Article pubs.acs.org/JPCB
Nanoparticle Solvation in Polymer−CO2 Mixtures Xiaofei Xu,†,‡ Diego E. Cristancho,§ Stéphane Costeux,§ and Zhen-Gang Wang*,‡ †
Center for Soft Condensed Matter Physics and Interdisciplinary Research, Soochow University, Suzhou 215006, China Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United States § The Dow Chemical Company, Midland, Michigan 48674, United States ‡
ABSTRACT: We study the solvation of a single nanoparticle in poly(methyl methacrylate)−CO2 mixture at coexistence by using statistical classical densityfunctional theory. In the temperature range where there is triple-phase coexistence, the lowest solvation free energy occurs at the triple point pressure. Beyond the end point temperature of the triple line, and for particle radii less than a critical value, there is an optimal pressure in the solvation free energy, as a result of the competition between the creation of nanoparticle−fluid interface and the formation of cavity volume. The optimal pressure decreases with increasing nanoparticle radius or the strength of nanoparticle attraction with the fluid components. The critical radius can be estimated from the pressure dependence of the interfacial tension between the fluid and the particle in the limit of infinitely large particle size (i.e., planar wall).
I. INTRODUCTION
There is a long history of theoretical study of particle solvation in liquids.9 Scaled particle theory (SPT)10 considers the solvation of a solute particle as a two-step process: First the formation of a cavity and then application of the solute−solvent attractions. The first step accounts for the excluded volume effect and the second step describes the interactions between solute and solvent molecules and its influence on the local solvent structure. The solvent density at the cavity surface is known exactly for the limits of small and large particle sizes.9 With an approximate expression for the contact density that smoothly bridges the small and large particle limts, the solvation free energy can be computed by integrating with respect to the contact density of solvent.10,11 SPT provides an analytical method to compute the solvation free energy and has been successfully applied to a number of systems.12 Another standard route for computing solvation free energies is by molecular simulation with an explicit molecular solvent model.13−15 The solute and solvent molecules are treated in a consistent way by a realistic molecular force field. Although molecular simulation provides atomistic details of solvation, the precise estimation of free energies by computer simulation remains extremely costly. Recently, alternative strategies have been developed for rapid calculation of solvation free energy by combining molecular simulation with density-functional theory (DFT).16 In this strategy, molecular simulation is used to calculate the microscopic structure of solvents surrounding solute and the DFT is used to connect the microscopic structure to solvation free energy.
Bubble nucleation in polymer−carbon dioxide (CO2) mixtures is a problem of great practical interest for manufacturing of polymer foams. It plays a crucial role in determining the cell size and pore density of the foam material.1,2 However, controlling the foam morphology is challenging because of the low solubility and high diffusivity of CO2 in the polymer matrix.3,4 Efforts have been made to modify the foaming process to increase the cell density and decrease the cell size.3−6 A small amount of well-dispersed nanoparticles in polymer matrix may serve as nucleation sites to facilitate and control bubble nucleation.7 Moreover, novel nanocomposite foams based on the combination of functional nanoparticles and foaming technology may lead to new classes of multifunctional materials.3,8 Most nanocomposite foams to date are synthesized via a twostep process: the nanocomposite is synthesized first and then followed by bubble nucleation.3,8 The blowing CO2 is dissolved in the synthesized nanocomposite to form a homogeneous mixture by pressurization. The nanoparticles spread out in the mixture and are surrounded by the solvent molecules. Nucleation is initiated by a pressure release, resulting in a state supersaturated with CO2. The initial state for bubble nucleation is the solvated state of nanoparticle in polymer− CO2 mixtures. Therefore, the solvation free energy of the nanoparticle plays a key role in the free energy barrier of nucleation, which determines the cell density and cell size of foam materials. As a first step, in this paper we study the equilibrium solvation of a single nanoparticle under the coexistence condition. We leave the study of a particle in CO2 supersaturated states and its effects on nucleation to future work. © 2014 American Chemical Society
Special Issue: James L. Skinner Festschrift Received: January 30, 2014 Revised: April 2, 2014 Published: April 3, 2014 8002
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and the fluids, we assume that the nanoparticle consists of a uniform distribution of “point particles” that interact with the solvent molecules via dispersion forces. The distance dependence of the interaction between a point at r′ in the nanoparticle and a fluid species (a CO2 molecule or a monomer on the polymer chain) at r is given by σ ⎧ d< i ⎪+∞ 2 ⎪ pi (s) = ⎨ 6 (σi /2) σ ⎪ s≥ i ⎪−ϵPi 6 ⎩ 2 s (2)
It has been shown that density-functional theory (DFT) provides an accurate description of the distribution of the solvent particles near a planar wall or around a nanoparticle.9,17 In the DFT, the grand potential for a fluid in the presence of an external potential field is written as a functional of the oneparticle density. Equilibrium is then obtained by minimizing the grand potential with respect to the density. To study solvation by DFT, the solvent molecules are modeled explicitly and the tagged solute molecule or particle serves as an external potential field. By minimizing the grand potential, we then obtain the solvation free energy from the excess grand potential between the equilibrium solvated state and the uniform bulk state. Recently, we have developed an accurate DFT that describes both the thermodynamic bulk and interfacial properties of polymer−CO2 mixtures in a wide temperature and pressure range.18 In this work, we use this DFT to study the solvation of a nanoparticle in the polymer−CO2 mixtures. We take our polymer to be poly(methyl methacrylate) (PMMA) as an example. We find that, at a given temperature, there is an optimal pressure corresponding to the lowest solvation energy. The effects of nanoparticle size, interaction strength, and temperature on this optimal pressure are examined in detail.
where d is the distance from the center of a CO2 molecule or a monomer to the nanoparticle surface and s = |r − r′|. The potential experienced by the fluid species due to interaction with the nanoparticle is then given by an integral average over the particle volume as ϕi(r) =
N2 − 1
∏ i=1
(3)
Figure 1. Interaction potential between a nanoparticle and CO2 molecules at ε = 1 kT. d is the distance of the center of the CO2 molecule (modeled as a sphere) to the nanoparticle surface.
nanoparticle radius R. We note that the net attraction increases with R and reaches a maximum for planar wall (R = +∞). In this limit, the attractive part of eq 3 reduces to ϕi(z) = −(σi/ 2)3/z3, where z is the distance to the wall surface.19 The thermodynamic behavior of polymer−CO2 mixture is described by a DFT that is extended from the perturbed-chain statistical associating theory equation of state.20 The DFT satisfactorily captures the phase and interfacial behavior of the mixtures.18 The Helmholtz free energy functional F for the system is expressed as a sum of an ideal-gas term and excess terms accounting for contributions from excluded volume effect, chain connectivity, association, and dispersion effect. The details for the free energy functional are given in our recent publication.18 The grand potential W of the system is related to the Helmholtz free energy functional as
δ(|ri + 1 − ri| − σ2) 4πσ2 2
∫V dr′ pi (|r − r′|)
where σ1 and σ2 are the diameters of CO2 and polymer segments, respectively, and V is the volume of the spherical body of the nanoparticle. The relative attraction strength between the nanoparticle and the two fluid species depends on the chemical identity of the particle. In this work, we assume that the attraction strength of CO2 molecules to nanoparticles is half of that of the polymer segments, i.e., ϵP1 = 0.5ϵP2 ≡ ϵ. The interaction potential between the nanoparticle and a CO2 molecule as a function of the distance to the nanoparticle surface d = |r| − R ≥ 0 is shown in Figure 1 for different
II. MODEL AND THEORY We consider solvation of a single nanoparticle in a compressible PMMA−CO2 mixture. The nanoparticle is modeled as a spherical particle of radius R with its center fixed at the origin (r = 0). The molecular units of all components of the mixture are coarse-grained as spherical particles with a hard core. The CO2 molecule is modeled as a single sphere. A weak association interaction is assumed between CO2 molecules. The association interaction between the attractive sites of carbon and oxygen atoms leads to the formation of dimers, trimers, tetramers, etc. We account for the strength of association interaction by the average size (N1) of the clusters. In principle, N1 should depend on the density of CO2 and temperature. However, for simplicity in this work we ignore such dependences, and assume N1 to be a constant. Best numerical fitting of experimental PVT data of pure CO2 yields N1 = 2. Note that this clustering effect is only included in the excess part of the free energy; it is not included in the ideal part. This is different from treating CO2 as a dimerized species. The polymer, PMMA, is modeled as a freely jointed chain with N2 identical tangentially connected spheres. We take N2 = 2855, corresponding to a molecular weight of 89 230 g/mol. The correspondence between the number of spheres in the chain and the molecular weight is obtained from numerical fitting of experimental bulk PVT data. Chain connectivity is enforced by the “bonding potential” between nearest-neighbor segments, exp[ −βVB(r N2)]] =
6 πσi 3
(1)
where σ2 is the diameter of PMMA segments and δ is the Dirac delta function. rN2 = (r1, r2, ..., rN2) and β−1 = kT with k the Boltzmann constant and T the temperature. The excluded volume between the species is represented by hard-core interactions. Energetic interactions are described by the attractive part of the Lennard-Jones potential. The details for the molecular model are given in our recent publication.18 The nanoparticle is modeled as a solid body of uniform density. To represent the interaction between the nanoparticle 8003
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penalty,18 making the polymers depleted from the particle surface. The monolayer of CO2 persists even in the presence of a preferential attraction between the particle and the polymers (ε2 = 2ε1) (Figure 2b). The solvation behavior is closely related to the bulk phase behavior of mixture. For compressible polymer−CO2 mixtures, there is a triple line, at which the polymer-rich phase coexists with the CO2-rich liquid and the CO2-rich vapor phase.18,21 Figure 3 shows the CO2 solubility in the polymer-rich phase for
W [ρ1(r),ρ2̂ (r N2)] = F[ρ1(r),ρ2̂ (r N2)]
∫ [ϕ1(r) − μ1]ρ1(r) dr + ∫ [Φ2(r N ) − μ2 ]ρ2̂ (r N ) dr N +
2
2
2
(4)
where μ1 and μ2 are the chemical potential of CO2 and polymer 2 chain, respectively. Φ2(rN2) = ∑i=N i=1 ϕ2(ri) is the total external potential exerted on the polymer. ρ1(r) is the density distribution of CO2. ρ̂2(rN2) is the multidimensional density distribution of the polymer, i.e., the joint density of all the N2 segments of the polymer, which is related to the segmental densities ρ2i (i = 1, 2, ..., N2) by N2
ρ2 (r) =
N2
∑ ρ2,i (r) = ∑ ∫ dr N
2
i=1
δ(r−ri) ρ2̂ (r N2)
i=1
(5)
The density profiles of polymer and CO2 around the nanoparticle are obtained by minimizing the grand potential, δW =0 δρ1(r)
δW =0 δρ2̂ (r N2)
With the equilibrium density profiles so obtained, solvation free energy for the nanoparticle is given by difference between the grand potential for the system in presence of the particle W and the grand potential of uniform bulk Wbulk. Ws = W − Wbulk
(6)
the the the the
Figure 3. Weight fraction (solubility) of CO2 in PMMA-rich fluid.
four temperatures. At T = 290, 300, and 305 K, increasing the pressure can take the system across the triple point. The solubility curves exhibits a discontinuity at the pressure of triple point (Ptriple). For P < Ptriple, the polymer-rich phase coexists with CO2-rich-vapor phase, whereas it coexists with CO2-richliquid phase for P > Ptriple. The triple points form a triple line in T−P phase diagram, which terminates at an end point of Te = 306.8 K, slightly above the critical temperature of CO2 (Tc = 304.7 K). See ref 18 for the phase diagram. When the temperature is above 306.8 K, there is no longer triple-phase coexistence. Thus, for T = 310 K, the solubility is a continuous curve as a function of pressure. The triple point greatly affects the solvation behavior of a nanoparticle in the polymer-rich phase. As is shown in Figure 4a, the solvation free energy shows a discontinuous drop at the triple point for T = 290 and 300 K; this is due to the discontinuous jump in the nanoparticle−fluid interfacial tension at the transition point. With increasing pressure, the solvation free energy first decreases for P < Ptriple and then increases for P > Ptriple. Thus, there is an optimal pressure for the solvation of the nanoparticle, at which it has the lowest solvation free energy. This optimal pressure coincide with the triple point pressure, except very close to the end point of the triple line (Te = 306.8 K), when the optimal pressure shifts to the branch of the CO2-rich vapor phase. For T > 306.8 K, the solvation free energy becomes a continuous function of pressure. This minimum of solvation free energy arises as a result of the competition between two contributions of solvation. The solvation free energy consists of volumetric contribution for formation the cavity and a surface free energy for the creation of nanoparticle−fluid interface, i.e.,
(7)
III. RESULTS The nanoparticle is solvated in the polymer-rich phase at coexistence. The equilibrium density profiles of PMMA segments and CO2 molecules around a nanoparticle are shown in Figure 2. For a hard sphere particle (Figure 2a),
Figure 2. Dimensionless density profiles (ηi ≡ (π/6)ρiσi3) of PMMA and CO2 around a nanoparticle at T = 310 K and P = 10 MPa. d is the distance to the nanoparticle surface.
the contact value for the CO2 density is very high, giving a sharp peak for the density profile, corresponding to a monolayer of CO2 at the particle surface. This monolayer forms as a result of packing effect: the diameter of CO2 (σ1 = 0.279 nm) is smaller than that of PMMA segments (σ2 = 0.310 nm), and the connectivity of the chains also make them less favorable to stay close to the hard wall because of entropic
Ws =
4πR3 P + 4πR2γR 3
(8)
where P is the pressure in bulk phase and γR is the tension for the nanoparticle−fluid interface. Equation 8 can be considered 8004
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5c). However, as can be seen from Figure 5a,b, the optimal pressure is relatively insensitive to whether γ∞ or γR is used in the solvation energy. We can thus estimate the optimal pressure by using γ∞ instead of γR. We now derive the condition for the existence of the optimal pressure for temperatures above Te = 306.8 K. The solvation free energy in this case is a continuous function of pressure. Therefore, the condition of an optimal pressure corresponds to the stationary point in the solvation free energy with respect to pressure, i.e.,
dWs(P) =0 dP
(10)
Using eq 8, we see that at the optimal pressure P*, the following relation is satisfied Figure 4. (a1)−(a4) Solvation free energy per unit area in the absence of attraction between the particle and the fluid species (ε = 0 kT). The solid circles indicate the minimum solvation energy. (b1)−(b4) Particle−fluid interfacial tension. The solid lines are the value of γR, and the dashed lines are the value of γ∞. The nanoparticle radius is R = 5 nm.
R = −3
4πR3 P + 4πR2γ∞ 3
dP
(11)
P*
As discussed earlier, the interfacial tension γR decreases as increasing the pressure; i.e., the slope in γR vs P is negative (Figure 4b). This ensures that the right-hand side in eq 11 is always positive. From eq 11, we can define a critical radius of nanoparticle as
a definition for the nanoparticle−fluid interfacial tension, which is shown in Figure 4b (solid lines). As P increases, the solubility of CO2 in the polymer-rich phase increases. More CO2 in the mixture reduces the tension value for the nanoparticle−fluid interface.23 The interfacial tension thus is a decreasing function of pressure. As a result, with increasing pressure, this decreased interfacial tension and the increased volume contribution gives to a minimum in the solvation free energy. The value of γR is in general dependent on the particle radius. For large particle sizes, we can use the tension at a planar wall (R = +∞) as an approximation. The solvation free energy thus can be approximated as Ws ≈
dγR (P)
⎧ d γ (P ) ⎫ d γ (P ) R * ≡ −3min⎨ R * ⎬ = −3 R * P ⎩ dP ⎭ dP
P=0
(12)
which can be used to determine the existence of optimal pressure. The second half of the equation in eq 12 follows from the observation in Figure 4b that the minimum (i.e., the most negative) slope in γR vs P occurs at P = 0. For R > R*, the nanoparticle radius is greater than the maximum of the righthand side of eq 11, so that the equality can never be met. That is to say, there is no optimal pressure. The minimum solvation free energy in this case is at 0 MPa. For R < R*, there is an optimal pressure whose value is determined by the solution of eq 10. Because γR and γ∞ as a function of pressure are almost parallel to each other (see the dashed and solid lines in Figure 4b), the optimal pressures obtained using γR and γ∞ should be very close. Making use of this property, we can estimate the critical radius from γ∞ by
(9)
The results of solvation free energy from both eqs 8 and 9 are shown in Figure 5a,b. The value of solvation free energy given by γ∞ is lower than that by γR. This is because the tension value decreases with increasing the nanoparticle radius (Figure
R* ≈ −3
dγ∞(P) dP
P=0
(13)
At T = 310 K, the critical radius estimated by eq 13 is R* = 28.3 nm; this is very close to the value of 28.4 nm obtained from solving eq 12 using the radius dependent interfacial tension. The results are shown in Figure 6. For particle sizes such that R < 28.4 nm, an optimal pressure exists and its value decreases with increasing nanoparticle radius. For R ≥ 28.4 nm, the solvation free energy monotonically increases with the pressure, whose minimum value is at 0 MPa. We note that the fact that we plot the solvation free energy per unit area of the particle surface has no effects on the existence and the position of the optimal pressure, because the optimal pressure is determined by the derivative with respect to the pressure. For T < 306.8 K, the solvation free energy has a discontinuity at the pressure of triple point, due to the discontinuous jump of interfacial tension γR at the transition point. The discussion about the existence of optimal pressure given above does not generally hold in this case. Figure 7 shows the solvation free
Figure 5. Solvation free energy per unit area in the absence of particle−fluid attraction (ε = 0 kT) (a) and in the presence of particle−fluid attraction (ε = 2 kT) (b) at T = 310 K. The solid lines are the results of using γR, and the dashed lines are approximations from using γ∞. The solid circles indicate the solvation energy minima. The nanoparticle radius is R = 5 nm. (c) Dependence of particle−fluid interfacial tension on particle curvature at T = 310 K and P = 10 MPa. 8005
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Figure 8. (a) Effect of the attraction strength between the particle and the fluid on the solvation free energy at T = 310 K. The solid circles indicate the free energy minima. (b) Optimal pressure as a function of the attraction strength. The particle radius is R = 5 nm.
Figure 6. (a) Effect of particle radius on solvation free energy at T = 310 K and ε = 0 kT. The solid square is the value of interfacial tension for planar wall at 0 MPa. (b) Optimal pressure as a function of the particle curvature.
Figure 7. Solvation free energy per unit area at T = 290 K and ε = 0 kT. The solid circles are the optimal pressure.
Figure 9. Loci of optimal pressure in the T−P plane. The particle radius is R = 5 nm. The red solid circles is the end point of triple line.
energy at T = 290 K for three different nanoparticle radius. For small nanoparticle, the optimal pressure is located exactly at the triple point. With increasing nanoparticle radius, the solvation free energy increases because of the increase in volume contribution. For R = 20 nm, the optimal pressure moves to the branch of the CO2-rich-vapor phase; the solvation free energy at the triple point is not a global minimum. The existence of optimal pressure in the branch of CO2-rich-vapor phase follows the same discussion as given by eqs 10−13. Attractive interaction of nanoparticle plays an important role in the solvation free energy. Figure 8 shows the effect of the strength of attraction on the solvation free energy. With increased interaction strength, more CO2 gets absorbed around its surface, leading to a lower interfacial tension. As a result, the optimal pressure decreases with increasing the strength of the attraction (Figure 8b). The critical radius R* of the existence of optimal pressure also becomes smaller than that for the case of the hard particle (ε = 0 kT) (Figure 6b). Finally, for a global picture of the solvation behavior, we present the locus for the minimum solvation free energy in T− P plane in Figure 9 for R = 5 nm. Below the curve, the formation of nanoparticle−fluid interface is the dominant contribution in the solvation free energy, and the solvation free
energy decreases with increasing pressure. Above the curve, the creation of cavity volume dominates, and the solvation free energy increases with pressure. The dot-dashed line is the triple line, which terminates at an end point at Te = 306.8 K, which is slightly above the critical temperature of CO2 (Tc = 304.7 K).18 Above this temperature, there is no triple phase coexistence. For temperatures well below Te = 306.8 K, the optimal pressure coincides with the pressure on the triple line for R = 5 nm. However, for larger nanoparticle sizes, the optimal pressure will move to the branch of CO2-rich-vapor phase; as shown by the curve for R = 20 nm in Figure 7. As we increase the temperature toward Te, the optimal pressure drops off from the pressure on the triple line. This is the behavior shown in Figure 4(a3). The optimal pressure as a function of temperature reaches a maximum value near the triple line and rapidly decreases as the temperature is further increased.
IV. CONCLUSIONS We have studied the solvation of a single nanoparticle in compressible polymer−CO2 mixtures by using a statistical mechanical density-functional theory. For nanoparticle radius smaller than a critical value, which is determined by the derivative of the interfacial tension between the particle and the 8006
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fluid mixture with respect to pressure, the solvation free energy has a minimum as a function of pressure. This minimum arises because of the competition between the volumetric and particle−fluid interfacial contributions. The CO2 density reaches a high contact value at the surface of nanoparticle, giving a monolayer of CO2 at the nanoparticle interface. The solvation behavior of a nanoparticle is closely related to the solubility of nanoparticles in the fluids and plays an essential role in the heterogeneous nucleation of CO2 bubble in polymer−CO2 mixtures induced by nanoparticles. For the nucleation, the evolution of density profiles along the minimum free energy path for nucleation initiated by the nanoparticles can be obtained by combining this work (extended to the metastable mixtures) with the string method,24−26 which will yield the necessary information on the pathways and barriers in heterogeneous nucleation. Such an effort will be undertaken in future.
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(13) Kollman, P. Free energy calculations: Applications to chemical and biochemical phenomena. Chem. Rev. 1993, 93, 2395−2417. (14) Chandler, D. Interfaces and the driving force of hydrophobic assembly. Nature 2005, 437, 640−647. (15) Bolhuis, P. G.; Chandler, D.; Dellago, C.; Geissler, P. L. Transition path sampling: Throwing ropes over rough mountain passes, in the dark. Annu. Rev. Phys. Chem. 2002, 53, 291−318. (16) Zhao, S. L.; Jin, Z. H.; Wu, J. Z. New theoretical method for rapid prediction of solvation free energy in water. J. Phys. Chem. B 2011, 115, 6971−6975. (17) Wu, J. Z. Density functional theory for liquid structure and thermodynamics. In Molecular Thermodynamics of Complex Systems; Lu, X., Hu, Y., Eds.; Structure and Bonding, Vol. 131; Springer-Verlag: Berlin, 2009; pp 1−73. (18) Xu, X.; Cristancho, D. E.; Costeux, S.; Wang, Z.-G. Densityfunctional theory for polymer-carbon dioxide mixtures: A perturbedchain SAFT approach. J. Chem. Phys. 2012, 137, 054902. (19) Israelachvili, Jacob N. Intermolecular and surface forces; Academic Press: Boston, 1992. (20) Gross, J.; Sadowski, G. Perturbed-chain SAFT: An equation of state based on a perturbation theory for chain molecules. Ind. Eng. Chem. Res. 2001, 40, 1244−1260. (21) Müller, M.; MacDowell, L. G.; Virnau, P.; Binder, K. Interface properties and bubble nucleation in compressible mixtures containing polymers. J. Chem. Phys. 2002, 117, 5480−5496. (22) Xu, X.; Cao, D. P.; Wu, J. Z. Density functional theory for predicting polymeric forces against surface fouling. Soft. Matter 2010, 6, 4631−4646. (23) Heni, M.; Löwen, H. Interfacial free energy of hard-sphere fluids and solids near a hard wall. Phys. Rev. E 1999, 60, 7057−7065. (24) Xu, X.; Cristancho, D. E.; Costeux, S.; Wang, Z.-G. Discontinuous bubble nucleation due to metastable phase transition in polymer-CO2 mixtures. J. Phys. Chem. Lett. 2013, 4, 1639−1643. (25) W.N, E.; Vanden-Eijnden, E. Transition-path theory and pathfinding algorithms for the study of rare events. Annu. Rev. Phys. Chem. 2010, 61, 391−420. (26) W.N, E.; Ren, W. Q.; Vanden-Eijnden, E. Simplified and improved string method for computing the minimum energy paths in barrier-crossing events. J. Chem. Phys. 2007, 126, 164103.
AUTHOR INFORMATION
Corresponding Author
*Electronic mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The Dow Chemical Company is acknowledged for funding and for permission to publish the results. The computing facility on which the calculations were performed is supported by an NSFMRI grant, Award No. CHE-1040558.
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REFERENCES
(1) Lee, S. T., Ramesh, N. S., Eds. Polymeric foams: Mechanisms and materials. Polymeric Foams Series; CRC Press: Boca Raton, FL, 2004; pp 1−318. (2) Lee, S. T., Park, C. B., Ramesh, N. S., Eds. Polymeric foams: Science and technology. Polymeric Foams Series; CRC Press: Boca Raton, FL, 2007; pp 1−218. (3) Lee, L. J.; Zeng, C. C.; Cao, X.; Han, X. M.; Shen, J.; Xu, G. J. Polymer nanocomposite foams. Compos. Sci. Technol. 2005, 65, 2344− 2363. (4) Tomasko, D. L.; Li, H.; Liu, D. H.; Han, X. M.; Wingert, M. J.; Lee, L. J.; Koelling, K. W. A review of CO2 applications in the processing of polymers. Ind. Eng. Chem. Res. 2003, 42, 6431−6456. (5) Zeng, C. C.; Han, X. M.; Lee, L. J.; Koelling, K. W.; Tomasko, D. L. Polymer-clay nanocomposite foams prepared using carbon dioxide. Adv. Mater. 2003, 15, 1743−1747. (6) Siripurapu, S.; DeSimone, J. M.; Khan, S. A.; Spontak, R. J. Controlled foaming of polymer films through restricted surface diffusion and the addition of nanosilica particles or CO2-philic surfactants. Macromolecules 2005, 38, 2271−2280. (7) Costeux, S.; Zhu, L. B. Low density thermoplastic nanofoams nucleated by nanoparticles. Polymer 2013, 54, 2785−2795. (8) Tomasko, D. L.; Han, X. M.; Liu, D. H.; Gao, W. H. Supercritical fluid applications in polymer nanocomposites. Curr. Opin. Solid State Mater. Sci. 2003, 7, 407−412. (9) Wu, J. Z. Solvation of a spherical cavity in simple liquids: Interpolating between the limits. J. Phys. Chem. B 2009, 113, 6813− 6818. (10) Reiss, H.; Frisch, H. L.; Lebowitz, J. L. Statistical mechanics of rigid spheres. J. Chem. Phys. 1959, 31, 369−380. (11) Helfand, E.; Reiss, H.; Frisch, H. L.; Lebowitz, J. L. Scaled particle theory of fluids. J. Chem. Phys. 1960, 33, 1379−1385. (12) Ashbaugh, H. S.; Pratt, L. R. Colloquium: Scaled particle theory and the length scales of hydrophobicity. Rev. Mod. Phys. 2006, 78, 159−178. 8007
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Nanoparticle Solvation in Polymer−CO2 Mixtures Xiaofei Xu,†,‡ Diego E. Cristancho,§ Stéphane Costeux,§ and Zhen-Gang Wang*,‡ †
Center for Soft Condensed Matter Physics and Interdisciplinary Research, Soochow University, Suzhou 215006, China Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United States § The Dow Chemical Company, Midland, Michigan 48674, United States ‡
ABSTRACT: We study the solvation of a single nanoparticle in poly(methyl methacrylate)−CO2 mixture at coexistence by using statistical classical densityfunctional theory. In the temperature range where there is triple-phase coexistence, the lowest solvation free energy occurs at the triple point pressure. Beyond the end point temperature of the triple line, and for particle radii less than a critical value, there is an optimal pressure in the solvation free energy, as a result of the competition between the creation of nanoparticle−fluid interface and the formation of cavity volume. The optimal pressure decreases with increasing nanoparticle radius or the strength of nanoparticle attraction with the fluid components. The critical radius can be estimated from the pressure dependence of the interfacial tension between the fluid and the particle in the limit of infinitely large particle size (i.e., planar wall).
I. INTRODUCTION
There is a long history of theoretical study of particle solvation in liquids.9 Scaled particle theory (SPT)10 considers the solvation of a solute particle as a two-step process: First the formation of a cavity and then application of the solute−solvent attractions. The first step accounts for the excluded volume effect and the second step describes the interactions between solute and solvent molecules and its influence on the local solvent structure. The solvent density at the cavity surface is known exactly for the limits of small and large particle sizes.9 With an approximate expression for the contact density that smoothly bridges the small and large particle limts, the solvation free energy can be computed by integrating with respect to the contact density of solvent.10,11 SPT provides an analytical method to compute the solvation free energy and has been successfully applied to a number of systems.12 Another standard route for computing solvation free energies is by molecular simulation with an explicit molecular solvent model.13−15 The solute and solvent molecules are treated in a consistent way by a realistic molecular force field. Although molecular simulation provides atomistic details of solvation, the precise estimation of free energies by computer simulation remains extremely costly. Recently, alternative strategies have been developed for rapid calculation of solvation free energy by combining molecular simulation with density-functional theory (DFT).16 In this strategy, molecular simulation is used to calculate the microscopic structure of solvents surrounding solute and the DFT is used to connect the microscopic structure to solvation free energy.
Bubble nucleation in polymer−carbon dioxide (CO2) mixtures is a problem of great practical interest for manufacturing of polymer foams. It plays a crucial role in determining the cell size and pore density of the foam material.1,2 However, controlling the foam morphology is challenging because of the low solubility and high diffusivity of CO2 in the polymer matrix.3,4 Efforts have been made to modify the foaming process to increase the cell density and decrease the cell size.3−6 A small amount of well-dispersed nanoparticles in polymer matrix may serve as nucleation sites to facilitate and control bubble nucleation.7 Moreover, novel nanocomposite foams based on the combination of functional nanoparticles and foaming technology may lead to new classes of multifunctional materials.3,8 Most nanocomposite foams to date are synthesized via a twostep process: the nanocomposite is synthesized first and then followed by bubble nucleation.3,8 The blowing CO2 is dissolved in the synthesized nanocomposite to form a homogeneous mixture by pressurization. The nanoparticles spread out in the mixture and are surrounded by the solvent molecules. Nucleation is initiated by a pressure release, resulting in a state supersaturated with CO2. The initial state for bubble nucleation is the solvated state of nanoparticle in polymer− CO2 mixtures. Therefore, the solvation free energy of the nanoparticle plays a key role in the free energy barrier of nucleation, which determines the cell density and cell size of foam materials. As a first step, in this paper we study the equilibrium solvation of a single nanoparticle under the coexistence condition. We leave the study of a particle in CO2 supersaturated states and its effects on nucleation to future work. © 2014 American Chemical Society
Special Issue: James L. Skinner Festschrift Received: January 30, 2014 Revised: April 2, 2014 Published: April 3, 2014 8002
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and the fluids, we assume that the nanoparticle consists of a uniform distribution of “point particles” that interact with the solvent molecules via dispersion forces. The distance dependence of the interaction between a point at r′ in the nanoparticle and a fluid species (a CO2 molecule or a monomer on the polymer chain) at r is given by σ ⎧ d< i ⎪+∞ 2 ⎪ pi (s) = ⎨ 6 (σi /2) σ ⎪ s≥ i ⎪−ϵPi 6 ⎩ 2 s (2)
It has been shown that density-functional theory (DFT) provides an accurate description of the distribution of the solvent particles near a planar wall or around a nanoparticle.9,17 In the DFT, the grand potential for a fluid in the presence of an external potential field is written as a functional of the oneparticle density. Equilibrium is then obtained by minimizing the grand potential with respect to the density. To study solvation by DFT, the solvent molecules are modeled explicitly and the tagged solute molecule or particle serves as an external potential field. By minimizing the grand potential, we then obtain the solvation free energy from the excess grand potential between the equilibrium solvated state and the uniform bulk state. Recently, we have developed an accurate DFT that describes both the thermodynamic bulk and interfacial properties of polymer−CO2 mixtures in a wide temperature and pressure range.18 In this work, we use this DFT to study the solvation of a nanoparticle in the polymer−CO2 mixtures. We take our polymer to be poly(methyl methacrylate) (PMMA) as an example. We find that, at a given temperature, there is an optimal pressure corresponding to the lowest solvation energy. The effects of nanoparticle size, interaction strength, and temperature on this optimal pressure are examined in detail.
where d is the distance from the center of a CO2 molecule or a monomer to the nanoparticle surface and s = |r − r′|. The potential experienced by the fluid species due to interaction with the nanoparticle is then given by an integral average over the particle volume as ϕi(r) =
N2 − 1
∏ i=1
(3)
Figure 1. Interaction potential between a nanoparticle and CO2 molecules at ε = 1 kT. d is the distance of the center of the CO2 molecule (modeled as a sphere) to the nanoparticle surface.
nanoparticle radius R. We note that the net attraction increases with R and reaches a maximum for planar wall (R = +∞). In this limit, the attractive part of eq 3 reduces to ϕi(z) = −(σi/ 2)3/z3, where z is the distance to the wall surface.19 The thermodynamic behavior of polymer−CO2 mixture is described by a DFT that is extended from the perturbed-chain statistical associating theory equation of state.20 The DFT satisfactorily captures the phase and interfacial behavior of the mixtures.18 The Helmholtz free energy functional F for the system is expressed as a sum of an ideal-gas term and excess terms accounting for contributions from excluded volume effect, chain connectivity, association, and dispersion effect. The details for the free energy functional are given in our recent publication.18 The grand potential W of the system is related to the Helmholtz free energy functional as
δ(|ri + 1 − ri| − σ2) 4πσ2 2
∫V dr′ pi (|r − r′|)
where σ1 and σ2 are the diameters of CO2 and polymer segments, respectively, and V is the volume of the spherical body of the nanoparticle. The relative attraction strength between the nanoparticle and the two fluid species depends on the chemical identity of the particle. In this work, we assume that the attraction strength of CO2 molecules to nanoparticles is half of that of the polymer segments, i.e., ϵP1 = 0.5ϵP2 ≡ ϵ. The interaction potential between the nanoparticle and a CO2 molecule as a function of the distance to the nanoparticle surface d = |r| − R ≥ 0 is shown in Figure 1 for different
II. MODEL AND THEORY We consider solvation of a single nanoparticle in a compressible PMMA−CO2 mixture. The nanoparticle is modeled as a spherical particle of radius R with its center fixed at the origin (r = 0). The molecular units of all components of the mixture are coarse-grained as spherical particles with a hard core. The CO2 molecule is modeled as a single sphere. A weak association interaction is assumed between CO2 molecules. The association interaction between the attractive sites of carbon and oxygen atoms leads to the formation of dimers, trimers, tetramers, etc. We account for the strength of association interaction by the average size (N1) of the clusters. In principle, N1 should depend on the density of CO2 and temperature. However, for simplicity in this work we ignore such dependences, and assume N1 to be a constant. Best numerical fitting of experimental PVT data of pure CO2 yields N1 = 2. Note that this clustering effect is only included in the excess part of the free energy; it is not included in the ideal part. This is different from treating CO2 as a dimerized species. The polymer, PMMA, is modeled as a freely jointed chain with N2 identical tangentially connected spheres. We take N2 = 2855, corresponding to a molecular weight of 89 230 g/mol. The correspondence between the number of spheres in the chain and the molecular weight is obtained from numerical fitting of experimental bulk PVT data. Chain connectivity is enforced by the “bonding potential” between nearest-neighbor segments, exp[ −βVB(r N2)]] =
6 πσi 3
(1)
where σ2 is the diameter of PMMA segments and δ is the Dirac delta function. rN2 = (r1, r2, ..., rN2) and β−1 = kT with k the Boltzmann constant and T the temperature. The excluded volume between the species is represented by hard-core interactions. Energetic interactions are described by the attractive part of the Lennard-Jones potential. The details for the molecular model are given in our recent publication.18 The nanoparticle is modeled as a solid body of uniform density. To represent the interaction between the nanoparticle 8003
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penalty,18 making the polymers depleted from the particle surface. The monolayer of CO2 persists even in the presence of a preferential attraction between the particle and the polymers (ε2 = 2ε1) (Figure 2b). The solvation behavior is closely related to the bulk phase behavior of mixture. For compressible polymer−CO2 mixtures, there is a triple line, at which the polymer-rich phase coexists with the CO2-rich liquid and the CO2-rich vapor phase.18,21 Figure 3 shows the CO2 solubility in the polymer-rich phase for
W [ρ1(r),ρ2̂ (r N2)] = F[ρ1(r),ρ2̂ (r N2)]
∫ [ϕ1(r) − μ1]ρ1(r) dr + ∫ [Φ2(r N ) − μ2 ]ρ2̂ (r N ) dr N +
2
2
2
(4)
where μ1 and μ2 are the chemical potential of CO2 and polymer 2 chain, respectively. Φ2(rN2) = ∑i=N i=1 ϕ2(ri) is the total external potential exerted on the polymer. ρ1(r) is the density distribution of CO2. ρ̂2(rN2) is the multidimensional density distribution of the polymer, i.e., the joint density of all the N2 segments of the polymer, which is related to the segmental densities ρ2i (i = 1, 2, ..., N2) by N2
ρ2 (r) =
N2
∑ ρ2,i (r) = ∑ ∫ dr N
2
i=1
δ(r−ri) ρ2̂ (r N2)
i=1
(5)
The density profiles of polymer and CO2 around the nanoparticle are obtained by minimizing the grand potential, δW =0 δρ1(r)
δW =0 δρ2̂ (r N2)
With the equilibrium density profiles so obtained, solvation free energy for the nanoparticle is given by difference between the grand potential for the system in presence of the particle W and the grand potential of uniform bulk Wbulk. Ws = W − Wbulk
(6)
the the the the
Figure 3. Weight fraction (solubility) of CO2 in PMMA-rich fluid.
four temperatures. At T = 290, 300, and 305 K, increasing the pressure can take the system across the triple point. The solubility curves exhibits a discontinuity at the pressure of triple point (Ptriple). For P < Ptriple, the polymer-rich phase coexists with CO2-rich-vapor phase, whereas it coexists with CO2-richliquid phase for P > Ptriple. The triple points form a triple line in T−P phase diagram, which terminates at an end point of Te = 306.8 K, slightly above the critical temperature of CO2 (Tc = 304.7 K). See ref 18 for the phase diagram. When the temperature is above 306.8 K, there is no longer triple-phase coexistence. Thus, for T = 310 K, the solubility is a continuous curve as a function of pressure. The triple point greatly affects the solvation behavior of a nanoparticle in the polymer-rich phase. As is shown in Figure 4a, the solvation free energy shows a discontinuous drop at the triple point for T = 290 and 300 K; this is due to the discontinuous jump in the nanoparticle−fluid interfacial tension at the transition point. With increasing pressure, the solvation free energy first decreases for P < Ptriple and then increases for P > Ptriple. Thus, there is an optimal pressure for the solvation of the nanoparticle, at which it has the lowest solvation free energy. This optimal pressure coincide with the triple point pressure, except very close to the end point of the triple line (Te = 306.8 K), when the optimal pressure shifts to the branch of the CO2-rich vapor phase. For T > 306.8 K, the solvation free energy becomes a continuous function of pressure. This minimum of solvation free energy arises as a result of the competition between two contributions of solvation. The solvation free energy consists of volumetric contribution for formation the cavity and a surface free energy for the creation of nanoparticle−fluid interface, i.e.,
(7)
III. RESULTS The nanoparticle is solvated in the polymer-rich phase at coexistence. The equilibrium density profiles of PMMA segments and CO2 molecules around a nanoparticle are shown in Figure 2. For a hard sphere particle (Figure 2a),
Figure 2. Dimensionless density profiles (ηi ≡ (π/6)ρiσi3) of PMMA and CO2 around a nanoparticle at T = 310 K and P = 10 MPa. d is the distance to the nanoparticle surface.
the contact value for the CO2 density is very high, giving a sharp peak for the density profile, corresponding to a monolayer of CO2 at the particle surface. This monolayer forms as a result of packing effect: the diameter of CO2 (σ1 = 0.279 nm) is smaller than that of PMMA segments (σ2 = 0.310 nm), and the connectivity of the chains also make them less favorable to stay close to the hard wall because of entropic
Ws =
4πR3 P + 4πR2γR 3
(8)
where P is the pressure in bulk phase and γR is the tension for the nanoparticle−fluid interface. Equation 8 can be considered 8004
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5c). However, as can be seen from Figure 5a,b, the optimal pressure is relatively insensitive to whether γ∞ or γR is used in the solvation energy. We can thus estimate the optimal pressure by using γ∞ instead of γR. We now derive the condition for the existence of the optimal pressure for temperatures above Te = 306.8 K. The solvation free energy in this case is a continuous function of pressure. Therefore, the condition of an optimal pressure corresponds to the stationary point in the solvation free energy with respect to pressure, i.e.,
dWs(P) =0 dP
(10)
Using eq 8, we see that at the optimal pressure P*, the following relation is satisfied Figure 4. (a1)−(a4) Solvation free energy per unit area in the absence of attraction between the particle and the fluid species (ε = 0 kT). The solid circles indicate the minimum solvation energy. (b1)−(b4) Particle−fluid interfacial tension. The solid lines are the value of γR, and the dashed lines are the value of γ∞. The nanoparticle radius is R = 5 nm.
R = −3
4πR3 P + 4πR2γ∞ 3
dP
(11)
P*
As discussed earlier, the interfacial tension γR decreases as increasing the pressure; i.e., the slope in γR vs P is negative (Figure 4b). This ensures that the right-hand side in eq 11 is always positive. From eq 11, we can define a critical radius of nanoparticle as
a definition for the nanoparticle−fluid interfacial tension, which is shown in Figure 4b (solid lines). As P increases, the solubility of CO2 in the polymer-rich phase increases. More CO2 in the mixture reduces the tension value for the nanoparticle−fluid interface.23 The interfacial tension thus is a decreasing function of pressure. As a result, with increasing pressure, this decreased interfacial tension and the increased volume contribution gives to a minimum in the solvation free energy. The value of γR is in general dependent on the particle radius. For large particle sizes, we can use the tension at a planar wall (R = +∞) as an approximation. The solvation free energy thus can be approximated as Ws ≈
dγR (P)
⎧ d γ (P ) ⎫ d γ (P ) R * ≡ −3min⎨ R * ⎬ = −3 R * P ⎩ dP ⎭ dP
P=0
(12)
which can be used to determine the existence of optimal pressure. The second half of the equation in eq 12 follows from the observation in Figure 4b that the minimum (i.e., the most negative) slope in γR vs P occurs at P = 0. For R > R*, the nanoparticle radius is greater than the maximum of the righthand side of eq 11, so that the equality can never be met. That is to say, there is no optimal pressure. The minimum solvation free energy in this case is at 0 MPa. For R < R*, there is an optimal pressure whose value is determined by the solution of eq 10. Because γR and γ∞ as a function of pressure are almost parallel to each other (see the dashed and solid lines in Figure 4b), the optimal pressures obtained using γR and γ∞ should be very close. Making use of this property, we can estimate the critical radius from γ∞ by
(9)
The results of solvation free energy from both eqs 8 and 9 are shown in Figure 5a,b. The value of solvation free energy given by γ∞ is lower than that by γR. This is because the tension value decreases with increasing the nanoparticle radius (Figure
R* ≈ −3
dγ∞(P) dP
P=0
(13)
At T = 310 K, the critical radius estimated by eq 13 is R* = 28.3 nm; this is very close to the value of 28.4 nm obtained from solving eq 12 using the radius dependent interfacial tension. The results are shown in Figure 6. For particle sizes such that R < 28.4 nm, an optimal pressure exists and its value decreases with increasing nanoparticle radius. For R ≥ 28.4 nm, the solvation free energy monotonically increases with the pressure, whose minimum value is at 0 MPa. We note that the fact that we plot the solvation free energy per unit area of the particle surface has no effects on the existence and the position of the optimal pressure, because the optimal pressure is determined by the derivative with respect to the pressure. For T < 306.8 K, the solvation free energy has a discontinuity at the pressure of triple point, due to the discontinuous jump of interfacial tension γR at the transition point. The discussion about the existence of optimal pressure given above does not generally hold in this case. Figure 7 shows the solvation free
Figure 5. Solvation free energy per unit area in the absence of particle−fluid attraction (ε = 0 kT) (a) and in the presence of particle−fluid attraction (ε = 2 kT) (b) at T = 310 K. The solid lines are the results of using γR, and the dashed lines are approximations from using γ∞. The solid circles indicate the solvation energy minima. The nanoparticle radius is R = 5 nm. (c) Dependence of particle−fluid interfacial tension on particle curvature at T = 310 K and P = 10 MPa. 8005
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Figure 8. (a) Effect of the attraction strength between the particle and the fluid on the solvation free energy at T = 310 K. The solid circles indicate the free energy minima. (b) Optimal pressure as a function of the attraction strength. The particle radius is R = 5 nm.
Figure 6. (a) Effect of particle radius on solvation free energy at T = 310 K and ε = 0 kT. The solid square is the value of interfacial tension for planar wall at 0 MPa. (b) Optimal pressure as a function of the particle curvature.
Figure 7. Solvation free energy per unit area at T = 290 K and ε = 0 kT. The solid circles are the optimal pressure.
Figure 9. Loci of optimal pressure in the T−P plane. The particle radius is R = 5 nm. The red solid circles is the end point of triple line.
energy at T = 290 K for three different nanoparticle radius. For small nanoparticle, the optimal pressure is located exactly at the triple point. With increasing nanoparticle radius, the solvation free energy increases because of the increase in volume contribution. For R = 20 nm, the optimal pressure moves to the branch of the CO2-rich-vapor phase; the solvation free energy at the triple point is not a global minimum. The existence of optimal pressure in the branch of CO2-rich-vapor phase follows the same discussion as given by eqs 10−13. Attractive interaction of nanoparticle plays an important role in the solvation free energy. Figure 8 shows the effect of the strength of attraction on the solvation free energy. With increased interaction strength, more CO2 gets absorbed around its surface, leading to a lower interfacial tension. As a result, the optimal pressure decreases with increasing the strength of the attraction (Figure 8b). The critical radius R* of the existence of optimal pressure also becomes smaller than that for the case of the hard particle (ε = 0 kT) (Figure 6b). Finally, for a global picture of the solvation behavior, we present the locus for the minimum solvation free energy in T− P plane in Figure 9 for R = 5 nm. Below the curve, the formation of nanoparticle−fluid interface is the dominant contribution in the solvation free energy, and the solvation free
energy decreases with increasing pressure. Above the curve, the creation of cavity volume dominates, and the solvation free energy increases with pressure. The dot-dashed line is the triple line, which terminates at an end point at Te = 306.8 K, which is slightly above the critical temperature of CO2 (Tc = 304.7 K).18 Above this temperature, there is no triple phase coexistence. For temperatures well below Te = 306.8 K, the optimal pressure coincides with the pressure on the triple line for R = 5 nm. However, for larger nanoparticle sizes, the optimal pressure will move to the branch of CO2-rich-vapor phase; as shown by the curve for R = 20 nm in Figure 7. As we increase the temperature toward Te, the optimal pressure drops off from the pressure on the triple line. This is the behavior shown in Figure 4(a3). The optimal pressure as a function of temperature reaches a maximum value near the triple line and rapidly decreases as the temperature is further increased.
IV. CONCLUSIONS We have studied the solvation of a single nanoparticle in compressible polymer−CO2 mixtures by using a statistical mechanical density-functional theory. For nanoparticle radius smaller than a critical value, which is determined by the derivative of the interfacial tension between the particle and the 8006
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fluid mixture with respect to pressure, the solvation free energy has a minimum as a function of pressure. This minimum arises because of the competition between the volumetric and particle−fluid interfacial contributions. The CO2 density reaches a high contact value at the surface of nanoparticle, giving a monolayer of CO2 at the nanoparticle interface. The solvation behavior of a nanoparticle is closely related to the solubility of nanoparticles in the fluids and plays an essential role in the heterogeneous nucleation of CO2 bubble in polymer−CO2 mixtures induced by nanoparticles. For the nucleation, the evolution of density profiles along the minimum free energy path for nucleation initiated by the nanoparticles can be obtained by combining this work (extended to the metastable mixtures) with the string method,24−26 which will yield the necessary information on the pathways and barriers in heterogeneous nucleation. Such an effort will be undertaken in future.
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(13) Kollman, P. Free energy calculations: Applications to chemical and biochemical phenomena. Chem. Rev. 1993, 93, 2395−2417. (14) Chandler, D. Interfaces and the driving force of hydrophobic assembly. Nature 2005, 437, 640−647. (15) Bolhuis, P. G.; Chandler, D.; Dellago, C.; Geissler, P. L. Transition path sampling: Throwing ropes over rough mountain passes, in the dark. Annu. Rev. Phys. Chem. 2002, 53, 291−318. (16) Zhao, S. L.; Jin, Z. H.; Wu, J. Z. New theoretical method for rapid prediction of solvation free energy in water. J. Phys. Chem. B 2011, 115, 6971−6975. (17) Wu, J. Z. Density functional theory for liquid structure and thermodynamics. In Molecular Thermodynamics of Complex Systems; Lu, X., Hu, Y., Eds.; Structure and Bonding, Vol. 131; Springer-Verlag: Berlin, 2009; pp 1−73. (18) Xu, X.; Cristancho, D. E.; Costeux, S.; Wang, Z.-G. Densityfunctional theory for polymer-carbon dioxide mixtures: A perturbedchain SAFT approach. J. Chem. Phys. 2012, 137, 054902. (19) Israelachvili, Jacob N. Intermolecular and surface forces; Academic Press: Boston, 1992. (20) Gross, J.; Sadowski, G. Perturbed-chain SAFT: An equation of state based on a perturbation theory for chain molecules. Ind. Eng. Chem. Res. 2001, 40, 1244−1260. (21) Müller, M.; MacDowell, L. G.; Virnau, P.; Binder, K. Interface properties and bubble nucleation in compressible mixtures containing polymers. J. Chem. Phys. 2002, 117, 5480−5496. (22) Xu, X.; Cao, D. P.; Wu, J. Z. Density functional theory for predicting polymeric forces against surface fouling. Soft. Matter 2010, 6, 4631−4646. (23) Heni, M.; Löwen, H. Interfacial free energy of hard-sphere fluids and solids near a hard wall. Phys. Rev. E 1999, 60, 7057−7065. (24) Xu, X.; Cristancho, D. E.; Costeux, S.; Wang, Z.-G. Discontinuous bubble nucleation due to metastable phase transition in polymer-CO2 mixtures. J. Phys. Chem. Lett. 2013, 4, 1639−1643. (25) W.N, E.; Vanden-Eijnden, E. Transition-path theory and pathfinding algorithms for the study of rare events. Annu. Rev. Phys. Chem. 2010, 61, 391−420. (26) W.N, E.; Ren, W. Q.; Vanden-Eijnden, E. Simplified and improved string method for computing the minimum energy paths in barrier-crossing events. J. Chem. Phys. 2007, 126, 164103.
AUTHOR INFORMATION
Corresponding Author
*Electronic mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The Dow Chemical Company is acknowledged for funding and for permission to publish the results. The computing facility on which the calculations were performed is supported by an NSFMRI grant, Award No. CHE-1040558.
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REFERENCES
(1) Lee, S. T., Ramesh, N. S., Eds. Polymeric foams: Mechanisms and materials. Polymeric Foams Series; CRC Press: Boca Raton, FL, 2004; pp 1−318. (2) Lee, S. T., Park, C. B., Ramesh, N. S., Eds. Polymeric foams: Science and technology. Polymeric Foams Series; CRC Press: Boca Raton, FL, 2007; pp 1−218. (3) Lee, L. J.; Zeng, C. C.; Cao, X.; Han, X. M.; Shen, J.; Xu, G. J. Polymer nanocomposite foams. Compos. Sci. Technol. 2005, 65, 2344− 2363. (4) Tomasko, D. L.; Li, H.; Liu, D. H.; Han, X. M.; Wingert, M. J.; Lee, L. J.; Koelling, K. W. A review of CO2 applications in the processing of polymers. Ind. Eng. Chem. Res. 2003, 42, 6431−6456. (5) Zeng, C. C.; Han, X. M.; Lee, L. J.; Koelling, K. W.; Tomasko, D. L. Polymer-clay nanocomposite foams prepared using carbon dioxide. Adv. Mater. 2003, 15, 1743−1747. (6) Siripurapu, S.; DeSimone, J. M.; Khan, S. A.; Spontak, R. J. Controlled foaming of polymer films through restricted surface diffusion and the addition of nanosilica particles or CO2-philic surfactants. Macromolecules 2005, 38, 2271−2280. (7) Costeux, S.; Zhu, L. B. Low density thermoplastic nanofoams nucleated by nanoparticles. Polymer 2013, 54, 2785−2795. (8) Tomasko, D. L.; Han, X. M.; Liu, D. H.; Gao, W. H. Supercritical fluid applications in polymer nanocomposites. Curr. Opin. Solid State Mater. Sci. 2003, 7, 407−412. (9) Wu, J. Z. Solvation of a spherical cavity in simple liquids: Interpolating between the limits. J. Phys. Chem. B 2009, 113, 6813− 6818. (10) Reiss, H.; Frisch, H. L.; Lebowitz, J. L. Statistical mechanics of rigid spheres. J. Chem. Phys. 1959, 31, 369−380. (11) Helfand, E.; Reiss, H.; Frisch, H. L.; Lebowitz, J. L. Scaled particle theory of fluids. J. Chem. Phys. 1960, 33, 1379−1385. (12) Ashbaugh, H. S.; Pratt, L. R. Colloquium: Scaled particle theory and the length scales of hydrophobicity. Rev. Mod. Phys. 2006, 78, 159−178. 8007
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