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Stabilization and Antifouling of Polymer Films on a Planar Surface by CO2 Pressurization Xiaofei Xu*,† and Yibing Dai†,‡ †

Center for Soft Condensed Matter Physics and Interdisciplinary Research, and ‡College of Physics, Optoelectronics and Energy, Soochow University, Suzhou, Jiangsu 215006, China ABSTRACT: In this article, we study the dewetting phenomenon of a polymer and carbon dioxide (CO2) mixture on a planar surface by combining density functional theory and the string method. It is found that dewetting is a first-order discontinuous phase transition. When the pressure is lower than the completely dewetting pressure (Pd), CO2 stabilizes the polymer films. The density fluctuation of the polymer decreases significantly with the inclusion of CO2. When the pressure is above Pd, the polymer film is depleted far away from the surface, leaving a thick layer of pure CO2 in the region near the surface. Pd is proportional to the surface energy strength. The CO2 molecules enhance the density fluctuation of the polymer during the dewetting process. The polymer-rich phase at the triple point dewets to a CO2-rich vapor film, as the CO2-rich liquid film near the surface is metastable. These results have promising application in the industry of fabricating polymer films and antifouling polymers on attractive surfaces.

I. INTRODUCTION Polymer films on solid surfaces have a wide range of applications in both industry and daily life.1−4 These films prefer dewetting on a hard surface or weakly attractive surface because the vapor density at saturation is very low. In this case, the system is close to a completely dewetted transition point (the contact angle of the polymer melt at the surface is 180°). The films would break by the creation of nonregular holes, proceeding either by nucleation or spinodal decomposition.5 In this case, it is hard to produce polymer films with a uniform thickness and a defect-free stable shape. One of the methods for preventing dewetting is to increase the tension of the surface-vapor or to decrease that of the surface-liquid interface.6 This will fulfill the requirements when we pressurize gas into the polymer melt. A small fraction of the gas solvent can stabilize the polymer films. Carbon dioxide (CO2) is suitable for this application because it is nontoxic and easy to control at room temperature. The first part of this work discusses the mechanism of stabilization of the polymer film by CO2 in molecular detail. It is also important to identify novel antifouling materials that are easy to fabricate and exhibit high efficiency, minimal ecotoxicity, and good durability.7 CO2 fulfills all of these requirements as an antifouling agent. At a high pressure, CO2 can deplete the polymer far away from the surface by dewetting. The reason for this is that pure carbon dioxide completely dewets a hard surface (low surface energy) at any temperature between the critical point and triple point.8−13 In polymer and CO2 mixtures, the polymer-rich phase would dewet the surface, as the fraction of CO2 exceeds the value at the completely dewetting point. This property can be utilized for antifouling polymers or © XXXX American Chemical Society

biopolymers on the surface. The second part of this work explores the antifouling behavior of CO2 at an attractive wall. A polymer and CO2 mixture shows a complex phase behavior at the triple point.14,15 At the triple point, the polymer-rich phase coexists with the CO2-rich liquid (CRL) phase and CO2-rich vapor (CRV) phase. It remains unclear how this triple point affects the dewetting behavior. We will explore this triple-point effect in the last part of this work. Polymer and CO2 mixtures have been widely studied16 through experiments,17 simulations,18,19 and statistical mechanics theory.15,17,20,21 Although these studies have given some very good and useful results, only few of them focus on the phase behavior of the nearby surface, particularly the interplay between CO2 compressibility and the wetting/dewetting behavior induced by the surface. In this work, we use classical density functional theory (DFT) to study the problems. DFT is a good method to study the interplay between CO2 compressibility and the wetting/dewetting behavior induced by the surface because it can successfully capture the necessary microscopic details for the compressible mixture.22 In DFT, the thermodynamics and phase behavior are described by the free-energy functional, which depends on the density profiles of the species. DFT is then combined with the string method,22,23 which is a numerical method to search for the most probable path of dewetting in the landscape of the free-energy functional. We can obtain key information on the dewetting dynamics, such as the restricted surface free energy and stable density profiles. In the following, Received: November 28, 2016 Revised: December 19, 2016 Published: December 20, 2016 A

DOI: 10.1021/acs.jpcb.6b11975 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B we first introduce the molecular model and DFT, then show the data calculated by the string method, and finally draw some conclusions.

ρ1(r ) =

−

1

1

ρ2 (r ) = exp[βλ 2(r )]∑ Ii(r )I N2 + 1 − i(r )

(2)

i=1

with λi(r ) = μi −

δF ex + Vi (r ) δρi (r )

(3)

The recursive function is given by ⎧1, ⎪ Ii(r ) = ⎨ 1 ⎪ 2 ⎩ 4πσ2

i=1

∫|r−r |=σ dr′Ii−1(r′) exp(−βλ2(r′)), ′

i>1 (4)

2

The key information in dewetting dynamics is the proceeding path from a uniform liquid bulk to a well-developed vapor film. In the literature, this information is usually obtained by the constraint method, fixing the location of the liquid−vapor interface.22 However, the method is neither general to a mixture nor rigorous for a thin vapor layer. Here, we use the string method to locate the dewetting path. In a mean-field framework, dewetting proceeds along the minimum free energy path (MFEP) on the functional surface of grand potential functional W[ρ1(r), ρ2(r)]. It is defined such that the tangent along the path is parallel to the free-energy gradient. The normalized reaction coordinate on the path is 0 ≤ s ≤ 1. The MFEP {ρj(r, s)|j = 1, 2; 0 ≤ s ≤ 1} connects the initial state {ρj(r, 0)} (the uniform liquid mixture) and the final state {ρj(r, 1)} (a well-developed vapor film). The equations of the MFEP are then solved by the string method, a modified steepest descent method. The numerical details of the string method are shown in our previous work.33 We assume that the density profiles depend only on the z direction, that is, ρ(r) ≡ ρ(z). The thickness of the vapor polymer film is selected as the order parameter, defined as L=

ρ2v

1 − ρ2l

∫ dz [ρ2 (z) − ρ2b]

(5)

ρb2

ρl2

ρv2

where is the bulk density of the polymer and and are the densities of the coexisting liquid and vapor, respectively, at the given T and P. The mobility of the polymer chain near the surface is described by the local density fluctuation. At a fixed temperature and pressure, the density fluctuation of component i is defined by34,35

⎛ ∂ρ (r ) ⎞ χi (r ) = ⎜⎜ i ⎟⎟ ⎝ ∂μi ⎠T , P

(6)

We can calculate χi by perturbing the bulk pressure (δP) and fixing the composition at the given T and P. Here, we use the dimensionless quantity χ/χb to show the data, where χb is the bulk value of χ at the same T, P, and composition. It should be mentioned that classical DFT can describe the density fluctuation by χi(r) to some extent. At the phase-transition point (e.g., the transition from liquid to vapor), the chemical potential is the same on both sides but the density changes

∫ dr ρ1(r )[μ1 − V1(r )]

∫ dr ρ2 (r )[μ2 − V2(r )]

∫|r−r |=σ exp[βλ1(r1)]

N

II. MODEL AND THEORY We consider a compressible mixture of a polymer and CO2 under isothermal and isobaric conditions. The mixture is described by a coarse-grained model. The polymer is modeled as a freely jointed chain of tangentially connected spheres. The CO2 molecule is modeled as a dimer of two identical spheres. The pairwise interaction between the segments is described by a LennardJones-like potential. The excluded volume of the species is represented by hard-core interactions, whereas energetic interactions are described by the attractive part of the LennardJones potential.14 We use DFT to describe the thermodynamics and phase behavior of the mixture. DFT is built from the perturbed-chain statistical associating fluid theory equation of state.24 In DFT, the Helmholtz free-energy functional, F, is expressed as the sum of an ideal-gas term and excess terms. The excess Helmholtz free energy comprises a local contribution of the equation-of-state effect and a nonlocal contribution form the long-range dispersion interaction. In the local term, we consider the contribution from the excluded-volume effect of hard spheres and the correlation due to chain connectivity and the dispersion effect. The excluded-volume effect is described by the modified fundamental measure theory (FMT). 25−27 The correlation of chain connectivity is modeled by combining the statistical associating fluid theory28−30 and the weighted density functional in FMT.31 The details of these expressions and the molecular model can be found in our previous publications.14,32 In our model, there are four parameters to describe a polymer chain and CO2 molecules: the ratio of chain length to molecular weight (N/M), the energy strength (ϵ), the segment diameter (σ), and kij to modify the cross-interaction of the polymer with CO2. These parameters are obtained by fitting the PVT data, binodal data, and CO2 solubility in the equation of state with the corresponding experimental data. Results show that the obtained values of the parameters give an accurate description of the phase behavior of the polymer and CO2 in both sub- and super-critical regions.32 Here, we take our polymer to be poly(methyl methacrylate) (PMMA), for which N/M = 0.0318, ϵ/kB = 223.79 K, σ = 2.890 Å, and kij = (1.25T − 422.689) × 10−4, where kB is the Boltzmann constant and T is the temperature. Because the thermodynamic behavior becomes insensitive to chain length for a long chain, we fix the chain length as N = 100 by considering the numerical efficiency. There is no particular reason for selecting PMMA as the studied species. It should be mentioned that the results shown here are also valid for other species of homopolymers. At a given temperature (T) and pressure (P), the equilibrium density distribution of a mixture in contact with a solid surface is obtained by minimizing the grand potential functional W [ρ1(r ), ρ2 (r )] = F −

1 exp[βλ1(r )] 2πσ1

(1)

discontinuously from liquid value to vapor. So,

where ρ1(r) and ρ2(r) are the density distributions of CO2 and polymer segments, respectively, and Vi(r) (i = 1, 2) is the external potential of the surface for component i. The minimization yields32

( ) ∂ρi (r ) ∂μi

has a

T ,P

singularity at the transition point, whose value is infinity. By measuring the closeness of χi(r) to infinity (the phase-transition point), we could describe the density fluctuation quantitatively. B

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Figure 1. Restricted surface free energy as a function of vapor-layer thickness of the polymer at T = 310 K. The red line in panel (c) is the value of restricted surface free energy at L = ∞.

The higher the absolute value of χi(r), the stronger is the density fluctuation.

parameter values (i.e., vapor layer thickness L). So we can calculate the restricted surface free energy by

III. RESULTS AND DISCUSSION III.I. Stabilization at a Hard Surface. First, we study how a small amount of CO2 stabilizes the polymer film. In this case, the

γs(L) =

surface does not favor adsorption of the polymer. Here, we suppose the surface to be a hard one for both components ⎪

⎪

∞

dz [g (z ; L) − gb]

(8)

where gb is the grand potential density in the bulk phase. We show the restricted surface free energies at various pressures in Figure 1. For a pure polymer on a hard surface, the vapor density is very low. The tension of the wall−vapor interface, γWV, will almost vanish, and the tension of the liquid in contact with the wall will be similar to the liquid−vapor interfacial tension, γLV. Following Young’s equation, the contact angle of the polymer at the surface will be close to 180°. Therefore, the restricted surface free energy is independent of the vapor-layer thickness of the polymer, as shown in Figure 1a. We know that a pure polymer completely dewets a hard surface. Further, pure CO2 also completely dewets a hard surface at any pressure.8−13 Thus, it will be interesting to explore what happens at the intermediate state of a mixture. For a polymer and CO2 mixture, the CO2 solubility is proportional to the pressure. At a sufficiently high pressure, the mixture behaves like pure CO2 and so completely dewets the surface (see Figure 1d). At a low pressure, the profiles in Figure 1b,c show that dewetting is a firstorder discontinuous transition. At P = 1 bar, the restricted surface energy has a global minimum at L ≈ 0.3 nm. The mixture partially dewets the wall. On increasing the pressure, the global minimum gradually increases. At P = 7.95 bar, it reaches the same value as that at L = ∞, that is, that of a well-developed vapor film. Above this pressure, the dewetting layer discontinuously changes its thickness from L ≈ 0.3 nm to L = ∞. This introduces the definition of the complete dewetting pressure (or complete drying pressure, Pd), at which the mixture starts to completely

Figure 2. P−x phase diagram of the polymer and CO2 mixture on a hard planar surface. x is the molar fraction of CO2 in the mixture.

⎧ ∞ , z < σi /2 Vi (z) = ⎨ ⎩ 0, z ≥ σi/2

∫0

(7)

The string method will give the density profiles (ρi(z; L)) and grand potential density profiles (g(z; L)) at various order C

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Figure 3. Density profiles of the polymer (blue lines) and CO2 (red lines) segments at the global stable point of Figure 1. The arrow in (d) means that the interface would move far away from the surface if the iteration is continued. ηi = (π/6)ρiσ3i is the packing fraction of component i.

the polymer is very low, the polymer-rich phase cannot move far away from the surface to leave a vacuum region near the surface due to the entropy effect. However, the density profile shown in Figure 3a is unstable if the thermal fluctuation is large enough (order of kT is enough). When considering the effect of thermal fluctuation, the interface may move away from the surface. As a result, the interface could fluctuate in the z direction near the surface. This is the reason behind the easy rupture of polymer thin films. Pressurizing CO2 into a polymer melt could prevent dewetting. Although the density profiles of the polymer are almost the same as those in the absence of CO2 (see Figure 3a− c), it becomes stable because of the decreased contact angle. Figure 4 shows the contact angle (θ) as a function of pressure. It is calculated by Young’s equation

cos θ =

Figure 4. Contact angle of polymer and CO2 mixtures on a planar hard surface at T = 310 K.

γwv − γwl γlv

(9)

where γwv and γwl are the interfacial tensions of the CO2-rich phase (vapor) and polymer-rich phase (liquid), respectively, in contact with planar wall, and γlv is the tension of the planar interface between the CO2- and polymer-rich phases at coexistence. As shown, the fluid completely dewets the wall at P = 0 bar. On increasing the pressure, the fluid partially dewets the wall and the contact angle decreases slightly, to a minimum of ∼169°. The contact angle increases on further pressurizing the gas, and the fluid starts to completely dewet the wall at about 8 bar. These data reveal that a small amount of CO2 at a low pressure could stabilize the polymer film at the surface. Although the contact angle is still close to 180°, we could obtain a more stable polymer film by pressurizing the system to a supersaturated state at the fixed composition. Figure 5 shows the data at a supersaturated state (P = 11 bar) at T = 310 K, at which the composition is fixed at that of equilibrium state at P = 1 bar. For the pure polymer (left panel), the vapor density is very low and hence the liquid−vapor interface can freely move away from the surface. In this case, the density fluctuation in theory is

dewet at the surface (the contact angle of the polymer melt at the surface is 180°). If the pressure is further increased beyond P = 7.95 bar in the profile of restricted surface free energy, the global minimum point at dewetting becomes a local one. This local minimum gives a predrying transition at undersaturation. In that case, there are two local minimal points for the restricted surface free energy. The dewetting layer discontinuously changes from a thin layer to a thick one at the predrying point. The slope of the cotangent line between these two local minima measures the undersaturation degree.36 Figure 2 shows the whole phase diagram for a planar hard surface. Below the complete dewetting point (Pd), the mixture only partially dewets the surface. The predrying line ends at a critical point (Ppre). As P > Ppre, the mixture completely dewets the surface at any composition of polymer-rich phase. The corresponding density profiles at equilibrium are shown in Figure 3. Although the pure polymer completely dewets the surface, it gives a finite depletion width. As the vapor density of D

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Figure 5. Data on the density profiles (a, b), density fluctuations (c, d), and restricted surface tensions (e, f) in the supersaturated state (T = 310 K and P = 11 bar). Left panel is for the pure polymer and right panel is for the polymer (blue solid lines) and CO2 (red dashed lines) mixture. The composition of the polymer and CO2 mixture in the right panel is fixed at a saturation value of T = 310 K and P = 1 bar.

Figure 6. Restricted surface free energy as a function of the vapor-layer thickness of the polymer at T = 310 K. The red solid points in (a) and (b) are the equilibrium stable states. The surface attraction of the polymer is ϵ2 = 2kT.

(see the data in (f)). Therefore, the polymer films are more stable in the supersaturated state than in the saturated state of the mixture. III.II. Antifouling on an Attractive Surface. In this section, we consider the antifouling behavior of CO2 against the

infinity. For the polymer and CO2 mixture (see right panel), the situation changes because of the effect of CO2. With the inclusion of CO2, the density fluctuation (χ/χb) of the polymer film decreases significantly, from infinity to less than 100. There is a great barrier for the polymer film to move away from the surface E

DOI: 10.1021/acs.jpcb.6b11975 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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Figure 7. (a, b) Corresponding density profiles at the equilibrium stable points of Figure 6a,b. (c, d) Density profiles of a well-developed vapor film. The blue lines are for the polymer segment, and the red lines are for the CO2 segment.

Figure 8. Maximum value of density fluctuation versus vapor-layer thickness of a polymer at T = 310 K and P = 47 bar.

to the order of kT. Figure 6 shows the restricted surface free energy for ϵ2 = 2kT. At P = 1 bar in the absence of CO2, the surface strongly adsorbs polymers with a finite vapor thickness. The stable solution has a finite vapor-layer thickness of L ≈ 0.1 nm. There is a high energy barrier for the state to be a welldeveloped vapor film. This energy barrier decreases significantly at P = 10 bar, and the stable solution becomes easy to be perturbed by thermal fluctuation. On further increasing the pressure to 47 bar, there would be no stable solution at finite L. The only stable solution is the vapor layer at L = ∞, a welldeveloped vapor film. In this case, the CO2 completely dewets the surface. Figure 7a,b shows the stable density profiles of Figure 6a,b, respectively. For the pure polymer at P = 1 bar, the density profile shows an obvious layered structure of liquid near the surface. On pressurizing CO2 at P = 10 bar, the depletion effect of CO2 starts

adsorption of a polymer on an attractive surface. The wall potential to the CO2 segments is still supposed to be a hard potential, whereas to the polymer segments, it is a shifted Lennard-Jones (9, 3) potential ⎧ ⎡ 9 ⎛ σ ⎞3 ⎤ ⎛ σ ⎞9 ⎛ σ ⎞3 ⎛ σ ⎞ ⎪ ⎪ ϵ2⎢⎜ 2 ⎟ − 3⎜ 2 ⎟ − ⎜ 2 ⎟ + 3⎜ 2 ⎟ ⎥, z ≤ zc ⎝z⎠ ⎝ zc ⎠ V2(z) = ⎨ ⎢⎣⎝ z ⎠ ⎝ zc ⎠ ⎥⎦ ⎪ ⎪ 0, z > zc ⎩ (10)

where zc is the cutoff length for the potential, whose value is fixed at zc = 5σ2 here, and ϵ2 is the energy strength of the wall potential. For a weakly attractive surface (ϵ2 ≪ 1kT), the dewetting behavior is similar to that of a hard surface (ϵ2 = 0kT). The complete dewetting pressure (Pd) is proportional to the attractive strength (ϵ2). The situation changes as ϵ2 increases F

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the triple point, the polymer-rich phase coexists with the CRL and CRV phases. The mixture shows complex dewetting behavior at the triple point when the polymer-rich liquid is in contact with the surface. We calculate the restricted surface free energy at a hard surface by selecting the final state in the string method as either the CRL or CRV phase. As shown in Figure 9, the CRV phase has a lower restricted surface free energy for a well-developed CO2 film. The polymer-rich phase in contact with the hard surface will first dewet to a CRV film and then further grow to a CRL film if L is not large. For a large L, the wall−CO2 interface cannot affect the interface of CO2 and the polymer-rich phase. We point out that the free-energy density of the pure CRV film is always lower than that of the pure CRL film. Therefore, the CRL film will collapse to form the CRV film when L becomes large. This means that the CRL film near the surface is metastable. Therefore, as the interface of the polymer-rich phase moves far away from the surface, the CRL film (see Figure 10b) will finally dewet to form the CRV film (see Figure 10a), which is the stable density profile. CRL can only exist far away from the surface at the interface of the polymer-rich phase. This gives a high peak for the CO2 density profile in Figure 10a.

Figure 9. Restricted surface free energy at a hard wall at the triple point (T = 300 K and P = 69.049 bar).

to affect the behavior of the polymer, leading to a wider interface between the surface and liquid polymer. The liquid polymer will be depleted further away from the surface if the pressure is increased further. When the pressure exceeds that at the predrying transition point, the mixture behaves like pure CO2. The liquid polymer starts to completely dewet the surface. The mobility of polymer molecules can be described by the local density fluctuation. Figure 8 shows the maximum density fluctuation (χ/χb) during the dewetting process. At the beginning, both CO2 and the polymer show low mobility. On adsorbing more CO2 onto the surface, the mobility of the polymer chain increases greatly. There is a stationary point in the density profile of the polymer (Figure 8b). We indicate that this stationary point corresponds to a saddle point on the free-energy surface. When the thermal fluctuation is lower than the freeenergy barrier, the surface adsorbs the polymer films in a stable state. When it is higher than the energy barrier, the polymer films would be depleted far away from the surface. The mobility of the polymer reaches a maximum value when it overcomes the energy barrier. The dewetting process proceeds forward to a welldeveloped vapor film after overcoming the energy barrier. III.III. Effect of the Triple Point. The polymer and CO2 mixture has a complex phase behavior at the triple point.14,15 At

IV. CONCLUSIONS One of the key issues in fabricating polymer films in industry is stabilizing the thin film on a surface. We propose a new method to prevent the thin film from dewetting by pressurizing a small amount of CO2. The contact angle of the polymer at the surface would decrease slightly in the presence of CO2. As a result, the film becomes stable. We can obtain more stable films by increasing the pressure to that of the supersaturated state and fixing the composition to that at P ≈ 1.5 bar. The density fluctuation of the polymer decreases with the addition of CO2. As CO2 is a green solvent for the environment, the method of stabilizing polymer films by CO2 pressurization should have promising applications in industry, particularly in the industry of fabricating polymer thin films. The second issue considered in this work is how to antifoul polymers on an attractive surface. It should be mentioned that pure CO2 always completely dewets a hard surface. So, the polymer and CO2 mixture will completely dewet the surface if the CO2 composition exceeds a certain value (i.e., the composition at Pd). Pd is proportional to the attractive energy strength of the

Figure 10. Density profiles of a well-developed layer of (a) the CRV phase and (b) the CRL phase. G

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surface. With an increase in the pressure above Pd, the adsorbed polymer would be depleted far away from the surface, leaving a thick layer of pure CO2 near the surface. This gives an efficient and nontoxic way to protect the surface from polymer or biopolymer fouling. Finally, we also studied the effect of the triple point on the dewetting behavior. The polymer-rich phase at the triple point first dewets to the CRV film and then further grows to the CRL layer near the surface. However, this CRL layer is metastable. Thus, it finally collapses to form a CRV film with an increase in the thickness of the vapor layer. Our results provide fundamental predictions on the issues related to fabricating polymer films and antifouling polymers on an attractive surface. Further research could extend to the dewetting and nucleation behaviors of the polymer and CO2 mixture on a nanoparticle surface.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +86-150-5147-8694. ORCID

Xiaofei Xu: 0000-0003-2459-7748 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China under Grant No. 21404078 and No. 21674077.

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REFERENCES

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DOI: 10.1021/acs.jpcb.6b11975 J. Phys. Chem. B XXXX, XXX, XXX−XXX