Effect of Dissolved Poly(lactic acid) on the ... - ACS Publications

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Effect of Dissolved Poly(lactic acid) on the Solubility of CO2, N2, and He Gases in Dichloromethane Jay Jesus Molino Cornejo,† Daichi Sakurai,‡ Hirofumi Daiguji,*,†,‡ and Fumio Takemura§ †

Department of Mechanical Engineering, Graduate School of Engineering, The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan ‡ Division of Environmental Studies, Graduate School of Frontier Sciences, The University of Tokyo 5-1-5 Kashiwanoha, Kashiwa 277-8563, Japan § Research Institute for Energy Conservation, National Institute of Advanced Industrial Science and Technology (AIST), 1-2-1 Namiki, Tsukuba 305-8564, Japan S Supporting Information *

ABSTRACT: The effect of dissolved poly(lactic acid) (PLA) on the solubility of gases (CO2, N2, and He) in dichloromethane (CH2Cl2) was investigated, between 288 and 303 K, by measuring the equilibrium pressure in a closed vessel containing a gas and CH2Cl2 solutions of different types (molecular weights) and concentrations of PLA. The results show that, in the absence of PLA, the order of the solubility of gases in dichloromethane was He < N2 < CO2. Furthermore, an increase in the concentration of PLA did not appreciably change the solubility of CO2 in the solutions but led to a higher solubility of N2 and He. The effect of PLA on the increase of gas solubility in CH2Cl2 was larger for less soluble gases. Semiempirical correlations of the dependence of the Henry’s law constant of the gases on the temperature and PLA concentration were also derived from the measured data. containing entrapped gas.11,12 Thus, vacuum and supersaturation will allow these cavities to grow. Yet, Han et al.13,14 demonstrated that this theory fails to describe nucleation during drying, especially in polymer solutions that have previously been degassed to avoid imperfections due to bubble nucleation. Furthermore, to the best of our understanding, none of the previous studies have considered that bubble nucleation during the drying process of a polymer solution is the result of the supersaturation of a gas already dissolved in the solution. This is the basis of our proposed method for the fabrication of hollow polymeric microcapsules by encapsulating microbubbles, that is, the bubble template method.15−17 However, our research group recently found that the presence of polymers not only aids the stabilization of the nucleated bubbles inside the droplet but also affects the solubility of the gas in the solution. Gas sorption in polymers is well known, and this phenomenon can be described by the Flory−Huggins equation,18−20 yet the effect of the polymer on the solubility of gases in polymer solutions requires further study. The understanding of this phenomenon is relevant for controlling the size of hollow polymeric microcapsules in the bubble template method. Moreover, the knowledge of the solubility of gases in polymer solutions is of vital importance in polymer processing, especially for polymer solutions containing dichloromethane (CH2Cl2), which is widely used for the

1. INTRODUCTION From a practical point of view, the solubility of gases in polymer solvents is an important consideration in removing volatile gases from the polymers and in designing microcellular/foam materials, such as absorbers for mechanical stress, thermal, electrical, and acoustic insulators, and optical reflectors.1−5 In the case of microcellular/foam materials, one of the most important production steps is the dissolution of a gas in a polymer solution for generating the bubbles that will become the cells of the material.6−9 Usually, two processes are employed for the formation of these materials; namely, injection molding or extrusion, and volatilization or drying. During foam processing, bubble nucleation is achieved by injecting a blowing agent or an inert gas that, upon reduction of pressure (usually to atmospheric pressure), leads to the nucleation of microbubbles in the solution. Bubble growth will continue to form bubbles of stable size. In general, supercritical fluids are employed as the physical blowing agents for foam processes, and the final microcellular/foam material has a uniform cell radius ranging from 1 to 50 μm. CO2 and N2 tend to be the preferred blowing agents.8,10 Nucleation during drying is another process employed to obtain foam materials. During drying, microbubbles are formed from the volatilization of residual solvent or the drying of a coating polymer solution, thus yielding microbubbles. However, the process of nucleation during drying is not well understood. For instance, classical nucleation theory proposes that during volatilization, bubble nucleation begins in cavities © 2015 American Chemical Society

Received: March 24, 2015 Accepted: November 17, 2015 Published: November 25, 2015 94

DOI: 10.1021/acs.jced.5b00268 J. Chem. Eng. Data 2016, 61, 94−101

Journal of Chemical & Engineering Data

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Table 1. Chemical Sample Table chemical name

source

polymer properties/initial purity

purification method none

poly(L-lactic acid) [MW 325 000−460 000] (300 kDa PLA)

Polysciences

dichloromethane carbon dioxide nitrogen helium

Wako Pure Chemical Industries Suzuki Shokan Suzuki Shokan Atox

molecular weight 1600−2400 polydispersity ∼1.8 inherent viscosity 0.10−0.20 crystallinity ∼70% molecular weight 40 000−70 000 polydispersity ∼1.8 inherent viscosity 0.80−1.20 crystallinity ∼70% molecular weight 325 000−460 000 polydispersity ∼1.5 inherent viscosity 4.00−5.00 crystallinity ∼70% ≥ 0.995 (mass fraction) 0.995 (mole fraction) 0.9999 (mole fraction) 0.9999 (mole fraction)

poly(L-lactic acid) [MW 1600−2400] (2 kDa PLA)

Polysciences

poly(L-lactic acid) [MW 40 000−70 000] (45 kDa PLA)

Polysciences

none

none

none none none none

10−2 mol·kg−1 were prepared. Here, the molecular weight of PLA, MPLA, was assumed to be 2 kg·mol−1 for 2 kDa PLA. CH2Cl2 solutions of 45 and 300 kDa PLA at 0.0075 mass fraction were also prepared. The values of bPLA for 2, 45, and 300 kDa PLA solutions at 0.0075 mass fraction were 3.78 × 10−3, 1.68 × 10−4, and 2.52 × 10−5 mol·kg−1, respectively. Samples of about 40 mL of each solution were placed in a pressure vessel and later degassed to measure the vapor pressure of each solution. It is noted that the total mass was measured before and after degassing and the mass of CH2Cl2 was determined by assuming that the change in total mass was due to the evaporation of CH2Cl2. The explosive boiling and the scattering of liquid, resulting in the loss of PLA, were prevented by opening the valves slowly. Because the total mass change was much smaller than the mass of PLA, the effect of the degassing process on bPLA was marginal. The solutions were subsequently pressurized using CO2, N2, or He to determine the solubility of each gas in the solutions. The setup is shown schematically in Figure 1.

aforementioned purposes. Nonetheless, data on the solubility of gases in polymer solutions is scarce, and the mechanism by which the inclusion of polymers in a solvent affects the final solubility of gases in polymeric solutions remains unclear. In this study, we employed a pressure decay technique to measure the solubility of a gas−polymer−solvent system at different temperatures. This technique relies on measuring the decrease of gas pressure inside a closed vessel at different temperatures. A reduction in gas pressure implies that the gas is dissolved/adsorbed in the solvent/polymer. The solubility was calculated from the total amount of gas in the solution. We focused on the solubility of N2, CO2, and He in the CH2Cl2 solutions of poly(lactic acid) (PLA). In order to perform a comparative analysis of the effect of PLA on the solubility of the gases, He was employed since it has a low solubility in CH2Cl2 compared with N2 and CO2. On the basis of the pressure and temperature measurements, the solubility of each of the three gases in the CH2Cl2 solutions was determined as a function of the PLA concentration. The effect of PLA on the solubility of the three gases in CH2Cl2 was clarified, and semiempirical correlations of the dependence of the Henry’s law constant of the gases on the temperature and PLA concentration were derived from the measured data.

2. EXPERIMENTAL SECTION 2.1. Materials. The polymeric shell material comprised PLA. In this study, L-type PLA of three different molecular weights, poly(L-lactic acid) [MW 1600−2400], poly(L-lactic acid) [MW 40 000−70 000], poly(L-lactic acid) [MW 325 000− 460 000] (Polysciences, Warrington, PA, U.S.A.) were purchased and employed. The three kinds of PLA were described as 2 kDa, 45 kDa, and 300 kDa PLAs, respectively. The solvent was CH2Cl2 (≥0.995 mass fraction purity, Wako Pure Chemical Industries, Osaka, Japan). The solute gases were CO2 (0.995 mole fraction purity, Suzuki Shokan, Tokyo, Japan), N2 (0.9999 mole fraction purity, Suzuki Shokan, Tokyo, Japan), and He (0.9999 mole fraction purity, Atox, Tokyo, Japan). All chemicals were reagent grade and used without further purification. They are summarized in Table 1. 2.2. Methods. In this study, the PLA concentration, bPLA (mol·kg−1), was defined as the mole of PLA per unit mass of the solvent CH2Cl2. The CH2Cl2 solutions of 2 kDa PLA at bPLA = 0, 3.78 × 10−3, 1.13 × 10−2, 1.89 × 10−2, and 2.64 ×

Figure 1. Schematic of the experimental setup for measurement of the vapor pressure and gas solubility in CH2Cl2 solutions of PLA. V1, V2, and V3 are valves. P1 measures the total pressure inside the vessel and P2 measures the saturation vapor pressure of the solution. T measures the temperature of the thermostat (TS). VAC is the vacuum pump and GAS is the gas cylinder.

We followed the procedure described by Shirono et al.21 The total volume of the vessel was 7.892 × 10−5 m3, which was calculated from its inner geometry. P1 is a digital pressure sensor (AP-53A, Keyence, Japan), the range and resolution of which are (0 to 1) × 106 Pa and 1 × 103 Pa, respectively. P1 was employed to measure the initial and equilibrium total pressures, Pi and P. P2 is a digital pressure sensor (AP-V85, Keyence, Japan), the range and resolution of which are (0.0 to 95



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2.0) × 105 Pa and 10 Pa, respectively. P2 was employed to measure the saturation vapor pressure of the solution before supplying a gas. Because the saturation vapor pressure of the solution was in the order of 104 Pa, it could be measured more precisely by P2 than by P1. Furthermore, a high precision thermometer (CAB-F201, CHINO, Japan) with a resolution of ±0.01 K in the temperature range from 73.15 to 1123.15 K was employed to record the corresponding temperature of the pressure measurements. At a given temperature, T, and PLA concentration, bPLA, the conservation equations are satisfied for the total pressure, P, and the partial pressure of the solvent and gases, P1 and P2 (eq 1), the total volume of the vessel, V, and the volume of the liquid and gas phases, V′ and V″ (eq 2), the total mole number of the solvent, n1, and the mole number in the liquid and gas phases, n1′ and n1″ (eq 3), and the total mole number of the gas, n2, and the mole number in the liquid and gas phases, n′2 and n″2 (eq 4) P = P1 + P2

(1)

V = V′ + V″

(2)

n1 = n1′ + n1″

(3)

n2 = n2′ + n2″

(4)

where R is the gas constant. Before supplying the gas to the vessel, only a CH2Cl2 solution of PLA was contained in the vessel. At given values of V, n1, T, and bPLA, five equations (eqs 2, 3, 5, 8, and 9) were solved simultaneously, and then five unknown parameters, P1, V′, V″, n′1, and n″1 were determined. The total mole number of the gas n2 was obtained as follows. Immediately after supplying the gas into the vessel, all the supplied gas was in the gas phase. Here, we assumed that the volume of the gas phases just after supplying the gas, that is, the initial volume of the gas phase, V″i , was the same as that before supplying the gas. It is noted that the value for V″i was calculated using the measured temperature just after supplying the gas; that is, the initial temperature, Ti. At given values of Pi, Ti, and V″i , three eqs (eqs 1, 5, and 10) were solved simultaneously, and then three unknown parameters, P1, P2, and n″2 were determined. The values of n″2 obtained here should be the same as the total mole number of the gas supplied to the vessel n2. After the system reached a state of equilibrium by dissolution of the supplied gas into the CH2Cl2 solution of PLA, the Henry’s law constant was determined. At given values of P, T, V, bPLA, n1, and n2, eight equations (eqs 1−5 and 8−10) were solved simultaneously, and then eight unknown parameters, P1, P2, V′, V″, n′1, n″1 , n′2, and n″2 were determined. The Henry’s law constant, H, is given by

The partial pressure of the solvent, P1, can be approximated as the saturation pressure of the pure solvent as a function of T

P1 = f (T )

H=

(5)

The validity of this assumption was proved experimentally, and it is demonstrated in section 3.1. The volume of the liquid phase, V′, was given by the following equation V′ =

n1′M1 ′ + VPLA ρ1

bPLA MPLA ρPLA

x2 = (6)

n1′M1 (1 + bPLA MPLA ) ρ

(7)

∂H ∂H ∂H ∂H u(n1′, n2′) + 2 u(n1′, P2) · · ∂n1′ ∂n2′ ∂n1′ ∂P2 ⎤1/2 ∂H ∂H u(n2′ , P2)⎥ +2 · ⎥⎦ ∂n2′ ∂P2 +2

In this study, the volume of the liquid phase, V′, was given by eq 8 as a function of bPLA. The value of ρ was given by the following prediction formula for the density of CH2Cl2 in kg· m−3 as a function of T in K: ρ = A − B(T − T0) − C(T − T0)2, where A = 1351.1, B = 1.26083, C = 0.00645833, and T0 = 273.15. The value of M1 was assumed to be 84.93 g·mol−1. The mole number of the solvent and gas in the gas phase is given by the ideal gas law PV 1 ″ RT

(9)

n2″ =

P2V ″ RT

(10)

(12)

⎡⎛ ⎞2 ⎞2 ⎛ ∂H ⎞2 ⎛ ∂H ∂H ⎢ ′ ′ u(H ) = ⎜ u(n )⎟ + ⎜ u(n2)⎟ + ⎜ u(P2)⎟ ⎢⎣⎝ ∂n1′ 1 ⎠ ⎠ ⎝ ∂P2 ⎠ ⎝ ∂n2′

(8)

n1″ =

n2′ n1′ + n2′

2.3. Uncertainty in the Calculated Henry’s Law Constant. The uncertainty introduced by the equipment used to measure the mass, temperature, and pressure is the major source of uncertainty in the calculated Henry’s law constant. The combination of eqs 11 and 12 yields the relationship of H = P2/{n′2/(n′1 + n′2)}. The propagation rule of the uncertainties is given by the following equation:22

where ρPLA is the density of PLA. By substituting eq 7 into eq 6, and assuming that the density of PLA, the solvent, and the solution are the same (ρPLA = ρ1 = ρ), eq 6 becomes V′ =

(11)

where x2 is the mole fraction of the dissolved gas in the solvent and is defined as

where M1 and ρ1 are the molar weight and density of the solvent, and V′PLA is the volume increment by adding PLA. In this study, VPLA ′ was modeled as follows ′ = n1′M1 × VPLA

P2 x2

(13)

where u(H), u(n1′ ), u(n2′ ), and u(P2) are the standard uncertainties of H, n1′ , n2′ , and P2, respectively. u(n1′ ,n2′ ), u(n1′ ,P2), and u(n2′ ,P2) are the covariances between (n1′ ,n2′ ), (n1′ ,P2), and (n2′ ,P2), respectively. Moreover, ∂H/∂n1′ = (H/n1′ )· (n1′ /n2′ )·x2, ∂H/∂n2′ = −(H/n2′ )·(n1′ /n2′ )·x2, and ∂H/∂P = H2/P2. Assuming that x2 ≈ n2′ /n1′ , eq 13 can be approximated by the following equation: 96



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2 ⎡ ⎛ u(n2′) ⎞2 ⎛ u(P2) ⎞2 u(H ) ⎢⎛ u(n1′) ⎞ ≈ ⎜ ⎟ +⎜ ⎟ +⎜ ⎟ ⎢⎣⎝ n1′ ⎠ H ⎝ n2′ ⎠ ⎝ P2 ⎠

P2 = P − P1

(15)

The measured gas pressure, P, was about 103 kPa with the standard uncertainty of about 1 kPa. The calculated P1 was about 30−70 kPa, and the error was less than 1 kPa, thus u(P2)/P2 was around 10−3. As a result, it could be said that u(n2′ )/n2′ ≫ u(n1′ )/n1′ and u(P2)/P2. Moreover, (u(n′2)/n′2)2 was much larger than the magnitude of the covariance of u(n1′ ,n2′ )/(n1′ n2′ ) because (u(n2′ )/ n2′ )2 ≫ u(n1′ )u(n2′ )/(n1′ n2′ ) ≥ |u(n1′ ,n2′ )/(n1′ n2′ )|. Similarly, it could be said that (u(n′2)/n′2)2 ≫ |u(n′1,P2)/(n′1P2)| and | u(n′2,P2)/(n′2P2)|. Thus, (u(n′2)/n′2)2 was only a significant term in the right-hand side of eq 14. In summary, the uncertainty of the Henry’s law constant, H, was dominated by the uncertainty of n′2, that is, u(H)/H ≈ u(n′2)/n′2, hence u(H) was 1−10% of H for the least soluble gas He.

(16)

3. RESULTS 3.1. Effect of Dissolved PLA on Saturation Vapor Pressure of CH2Cl2. Figure 2 shows the measured saturation

⎤1/2 u(n1′, n2′) u(n1′, P2) u(n2′ , P2) ⎥ +2 +2 +2 n1′n2′ n1′P2 n′P2 ⎥⎦ (14)

It is worth noting that |u(n1′ ,n2′ )| ≤ u(n1′ )u(n2′ ), and the same can be said about the combinations of (n′1,P2) and (n′2,P2). Regarding u(n′1), the value of n′1 was calculated from eqs 2, 3, 5, 8, and 9 ⎧⎛ PV ⎞ n1′ = n1⎨⎜1 − 1 ⎟ n1RT ⎠ ⎩⎝ ⎪

⎪

⎫ ⎛ Pv 1 ⎞ ⎜1 − ⎟⎬ ⎝ RT ⎠⎭ ⎪

⎪

where v=

M1 (1 + c PLAMPLA ) ρ

(20)

The value of n1 was calculated from n1 = ((m − mPLA − mPV)/M1), where m, mPLA, and mPV were the measured mass of the total, PLA, and the pressure vessel, respectively. The uncertainty in the mass measurement was negligible in comparison with that in the pressure measurement. Since the values of P1V/n1RT and P1ν/RT were in the order of 10−3, thus u(n1′ )/n1′ was in the order of 10−3 at a maximum. Regarding u(n′2), the value of n2 was calculated from eqs 1, 5, and 10 n2 =

{Pi − P1(Ti)}V i″(Ti) RTi

(17)

where Pi, Ti, and Vi″ were the initial pressure, temperature and volume of a gas phase, respectively. In the same way, n2″ was calculated as follows n2″ =

{P − P1(T )}V ″(T ) RT

(18)

Figure 2. Saturation vapor pressure of CH2Cl2 solutions of PLA as a function of temperature. The CH2Cl2 solutions of 2 kDa PLA with different concentrations and different molecular weights PLA at 0.0075 mass fraction were prepared. The dotted curve was the prediction of pure CH2Cl2.

where P and V″ were the equilibrium pressure and volume of a gas phase at T, respectively. The initial or equilibrium gas pressure, Pi or P, was about 103 kPa and the saturation vapor pressure of CH2Cl2, P1, was about 30−70 kPa. If the measured temperature, T, was the same as the initial temperature, Ti, the pressure change due to the dissolution of He gas, Pi − P, was in the order of 10 kPa; that is, 1−10% of the supplied He gas was dissolved in the solution. The standard uncertainty of the measured initial or equilibrium He gas pressure, u(Pi) or u(P), was approximately 1 kPa based on preliminary experiments. Defining the relative standard uncertainty of Pi − P as u(Pi − P)/(Pi − P), u(Pi − P)/(Pi − P) was 10−2−10−1. The volume of a gas phase, V″, changed by the dissolution of He gas, and V″ was calculated from eqs 2, 3, 5, 8, 9, and 16 ⎧⎛ n v⎞ ⎛ Pv ⎞⎫ V ″ = V ⎨⎜1 − 1 ⎟ ⎜1 − 1 ⎟⎬ ⎩⎝ V ⎠ ⎝ RT ⎠⎭

vapor pressure of CH2Cl2 solutions of 2 kDa PLA with different concentrations, and that of CH2Cl2 solutions of different molecular weights PLA at 0.0075 mass fraction in the temperature range between 288 and 303 K. The values of bPLA for 2, 45, and 300 kDa PLA solutions at 0.0075 mass fraction were 3.78 × 10−3, 1.68 × 10−4, and 2.52 × 10−5 mol· kg−1, respectively. The vapor pressure of the solutions did not change as the polymer concentration or molecular weight was increased. The dotted curve shows the saturation pressure of pure CH2Cl2, as predicted by the Antoine equation: log p = A − (B/T + C − 273.15), where A = 6.07622, B = 1070.070, and C = 223.240 with p in kPa and T in K.23 The experimental data presented in Figure 2 are summarized in Table S1 (see Supporting Information). 3.2. Repeatability Test. In the experiment, all the samples were pressurized at room temperature. The pressure vessel containing the CH2Cl2 solution of PLA was introduced into a thermostatic bath to change its temperature, and after some minutes, the equilibrium pressure and temperature were

(19)

The change in V″ due to the dissolution of He gas was negligibly small compared to the magnitude of u(Pi − P)/(Pi − P). In summary, the uncertainty of n′2 was dominated by that of the measured pressure, and u(n′2)/n′2 was 10−2−10−1. Regarding u(P2), the value of P2 was calculated from eqs 1 and 5 as follows 97



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concentration. The measured data were fitted by the flowing function for each concentration of PLA

measured. Table 2 shows the calculated Henry’s law constant and the coefficient of variation (CV) for the repeatability tests

⎛H⎞ ⎛ T⎞ ln⎜ ⎟ = A⎜1 − 0 ⎟ ⎝ T⎠ ⎝ H0 ⎠

Table 2. Calculated Henry’s Law Constant and CV for Three Measurements for the Solubility of He in CH2Cl2 Solutions of 2 kDa PLA at bPLA = 0 and 3.78 × 10−3 mol·kg−1 between 288 and 303 Ka test 1 bPLA/mol·kg−1 0

3.78 × 10−3

a

test 2

where H0 and A are fitting parameters, and T0 is given by 298.15 K.21 The fitted parameters were summarized in Table 3. The experimental data presented in Figure 3a−c are summarized in Table S3 (see Supporting Information). 3.4. Effect of PLA Molecular Weight on N2 Solubility. The solubility of N2 was also found to increase as the molecular weight of the polymer increases. With the mass fraction of PLA in the CH2Cl2 solutions fixed at 0.0075, H increases as the number of molecules increases (i.e., the molecular weight decreases), which implies that the saturation concentration of N2 increases with increasing molecular weight. These results are shown in Figure 4. The measured data were also fitted by eq 21, and the fitted parameters were summarized in Table 4. The experimental data presented in Figure 4 are summarized in Table S4 (see Supporting Information).

test 3

T/K

H/MPa

H/MPa

H/MPa

CV

302.1 297.2 292.4 287.4 302.1 297.2 292.3 287.4

1048.7 1153.9 1220.7 1344.3 910.94 988.71 1012.0 1083.3

1120.0 1237.4 1319.4 1376.8 957.56 983.20 1069.2 1139.0

1127.6 1231.3 1331.4 1454.0 866.46 941.11 1033.5 1112.3

0.03 0.03 0.04 0.03 0.04 0.02 0.02 0.02

(21)

Standard uncertainties u are u(T) = 0.3 K and ur(H) = 0.04.

for the solubility of He in CH2Cl2 solutions of 2 kDa PLA at bPLA = 0 and 3.78 × 10−3 mol·kg−1 between 288 and 303 K. CV was defined as the ratio of the standard deviation to the mean. Because He is the least soluble of the three gases, we employed He for the repeatability test. As the solubility of a gas decreases, the uncertainty of n′2 increases, because the value of n′2 is calculated as a smaller difference between n2 and n2″. As mentioned in section 2.3, the uncertainty of H was dominated by the uncertainty of n′2; thus, the uncertainty of H increases with an increase in the uncertainty of n′2. A fresh solution of PLA was employed for each test. Because CV for the Henry’s Law constant at each condition was below 0.04, we employed the data from a single test for the theoretical models. The experimental data presented in Table 2 are summarized in Table S2 (see Supporting Information). 3.3. Effect of PLA Concentration on Gas Solubility. Figure 3a−c shows the Henry’s law constant, H, of the three gases at different polymer concentrations in the temperature range between 288 and 303 K. H for CO2 is much lower than that for either N2 or He, suggesting that CO2 is more soluble in CH2Cl2 than either N2 or He. H for CO2 does not depend on the concentration of PLA. In contrast, H for both N2 and He decreases as the concentration of PLA increases, suggesting that the solubility of N2 and He increases with increasing PLA

4. DISCUSSION Shirono et al.21 proposed a semiempirical equation for correlating H and T; that is, an expression for the solubility of gases in pure CH2Cl2 as a function of temperature (eq 21). Their predicted values of H0 and A in pure CH2Cl2 for each gas are also summarized in Table 3. From this equation, a variable, χ, which was the difference of the left-hand side and the righthand side of eq 21, was defined as follows: ⎛H⎞ ⎛ T⎞ χ = ln⎜ ⎟ − A⎜1 − 0 ⎟ ⎝ T⎠ ⎝ H0 ⎠

(22)

When we calculated χ, we employed the values of H0 and A in pure CH2Cl2 for each gas. Figure 5a−c shows the value of χ as a function of PLA molar concentration, cPLA (mol·L−1). From eq 8, cPLA is given by c PLA =

bPLA n1′M1 V′

⎛ ρ ⎞ bPLA MPLA =⎜ ⎟ ⎝ MPLA ⎠ 1 + bPLA MPLA

(23)

Figure 3. Henry’s law constant of gases vs temperature at different concentrations of 2 kDa PLA, bPLA: (a) CO2, (b) N2, and (c) He. Solid curves are the fitted curves of the function H = H0exp[A × (1 − T0/T)], where H0 and A are fitting parameters, and T0 is given by 298.15 K. The dotted curve is the prediction for Henry’s law constant of gases in pure CH2Cl2.21 98



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Table 3. Fitted Parameters with Their Standard Uncertaintiesa CO2 bPLA/mol·kg−1 0 3.78 1.13 1.89 2.64 0b a

× × × ×

10−3 10−2 10−2 10−2

N2

He

H0/MPa

u(H0)/MPa

A

u(A)

H0/MPa

u(H0)/MPa

A

u(A)

H0/MPa

u(H0)/MPa

A

u(A)

12.54 12.94 11.68 12.76 11.72 12.3

0.02 0.03 0.01 0.05 0.02 0.10

4.60 3.95 4.36 3.54 3.72 4.43

0.08 0.11 0.06 0.20 0.11 0.13

295.34 282.79 266.39 253.00 244.50 294

0.42 0.08 0.09 0.16 0.25 2

−2.34 −2.44 −2.32 −2.38 −2.36 −2.38

0.06 0.01 0.01 0.03 0.05 0.13

1203 932.8 704.6 582.6 414.2 1200

16 3.5 11.7 3.7 6.1 33

−3.94 −4.95 −7.86 −9.03 −9.35 −4.52

0.58 0.16 0.70 0.26 0.59 0.62

The fitting function is H = H0exp[A(1 − T0/T)] at T0 = 298.15 K. bRef 21.

0, 4.96 × 10−3, 1.47 × 10−2, 2.41 × 10−2, and 3.32 × 10−2 mol· L−1, respectively. For CO2, the measured data were scatted around χ = 0. A dependence of the Henry’s law constant on PLA concentration was not observed. In contrast, for N2 and He, a dependence on PLA concentration was clearly evident. Here, we considered a model to explain the dependence of the Henry’s law constant on PLA concentration. Assuming that X (the molar concentration of either N2 or He) in the gas phase is equilibrated with that in the liquid phase (XG ⇌ XL), and X in the liquid phase adsorbs on or desorbs from the adsorption sites of PLA molecules, Y, at equilibrium (XL + Y ⇌ XLY). The equilibrium constants are defined as

Figure 4. Henry’s law constant of N2 for different molecular weights of PLA. The mass fraction of PLA in the CH2Cl2 solutions was fixed at 0.0075. Solid curves are the fitted curves of the function: H = H0exp[A(1 − T0/T)], where H0 and A are fitting parameters and T0 is given by 298.15 K.

[XL] [X G]

(24)

K2 =

[XLY] [XL][Y]

(25)

If no PLA is included in this model system, then from eqs 10−12, K1 is given as follows:

Table 4. Fitted Parameters with Their Standard Uncertaintiesa

a

K1 =

K1 =

type of PLA

H0/MPa

u(H0)/MPa

A

u(A)

2 kDa 45 kDa 300 kDa

282.79 230.13 192.66

0.08 0.31 0.43

−2.44 −1.86 −1.62

0.01 0.06 0.10

Here, ρ = 1315.5 kg/m3 at 298.15 K and MPLA = 2 kg·mol−1 for 2 kDa PLA. The values of cPLA equivalent to bPLA = 0, 3.78 × 10−3, 1.13 × 10−2, 1.89 × 10−2, and 2.65 × 10−2 mol·kg−1 were

n2″ V″

V′

P2 RT

=

n2′ V′

⎛H n2′ ⎞ ⎜ ⎟ ⎝ RT n1′ + n2′ ⎠

=

RT (c1′ + c 2′) H

=

The fitting function is H = H0exp[A(1 − T0/T)] at T0 = 298.15 K.

n2′ V′ n2′

(26)

where c′1(= n′1/V′) and c′2(= n′2/V′) are the molar concentrations of the liquid solvent and gas, respectively. When no

Figure 5. χ as a function of cPLA: (a) CO2, (b) N2, and (c) He. Solid curve is the fitted curve. 99



Article

Figure 6. (a) Fitting parameter A as a function of cPLA, and (b) χ as a function of cPLA when A is given by the function of cPLA, A = −10.56 + 6.48exp(−57.5cPLA), in eq 22. The solid curve is the fitting curve.

in CH2Cl2 and did not adsorb on PLA. In contrast, in the case of N2 and He, B was not near to zero. The fitted values of B for N2 and He were 6.49 and 49.9 mol−1·L, respectively. The standard uncertainties of B, u(B), for N2 and He were 0.23 and 3.4 mol−1·L, respectively. The fitted value for He is about one order larger than that for N2. The equilibrium constant K2 increased in the order of CO2, N2, and He. The addition of PLA increases the solubility of gases in CH2Cl2, and this effect is larger for lower solubility gases. From eqs 22 and 32, a semiempirical relationship of the Henry’s law constant of gases at T and cPLA can be derived as follows:

PLA is included in the model system, the Henry’s law constant is given by H=

RT (c1′ + c 2′) RTc1′ RTρ ≈ = K1 K1 K1M1

(27)

When PLA is included in the model system, eqs 24 and 25 are satisfied simultaneously. Here, the molar concentrations of XG, XL, XLY, and Y are expressed by c2″(= n2″/V″), c2′ − c′(= (n2′ − n′)/V′), c′(= n′/V′), and αcPLA(= αnPLA ′ /V′), respectively, where α is the number of adsorption sites in one mole PLA. From eqs 24 and 25, the molar concentrations of XG is given by c 2″ =

c 2′ K1(1 + K 2αc PLA )

⎡ ⎛ T ⎞⎤ H 1 exp⎢A⎜1 − 0 ⎟⎥ = ⎣ ⎝ H0 1 + Bc PLA T ⎠⎦

(28) G

From eqs 10−12, the molar concentrations of X is also given by

where H0 is the Henry’s law constant at T = T0 and cPLA = 0. It is noted that the parameter B should depend on the type of PLA as well as the type of gases. From the Henry’s law constant of N2 for different molecular weights of PLA, as is shown in Figure 4, the parameter B should increase with increasing molecular weight of PLA. When the molecular weight increased under the same mass fraction of PLA, the solubility of N2 also increased. In this model, the effects of temperature and PLA concentration on the Henry’s law constant were considered independently. The parameter for temperature dependency, A, was determined at cPLA = 0, and the parameter for PLA concentration dependency, B, was determined at T = T0. H0 is the Henry’s law constant at T = T0 and cPLA = 0. However, in eq 30, the equilibrium constant, K1, should depend on PLA concentration as well as temperature because gases in the liquid phase cannot be clearly separated into the gases dissolved in the solvent and the gases adsorbed on the PLA polymer. As the PLA concentration increases, gases dissolved in the solvent could also be influenced by the dissolving PLA. Therefore, the parameter A should also depend on the PLA concentration. On the other hand, the equilibrium constant, K2, should depend on temperature because the adsorption equilibrium is balanced between the energetic and entropic effects. Therefore, the parameter B should also depend on temperature. In order to obtain more accurate semiempirical relationships of the Henry’s law constant of gases at T and cPLA, the parameters A, B, and H0 should be tuned by considering the dependency of temperature and PLA concentration on the equilibrium constants, K1 and

n2″ V″ P = 2 RT

c 2″ =

=

c 2′ H RT c1′ + c 2′

(29)

From eqs 28 and 29, the Henry’s law constant is given by H=

RT (c1′ + c 2′) RTc1′ 1 ≈ K1(1 + K 2αc PLA ) K1 (1 + K 2αc PLA )

(30)

From eq 27, eq 30 can be transformed into eq 31 H=

Hc0(T ) (1 + K 2αc PLA )

(31)

where Hc0(T) is the Henry’s law constant at cPLA = 0. Therefore, in Figure 5a−c, the quantity χ plotted as a function of PLA concentration can be fitted by the following equation χ = −ln(1 + Bc PLA )

(33)

(32)

where B was a fitting parameter. The curve fitting was performed for the data at 298 K. In the case of CO2, B was almost zero; that is, the equilibrium constant, K2, was very small. This suggests that most CO2 in the liquid phase dissolved 100



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K2. For example, in the case of He, if the fitting parameter A is given as a function of cPLA (see Table 3), that is, A is fitted by A = D + Eexp(−FcPLA), where D, E, and F are fitting parameters, as shown in Figure 6a, the parameters D, E, and F were determined to be −10.56, 6.84, and 57.5 mol−1·L, respectively. The standard uncertainties of D, E, and Fu(D), u(E), and u(F)were 1.27, 1.18, and 24.0 mol−1·L, respectively. If A is a function of cPLA, namely A = −10.56 + 6.48exp(−57.5cPLA), instead of a constant (A = −4.52), in eq 22, χ as a function of cPLA changes from Figure 5c to Figure 6b. When χ as a function of cPLA is fitted by eq 32, the parameter B was determined to be 51.3 mol−1·L. The standard uncertainty of B, u(B), was 3.1 mol−1·L. For the theoretical prediction of the Henry’s law constant in a gas−PLA−CH2Cl2 system, it is necessary to understand the structural and dynamic properties of PLA dissolving in the solvent and to clarify the gas state in the liquid phase.24,25

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.5b00268.

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REFERENCES

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5. CONCLUSION The effect of dissolved PLA on the solubility of CO2, N2, and He in CH2Cl2 was investigated by measuring the equilibrium pressure in a closed vessel containing a gas and a CH2Cl2 solution of PLA. Measurements were performed with different types (molecular weights) and concentrations of PLA in the temperature range between 288 and 303 K. The following conclusions can be drawn from this study: 1. The saturation vapor pressure of CH2Cl2 solutions of PLA was close to that of pure CH2Cl2. The vapor pressure of the solutions did not change as the PLA concentration or the molecular weight of PLA was increased. 2. In the absence of PLA, the order of the solubility of gases in CH2Cl2 was He < N2 < CO2. An increase in the concentration of PLA did not appreciably change the solubility of CO2 but yielded a higher solubility of N2 and He. The effect of PLA on the increase of gas solubility in CH2Cl2 was larger for less soluble gases. 3. Semiempirical correlations of the Henry’s law constant of gases to temperature and PLA concentration were derived from the measured data.

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Article

Experimental data presented in Figures 2−4 and Table 2. (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +81 3 5841 6971. Fax: +81 3 5841 6971. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by Grant-in-Aid for JSPS Fellows Grant Number 25-6917 (J. J. Molino Cornejo). 101


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