2 molecules, and a fluorinated SiO2 particle contributes only the interaction sites of CF2. These energy and distance parameters are σO = 3.00 Å, εO = 1.91 kJ/mol, σCF2 = 4.65 Å, and εCF2 = 0.25 kJ/mol. By minimizing of the grand potential, the density profiles can be determined with

γ = ΔΩ/A = [Ω[ρ(r)] + PV ]/A

2πr(r − R cos θ − t /2)[f [ρpv (r )] − ρpv (r )μ

pv

7

⎛ ⎞ δ(F ex[ρ1(r), ρ2 (r)]) ρα (r) = exp⎜⎜μα − − βVα(r)⎟⎟ δρα (r) ⎝ ⎠

∞

∞

Evl =

|r| + 7|r|(R α − Rβ) + 6(R α 2 − 7R αRβ + Rβ 2) |r|2 − 7|r|(R α − Rβ) + 6(R α 2 − 7R αRβ + Rβ 2) ⎤ ⎥ ⎥⎦ (|r| − R α + Rβ)7

(7)

+ ρpv (r )Vext(r ) + P] dr

|r|2 + 7|r|(R α + Rβ) + 6(R α 2 + 7R αRβ + Rβ 2) 2

−

∞

+ ρpl (r )Vext(r ) + P] dr

⎛ |r|2 − (R + R )2 ⎞⎤ α β ⎟⎥ + ln⎜⎜ 2 2⎟ ⎝ |r| − (R α − Rβ) ⎠⎥⎦

+

∫R

ΔΩ = 2π(1 − cos θ) ⎧ ⎫ ∞ r 2⎨∑ {f [ρα (r )] + ρα (r ){V αext(r ) − μα }} + P ⎬ dr Rcos θ ⎩ α ⎭ (12)

∫

(6) 26843

⎪

⎪

⎪

⎪


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The Journal of Physical Chemistry C

Figure 2. Pressure−density curves in the systems of CO2−PS (a) and CO2−PMMA (b) with different mass fractions of CO2 in polymer phase. The dotted lines switch to the foaming pressure (0.1 MPa).

Table 1. Supersaturation and Subsaturation Ratios in the CO2−PS System after Pressure Declines from the Initial State to 0.1 MPaa T (K)

P (MPa)

Ssup,homo

Ssup,SiO2

Ssup,F‑SiO2

Ssub,homo

Ssub,SiO2

Ssub,F‑SiO2

313.20 338.20 363.20 313.20 313.20 313.20 338.20 363.20

0.100 0.100 0.100 7.30 13.80 17.90 17.90 17.90

1.00 1.00 1.00 326.32 526.32 631.58 544.44 464.71

1.00 1.00 1.00 578.95 947.37 1157.89 944.44 764.71

1.00 1.00 1.00 736.84 1000.00 1210.53 1055.56 882.35

0.97 0.98 0.99 0.88 0.84 0.82 0.83 0.86

0.97 0.98 0.99 0.86 0.73 0.72 0.78 0.84

0.97 0.98 0.99 0.75 0.65 0.63 0.69 0.73

a

Note that Ssup,homo, Ssup,SiO2, and Ssup,F‑SiO2 are the supersaturation ratios of CO2 in the homogeneous system and in inhomogeneous systems with addition of SiO2 or fluorinated SiO2, respectively. Ssub,homo, Ssub,SiO2, and Ssub,F‑SiO2 are the corresponding subsaturation ratios of PS matrix.

Table 2. Supersaturation and Subsaturation Ratios in the CO2−PMMA System after Pressure Declines from the Initial State to 0.1 MPaa T (K)

P (MPa)

Ssup,homo

Ssup,SiO2

Ssup,F‑SiO2

Ssub,homo

Ssub,SiO2

Ssub,F‑SiO2

313.20 338.20 363.20 313.20 313.20 313.20 338.20 363.20

0.10 0.10 0.10 7.30 13.80 17.90 17.90 17.90

1.00 1.00 1.00 590.91 863.64 1000.00 904.76 700.00

1.00 1.00 1.00 909.09 1181.82 1409.09 1238.10 1000.00

1.00 1.00 1.00 1045.45 1227.27 1454.55 1285.71 1050.00

0.97 0.97 0.97 0.78 0.72 0.67 0.74 0.81

0.97 0.97 0.97 0.78 0.63 0.59 0.65 0.71

0.97 0.97 0.97 0.67 0.56 0.53 0.58 0.63

a

Note that Ssup,homo, Ssup,SiO2, and Ssup,F‑SiO2 are the supersaturation ratios of CO2 in the homogeneous system and in inhomogeneous systems with addition of SiO2 or fluorinated SiO2, respectively. Ssub,homo, Ssub,SiO2, and Ssub,F‑SiO2 are the corresponding subsaturation ratios of PS matrix.

Finally, the stationary nucleation rate J[cm−3·s−1] is expressed as21 J = J0 exp( −ΔΩ/kBT )

3. RESULTS AND DISCUSSION It is well-known that the glass transition temperature, Tg, of PMMA or PS decreases with more and more CO2 molecules dissolved. For PMMA, Tg can be reduced from 384 K to about 280 K when saturated with CO2 in the pressure range of 5−25 MPa.43,44 For PS, with saturation pressure of CO2 in the range of 10−30 MPa, Tg can be reduced from 376 K to about 305 K.43,44 Therefore, it is possible for us to consider PMMA and PS as amorphous polymers above 313 K. CO2 solubilities and polymer swelling at high pressure have been well addressed in our previous work.31 Here we recalculate these values at given temperatures to provide the necessary inputs for the research of bubble formation. Figure 2 illustrates the density variations, where the pressure declines from different initial pressures to the foaming pressure,

(13)

where the preexponential factor J0 depends upon the particular kinetics of cluster formation of nuclei with J0 = C0f0 (2ΔΩ/(πA sm))1/2

(14)

Here C0 is the number of molecules per cm , f 0 is a fitting parameter from the experimental homogeneous nucleation42 (f 0 = 2.23 × 10−20), As is the surface area of a bubble, and m is the mass of a CO2 molecule. For simplicity, the bubble nucleation process is finished within 1 s. Accordingly, the total number of nuclei N[nuclei/cm3 ] is determined by integrating the nucleation rate over the time period. 3

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Figure 3. Density profiles of CO2 and backbone site of polymer chain on different particle surfaces. The temperature and pressure are 313.2 K and 0.10 MPa, and the mass fractions of CO2 and polymer are determined at 17.9 MPa.

Figure 4. Determination of Tolman length as a function of the surface curvature of particles for CO2−PS (a) and CO2−PMMA (b) systems. The calculation conditions are the same as those of Figure 3.

such as the ambient pressure (0.1 MPa). The detailed values are listed in Tables 1 and 2. In the process of decompression, CO2 mass fraction is a constant, but the densities of CO2 and polymer decrease. Nevertheless, the concentration of CO2 is several orders larger than the equilibrium concentration at the foaming pressure. Therefore, the polymer matrix swells, resulting in a further decrease of density and an increase of swelling ratio. In this case, the overswelled polymer matrix provides enough free volume for bubble nucleation and growth. As a consequence, CO2 concentration shows an extremely high supersaturation ratio, but the polymer concentration declines to subsaturation, which is imperative for the metastable bubble nucleation. It is shown that a higher initial pressure and a lower temperature are

more conductive to improve the CO2 supersaturation ratio, which facilitates the bubble nucleation. Moreover, the state lines in the CO2−PMMA system show more obvious deviations from the equilibrium state than those in the CO2−PS system, providing much higher supersaturation ratios of CO2. Since the particle concentration is very low, its influence on phase equilibrium can be overlooked. However, particles affect the local densities of CO2 and polymer and regulate the local supersaturation and subsaturation. It is necessary to consider such influence in heterogeneous nucleation. Figure 3 shows the density distributions of the dissolved CO2 and the backbone sites of PS or PMMA on the particle surface at 313.2 K. Here the particle size is 150 nm, the total density of binary melt is given at 26845


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Figure 5. Bubble contact angles versus the surface curvatures of particles in the systems containing CO2−PS (a) and CO2−PMMA (b). The temperature and pressure are 313.2 K and 0.10 MPa, the initial mass fractions of CO2 and polymer are determined at 17.9 MPa, and final mass fractions are corrected with the local supersaturation and subsaturation.

Figure 6. Constrained free-energy curves of the growing bubbles in CO2−PS (a) and CO2−PMMA (b) mixtures with and without addition of particles. The calculation conditions are the same as those in Figure 5.

spherical in the homogeneous system. In polymer nanocomposites, sphere-cap bubbles are formulated on the surface of particles; thus the physicochemical properties and sizes of particles have a big effect on the contact angle and finally on the cell size and density. Here we derive the contact angles on particles by combing eqs 5 and 6. Figure 5 displays that the contact angle decreases as the surface curvature increases. Given the same polymer matrix and particle size, the contact angle on unmodified nanosilica is slightly larger than that on fluorinated nanosilica. For the same particle, the contact angle in the CO2− PS system is relatively smaller than that in the CO2−PMMA system. This is the result of the competition between different interfacial tensions of bubble−melt, bubble−particle, and melt− particle. The formations of new bubbles nucleated from the mixture of CO2 and polymer without or with addition of particles are then investigated to evaluate the difference between homogeneous and heterogeneous nucleation, which is crucial to the quality of the final product. With the defined contact angle and the densities of compositions as the initial inputs, we calculate the constrained free-energy curves in the process of bubble nucleation. Figure 6 shows the results for the two systems with the addition of 150 nm particles. During bubble nucleation, it is necessary for the dissolved CO2 molecules to overcome the energy barrier and accumulate together (via local density and energy fluctuation) to form embryos of the new phase. As displayed in Figure 6, each free-energy curve has a local

0.1 MPa, but the mass fractions of CO2 and polymer are determined by the initial pressure of 17.9 MPa. In the CO2−PS systems, SiO2 particles display a weak repulsion to polymer sites, whereas fluorinated SiO2 particles show a strong repulsion, leading to different enrichments of CO2 molecules around the particles. In the CO2−PMMA systems, the repulsive interactions given by the SiO2 particles and fluorinated SiO2 particles show a more obvious difference and therefore an increased discrepancy of CO2 enrichments. In other words, the fluorinated SiO2 particles can further enhance the supersaturation of CO2 and the subsaturation of polymer. Accordingly, the average excess densities in the areas of bubble formation are obtained to correct the foaming density. Detailed information is summarized in Tables 1 and 2. For a spherical nanoparticle, Tolman length gives deep insight into the influence of surface curvature on fluid structure and properties. Figure 4 depicts the Tolman lengths in the CO2−PS and CO2−PMMA systems with addition of different particles. As the particle radius decreases, the value first declines to its minimum and then rises slightly. It is obvious that Tolman length is size-dependent. On the other hand, the Tolman length is also related to the particle chemistry. The absolute values of Tolman length on the fluorinated silica surface are slightly smaller than the ones on bare silica surface. This is also induced by the stronger repulsion of fluorinated silica to polymer chains. The contact angle of bubble on particle surface (see Figure 1b) is of vital importance. Obviously, the nucleated bubbles are 26846


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Figure 7. Density profiles of critical bubbles in the systems of CO2−PS (a), CO2−PS with addition of SiO2 (b), and CO2−PS with addition of fluorinated SiO2 (c). The calculation conditions are the same as those in Figure 5.

Figure 8. Nucleation free-energy barriers and the critical radii of bubbles for homogeneous and heterogeneous nucleation in the systems containing CO2−PS and CO2−PMMA.

and 36.0% or 52.0% in critical radius. In the CO2−PMMA systems, there is 67.7% or 83.9% reduction in energy consumption and 50.0% or 61.1% shrinkage of critical radius. In summary, addition of SiO2 particles into the two polymer systems can result in great discounts of nucleation energy barrier and critical radius, and the effect of fluorinated SiO2 particles is even more obvious. The above phenomena can be interpreted by their different microstructures or the density profiles along the central height direction of bubbles. From Figure 7, one can immediately see large density fluctuations at bubble−melt interface and small variations at bubble−particle interface during bubble formation. In general, larger density fluctuation correlates to higher interfacial energy or interfacial tension; thus the bubble−melt interface tension plays a more important role in the free-energy

maximum at R = Rc, which represents the nucleation energy barrier, and Rc is the critical radius. When an embryo has a radius less than the critical value, any increase of the radius corresponds to an increase of free-energy consumption, which means the embryo would shrink. On the other hand, when an embryo has a radius greater than the critical value, any increase in radius corresponds to a decrease of free-energy, and the bubble would grow. In the CO2−PS systems, the nucleation energy barriers without particle addition, with addition of SiO2 and with fluorinated SiO2 are 8.5, 2.9, and 1.7 × kBT, and the corresponding critical radii are 2.5, 1.6, and 1.2 nm, respectively. In the CO2−PMMA system, the values are 6.2, 2.0, and 1.0 × kBT, and 1.8, 0.9, and 0.7 nm, respectively. Therefore, the addition of SiO2 or fluorinated SiO2 particles in the CO2−PS system leads to a decrease of 65.9% or 80.0% in nucleation energy 26847


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Figure 9. Predicted nuclei number as a function of pressure. The spots in panel d are the experimental data.45 The blue squares, red triangles, and black dots represent the nucleation on fluorinated SiO2, SiO2, and in bulk system, respectively.

Figure 10. Predicted nuclei numbers versus particle diameters in the systems containing CO2−PS (a) and CO2−PMMA (b). The calculation conditions are the same as those in Figure 5.

increasing pressure. In the low pressure region, a slight increase of pressure could result in an extreme enhancement of CO2 solubility and reduction of energy barrier. As the CO2 loading increases, although more CO2 molecules and free volume are available for bubble nucleation, the improving amplitude of local supersaturation declines, resulting in the slow decline of energy and critical radius. On the other hand, lower temperature can reinforce CO2 dissolution and therefore provides larger supersaturation ratio, resulting in much lower nucleation energy. In addition, one can see that the energy barrier and critical radius of bubbles in PMMA matrix are relatively smaller. Apart from the higher CO2 composition in PMMA matrix available for bubble nucleation, PMMA has more free volume and more flexible chains. According to the free-energy barrier and critical radius, the nucleation numbers are calculated for the homogeneous and

barrier. In homogeneous nucleation (Figure 7a), the energy barrier is entirely contributed by the bubble−melt interfacial energy, while in heterogeneous nucleation, due to the direct contact of bubble with the particle surface, part of the bubble− melt interface has been replaced by the bubble−particle interface, leading to a lower interfacial energy. Comparing the bubbles nucleated on the SiO2 (Figure 7b) and fluorinated SiO2 (Figure 7c) particles, one can find that less CO2 molecules accumulated on the fluorinated SiO2 surface due to the relatively weak attraction. As a result, the much lower density fluctuations on the fluorinated SiO2 surface lead to a further decrease of the total interfacial energy. The nucleation energy barrier and critical radius in the two systems with addition of 150 nm particles at different temperatures are summarized in Figure 8. It is noticeable that both the energy barrier and the critical radius decrease with 26848



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■

APPENDIX For CO2−polymer mixture, the excess free-energy can be generalized as

heterogeneous foaming process. Here we get the total nuclei number by integrating the nucleation rate over the time period of nucleation. The results are summarized in Figure 9. At 313.2 K, the bubble has difficulty forming in the CO2−PS system below 11.4 MPa, whereas the addition of SiO2 or fluorinated SiO2 particles makes the pressure limitation decline to 6.5 or 5.1 MPa. In comparison, the critical pressures for bubble appearance in the bulk CO2−PMMA or inhomogeneous CO2−PMMA with addition of SiO2 and of fluorinated SiO2 are 9.6, 4.9 and 4.1 MPa, respectively. The number of bubbles and their sizes are determined by the competition between bubble nucleation and growth rates. In eqs 13 and 14, the free-energy appears in both exponent and preexponential factor terms; hence, a slight decrease of energy in the low pressure region could lead to a huge increase of nucleation rate. In the high pressure region, the increasing amplitude of nuclei number slows down. From Figure 9, one can draw a general conclusion: SiO2 or fluorinated SiO2 particles can improve the nucleation rate about 2 or 3 orders of magnitude. Such enhancement is even more effective as the foaming temperature increases. At 363.2 K, the effect given by the fluorinated SiO2 particles can be enhanced up to 5 orders of magnitude. The results are partially examined by the available experimental data,45 as shown in Figure 9d. The good agreement between theoretical predictions and experimental measurements indicates that the current theoretical model provides a reasonable evaluation of heterogeneous bubble nucleation rate in polymer matrix doped with nanoparticles. The nuclei numbers in the systems with the different particle additions are shown in Figure 10. When the nuclei numbers of 15 and 150 nm bare and fluorinated silica containing CO2−PS system are compared, ratios of 2.15 and 2.23 are obtained for samples soaked at 17.9 MPa. The corresponding values become 1.79 and 1.83 in the CO2−PMMA systems. It is clear that the nucleation efficiency on a small particle is less than that on large particle. Such reduction is more obvious on the fluorinated SiO2 particle surface.

F ex[ρ1(r), ρ2 (r)] = F hs[ρ1(r), ρ2 (r)] + F att[ρ1(r), ρ2 (r)] + F chain[ρ1(r), ρ2 (r)] + F stiff [ρ1(r), ρ2 (r)] (15)

to account for the free-energy contributions arising from hardsphere repulsion, dispersive attraction, chain connectivity, and chain stiffness, respectively. The fundamental measure theory is commonly used to describe the contribution of hard-sphere repulsion, which is expressed as46 F hs[ρ1(r), ρ2 (r)] = kBT

∫ Φhs[nγ (r)] dr

(16)

where the detail information Φhs[nγ(r)] can be seen elsewhere.47 Using the weighted density approximation, the attractive part can be expressed as48 F att[ρ1(r), ρ2 (r)] = kBT ∑

∫ (ρα (r){F1[ρα̅ (r)] + F2[ρα̅ (r)]}) dr

α

(17)

in which the expressions of F1[ρ̅α(r)] and F2[ρ̅α(r)] are given by the first-order mean spherical approximation expansion48 F1[ρα̅ (r)] = − 2πρα̅ (r)β ∑ ∑ xαxα ′εαα ′ α

α′

⎡ ⎛ 1 + z1, αα ′R αα ′ ⎞ ⎢k1, αα ′⎜G0, αα ′(z1, αα ′) e z1,αα ′R αα ′ − ⎟⎟ ⎜ ⎢⎣ z1, αα ′2 ⎝ ⎠ ⎛ 1 + z 2, αα ′R αα ′ ⎞⎤ ⎟⎟⎥ − k 2, αα ′⎜⎜G0, αα ′(z 2, αα ′) e z 2,αα ′R αα ′ − z 2, αα ′2 ⎝ ⎠⎥⎦ + 8πρα̅ (r)β ∑ ∑ xαxα ′εαα ′R αα ′3Iαα ′ , ∞ α

α′

− 8πρα̅ (r)β ∑ ∑ xαxα ′εαα ′g0, αα ′(R αα ′)R αα ′3Iαα ′ ,1

4. CONCLUSION In summary, the thermodynamic behaviors of homogeneous and heterogeneous nucleation of CO2 bubbles in the PS and PMMA matrixes have been calculated using a density functional approach. The effects of particle entrance on the density distributions of CO2 and polymer have been taken into account to correct the local CO2 supersaturation and polymer subsaturation. The constrained free-energy curves for homogeneous and heterogeneous nucleation have been estimated to derive out the nucleation free-energy barriers and the sizes of critical nuclei with consideration of fluid wettability. At meantime, the preexponential factors for calculations of nucleation rates have been modified with the free-energy barriers. As such, preexponential factor is no longer invariable. Finally, the nuclei numbers in the CO2−PS and CO2−PMMA systems with or without particle addition are compared to evaluate the particle effect on bubble nucleation efficiency. It is shown that addition of SiO2 or fluorinated SiO2 particles can dramatically enlarge the pressure range for polymer foaming. Addition of SiO2 or fluorinated SiO2 particles can improve the nucleation rate about 2 or 3 orders of magnitude at low temperature, and the critical radii shrink to about half of homogeneous nuclei. At high temperature, the magnitude can be improved to 4 or 5 orders. As a result, this work provides a reasonable theoretical method to evaluate the contributions of nanoparticles on bubble size and cell density in polymer foaming.

α

α′

(18) F2[ρα̅ (r)] = − πρα̅ (r)β ∑ ∑ xαxα ′εαα ′[k1, αα ′(G1, αα ′(z1, αα ′) α

α′

e z1,αα ′R αα ′) − k 2, αα ′(G2, αα ′(z 2, αα ′) e z 2,αα ′R αα ′)] − 4πρα̅ (r)β ∑ ∑ xαxα ′εαα ′g1, αα ′(R αα ′)R αα ′3Iαα ′ ,1 α

α′

(19)

Here the details of calculation of F1[ρ] and F2[ρ] can be seen elsewhere.48,49 The weighted density ρ̅α(r) is defined as ρα̅ (r) =

∑ ∫ ρα ′(r′)ωααatt′(|r − r′|) dr′ α′

(20)

att att att where ωatt ij (r) is the weight function ωαα′(r) = uαα′(r)/∫ uαα′(r) dr, and uatt (r) is the LJ attractive interaction potential between αα′ any two sites of species α and α′. The free-energy contribution from the molecular connectivity is based on the associating theory50 Nsite

F chain[ρ1(r), ρ2 (r)] =

∫ dr1 ∑ ραsite (r1) α=1

×

26849

⎛ Xaα(r1) 1⎞ α + ⎟ ⎜ln Xa (r1) − 2 2⎠ a ∈Γ α ⎝

∑

(21)


Article

The Journal of Physical Chemistry C where Γα means the set of all the associating points on site α, Xαa corresponds to the fraction of site α which are not bonded at point a, and Nsite indicates the total site number of a polymer chain. The first summation of Nsite is over all the site α, and the second over all the association points on site α. The free-energy functional due to the contribution of chain conformation is constructed to account for the polymer configurational entropy F stiff [ρ1(r), ρ2 (r)] = −

1 2

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∑ ∑ ∫ dr′ ∫ dr″ρα (r′)ρα ′(r″) α

α′

stiff cαα ′ (|r′‐r″|)

(22)

with the approximation c (r) = c (r) − c (r). Here csemiflexible(r) and cflexible(r) stand for the direct correlation functions of semiflexible and flexible polymer chains. They are calculated from the polymer reference interaction site model integral equation51 stiff

h(r ) =

semiflexible

flexible

∫ d r ′⃗ ∫ d r ″⃗ ω(| r ⃗ − r ′⃗ |)c(| r ′⃗ − r ″⃗ |) [ω(r″) + ρ h(r″)]

(23)

where h(r) is the total correlation function and ω(r) the intramolecular correlation function. ω(r) for flexible or semiflexible chains is represented by the Koyama model.52,53 The Kovalenko−Hirata approximation54 is adopted to solve the equation.

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Telephone: 010-64412597. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS This work is supported by the Natural Science Foundation of Beijing (Nos. 2162012 and 2164058). REFERENCES

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Evaluation of Nanoparticle Effect on Bubble Nucleation in Polymer

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