Melting of solvents nanoconfined by polymers and networks

Thermodynamic Analysis

Melting and Thermodynamics of Mixing

In the case of polymer networks, thermoporosimetry is based on two thermodynamic relationships known as the FH theory15, 16 and the GT equation.14 The FH thermodynamics describes the melting point depression of the diluent and the polymer in polymer–diluent mixtures15, 16 as expressed in eqs 1, 2.

(1)

(2)

where the subscripts 1, 2 represent the diluent and polymer component, respectively. T_m and T are the melting temperatures in the solution and in the bulk and ΔH_u and V_u refer to the heat of fusion and volume per mole repeating unit, respectively. v refers to the volume fraction, x is the degree of polymerization of the polymer, and χ is the polymer–diluent interaction parameter. At equilibrium between liquid and crystalline diluent in the polymer solution, the chemical potentials of the diluent in the two phases must be equal. At the melting temperature of the pure diluent, its chemical potential equates to that in the standard state.17 Importantly, the melting point depression of the diluent occurs upon mixing with the polymer because the chemical potential of the diluent in solution is less than that of the pure solvent. This affects the melting of the solvent in that system.

Melting and Finite Size Effects

The GT equation predicts a lowering of melting temperature of liquids confined to a porous medium because of a small crystal size, which is generally written as:

(3)

where σ_sl is the solid–liquid interface energy, ΔH_f is the bulk heat of fusion, d refers to the crystal size, ρ_s is the crystal density. A is geometry factor, which takes a value of 4 for cylindrical pores and 2 for spherical pores.14

Materials

The solvents used in this study are benzene and n-hexadecane, both of which are of GC grade and obtained from Sigma-Aldrich, Co. without further treatment. cis-1,4-polyisoprene was obtained from Polysciences with weight average molecular weight of 305,000 g/mol and polydispersity of 1.05. Polyisoprene was crosslinked by dicumyl peroxide (DCP) at 149 °C for 2 h, which results in virtually complete reaction of the peroxide. The crosslink density was controlled by varying the amount of added dicumyl peroxide, which was represented as parts of DCP per hundred parts of polyisoprene (phr). Samples are designated phr1, phr5, and phr 10 for DCP quantities of 1, 5, and 10 phr, respectively. DCP is considered a quantitative crosslinker for polyisoprene,18 and the average molecular weight between crosslinks is estimated to be 13.95, 2.76, and 1.36 kg/mol for the phr1, phr5, and phr10 crosslinked polyisoprene, respectively.19

To calibrate the relationship between melting point depression and pore size, controlled pore glasses (CPGs) from Millipore with known mean pore size and narrow pore size distribution were used. The mean pore diameter of the CPGs ranged from 8 to 110 nm. The CPGs were cleaned and surface treated following the procedures in ref.20 to promote wetting by organic liquids.

Methods

The melting behavior of solvents in uncrosslinked and crosslinked polyisoprene was studied using a Perkin–Elmer Pyris 1 differential scanning calorimeter (DSC). Polymer and solvent were weighed separately into hermetic pans, and the sealed pans were kept at room temperature for at least 24 h before experiments were carried out to obtain a homogeneous solution. Pan weights were checked during this time as well as before and after experimental runs to ensure no solvent leakage, which is especially important for the benzene-polyisoprene mixtures. For n-hexadecane in controlled pore glasses, DSC samples were prepared in the same way except that hermetically sealed pans were heated at 100 °C for an additional 3 h to assure the filling of the nanopores with the n-hexadecane.

The temperature and heat flow of the DSC were calibrated using standard references. A cooling scan was first run at 5 K/min to −50 °C and the instrument held isothermally at −50 °C for 15 min. This was followed by a heating scan at 5 K/min. Melting temperatures were determined from the onset melting point during heating runs.

For the glass-transition temperature measurements of the benzene-polyisoprene mixtures, a cooling scan at 20 K/min down to −140 °C was first performed, which was immediately followed by a heating scan at 10 K/min to obtain the limiting fictive temperature, which is equivalent to glass-transition temperature.21

To obtain the “pore” size distribution of crosslinked polyisoprene, the calibration of “pore” diameter was performed using the CPGs with known pore size and narrow pore size distribution. By means of the Gibbs-Thomson (GT) equation, the melting point depression of confined n-hexadecane in CPG was related to the size of confinement, which is the “pore” size of the CPG. After determining the relationship between melting point depression and the “pore” size of the confinement, the “pore” size distribution of the crosslinked polyisoprene was obtained from the melting thermograms of the polyisoprene-confined n-hexadecane.

The weight average “pore” size distribution of crosslinked rubber, as its name suggests, is the weight fraction of “pores” of the same size. It was calculated by dividing the melting thermograms into narrow temperature intervals, each of which corresponds to a certain “pore” size. The weight of “pores” of a given diameter at which the melting transition occurs is directly proportional to the energy in the temperature interval and inversely proportional to the apparent heat of fusion:

(4)

where the Area refers to that covered by the thermogram after subtracting the baseline, k is a calibration factor, and ΔH_d is the heat of fusion of a crystal of diameter d, which latter was obtained from the controlled pore glass calibration data. In addition, by assuming a spherical “pore” geometry for the polyisoprene network, the number average “pore” size distribution was obtained from the weight average size distribution.

Solvent Melting in Uncrosslinked Polyisoprene

The DSC thermograms for the benzene/polyisoprene system at different concentrations are shown in Figure 1. As the volume fraction of polyisoprene increases, the melting endotherm of the benzene shifts to lower temperatures and broadens. The results for the melting temperature and the heat of fusion of benzene in the uncrosslinked polyisoprene are shown in Figure 2. Figure 2(a) also includes the FH prediction for melting point depression. A concentration independent FH parameter of χ = 0.4319 was used for the theoretical estimation. If peak melting temperature is used to determine ΔT_m, the melting point depression is close to, but still greater than the estimation based on FH theory. If onset melting temperature is chosen, however, one can see a large deviation from the FH theory in Figure 2(a). Here, because of the broad endotherm and our interest in deviations from FH theory, we use the onset melting point, rather than the just the peak maximum, to characterize the melting behavior of the diluent. We remark that when the melting endotherm is narrow, there is little difference between the two. Furthermore, it is a common practice to use the peak melting to describe size dependence of the melting behavior.19, 20, 22-24 In such cases, the interest has been in the general trends and not in the details of broadening, per se. In the case of thermoporosimetry, the entire thermogram is of interest and one can see from the above discussion that the peak is related most closely to the weight average pore diameter while the onset emphasizes the smallest crystals (or other event in the case of deviations from FH theory). As shown in Figure 2(b), there is a decrease in the heat of fusion of benzene at high polymer concentrations. To further examine this issue, plasticization effects due to noncrystallizing benzene were studied by investigating the glass-transition temperature of the benzene/polyisoprene mixture. The results indicate that there is some plasticization caused by the benzene at high polymer concentrations, but this accounts only partially for the decrease in the heat of fusion. This is discussed further subsequently.

We also examined n-hexadecane as the diluent in the polyisoprene. Its DSC thermograms and melting behavior in the presence of uncrosslinked polyisoprene are shown in Figures 3 and 4, where again we show melting point depression as determined by onset value. Once again melting point depression does not follow the FH prediction and the deviation depends on the volume fraction of polyisoprene v₂. As v₂ increases, the deviations from the FH estimation increase. In terms of the heat of fusion, the n-hexadecane displays a quite different behavior from the benzene. Instead of decreasing, the heat of fusion for the n-hexadecane remains almost constant over the full concentration range investigated. This observation satisfies the assumption underlying FH theory, which assumes that the heat of fusion in the solution is equal to that in the bulk. In spite of this, the n-hexadecane/polyisoprene system still exhibits an evident deviation from the FH prediction for the melting point depression.

Here, because of the unavailability of a literature value for the χ parameter for the n-hexadecane and polyisoprene mixture, we took χ = 0.5, which is reasonable for n-alkane and polyisoprene systems.25 Although we did not take into account the possible dependence of χ on concentration, it is not expected to have a great impact here.

A common method of using the FH theory to obtain an “empirical” fit to the behavior of polymer solutions is to make the χ parameter an adjustable parameter.26 If we do this here, we find that for the n-hexadecane, an unreasonable value for the interaction parameter (χ ≈ −0.4) would be required. Similarly, in the case of the benzene/polyisoprene system, to make the onset melting point consistent with the FH prediction, the FH interaction parameter χ needs to be zero (χ ≈ 0), which is not reasonable for polymer–diluent systems.

“Phase” Diagrams

Considering the complex melting behavior, especially the decrease in the heat of fusion shown by the benzene/polyisoprene system, we constructed its “phase” diagram as shown in Figure 5. The diagram includes the nonequilibrium glassy and supercooled liquid states in addition to the true thermodynamic states. In Figure 5, the squares represent glass-transition temperature data for benzene/polyisoprene solutions, which is accounted for by the appreciable plasticization effect from the vitrified benzene/polyisoprene “supercooled” liquid phase in the high polymer concentration regime. The dotted line in the lower part of “phase” diagram is the prediction for T_g of the mixtures based on the empirical Fox equation.28 As can be seen, at a volume fraction of polymer less than around 0.6, the experimentally observed T_g data depart significantly from the Fox relation. In fact, the T_g from v₂ ≈ 0.17 to v₂ ≈ 0.53 is almost constant and consistent with a volume fraction benzene of about 10% being dissolved in the polyisoprene. Importantly, the phase diagram glass transition data do not show inconsistencies with the literature29, 30 because we do not see, e.g., two glass transitions in our system, perhaps because of the difficulty of glass formation in benzene. The left side of the phase diagram in Figure 5 shows a surprising jump in the T_g of the system at a composition of around 53% polymer, i.e., we see that the jump is to a T_g that is very close to the bulk T_g of polyisoprene indicating that at concentrations lower than about 53% polymer, the benzene phase separates nearly completely and, therefore, the T_g observed is that of the polyisoprene rather than of plasticized polyisoprene. In contrast, at the higher polymer concentrations or, perhaps, in the small confinements, the glass forming ability of benzene improves. As a result above 53% polymer, as shown in Figure 5, the glass transition of the mixture exhibits evident composition dependence, which suggests the forming of the glassy or supercooled benzene/isoprene mixture.

We did not measure the melting of polyisoprene as it is a slow crystallizer, therefore our FH estimate for the melting of polyisoprene, shown by the dashed line in Figure 5, was based on literature data31 for the polyisoprene parameters required by eq 2. Since there is not a large difference between the melting temperatures of benzene and polyisoprene, it is theoretically expected to have a eutectic point32-35 at v₂ ≈ 0.5, where the two solid phases of ployisoprene and benzene are simultaneously in equilibrium with the isotropic liquid phase. We present this in Figure 5 for completeness; in the hypoeutectic concentrations, there is crystallized benzene and supercooled polyisoprene/benzene in our system. Interestingly, it appears that this hypoeutectic system is a supercooled mixture of 10% benzene with polyisoprene plus pure benzene crystals. Above the eutectic composition, the benzene in the supercooled mixture takes on the composition of the glassy system, which now decreases from about 35% to 15% benzene in the hypereutectic range investigated. We comment that the details of this behavior could conceivably be affected by the kinetics of crystallization of the benzene during cooling.

For n-hexadecane and uncrosslinked polyisoprene, because of the nearly constant heat of fusion, plasticization is not expected, which makes its “phase” diagram much simpler than that for the benzene/polyisoprene system as illustrated in Figure 6.

Solvents Nanoconfined in Controlled Pore Glasses

To investigate whether or not finite size effects provide an explanation for the observed breakdown of the FH prediction for diluent melting in uncrosslinked polyisoprene, we employed CPGs as “calibrants” for the diluent melting behaviors. The results for n-hexadecane in the CPGs are presented in Figure 7, which shows a linear relationship between melting point depression and inverse crystal size, as predicted by the GT eq 3. Assuming cylindrical pores, the surface energy of the solid–liquid interface was determined to be 13.9 erg/cm², which is in reasonable agreement with literature values for other alkanes (σ_sl = 7.2 erg/cm² for n-heptadecane and σ_sl = 9.6 erg/cm² for n-octadecane36).

The heat of fusion of the n-hexadecane in the CPGs exhibits a strong dependence on crystal size, which is consistent with literature findings18, 22, 37, 38 for small molecules confined in CPGs and the melting of some metallic nanocrystals. Generally, the size dependence of the heat of fusion for cylindrical small crystals can be described by the following equations:

(5)

(6)

By fitting our experimental data to eq 5 as shown in the inset of Figure 7(b), we obtain a surface energy σ_sl = 112 erg/cm². Similar to Sun's results,38 the surface energy estimated by the heat of fusion is around an order of magnitude larger than that predicted by the melting point depression, which might indicate that there is an additional contribution to the decrease of the heat of fusion not accounted for in eq 5.38 Equation 6 is also known as the liquid layer model,39 where t₀ refers to the thickness of the liquid layer. The fitting of our results for the liquid layer model gives t₀ ≈ 0.9 nm. The calculated surface energies and thicknesses of the liquid layers are summarized in Table 1. Interestingly, Table 1 shows that the solid–liquid surface energies estimated from eq 5 for the reduced heat of fusion are significantly higher than those estimated by the melting point depression, which suggests that the classical GT model can only account for melting point depression and not for the reduced heat of fusion.

The melting point depression and the heat of fusion of benzene confined in the CPGs are also shown in Figure 8. Except for the smallest pore size studied (d = 8 nm), the melting point depression of the confined benzene follows a linear relationship with respect to reciprocal of the pore diameter. The heat of fusion also decreases with decreasing pore diameter. For the benzene in the uncrosslinked polyisoprene, we found that the crystal size estimated from melting point depression and heat of fusion calibration gave similar results, i.e., crystal sizes range from 8 to 40 nm.

Both benzene and n-hexadecane exhibit melting point depressions and heat of fusion depressions upon confinement in controlled pore glasses. However, only benzene shows a depression of the heat of fusion in the uncrosslinked polyisoprene. Hence, while finite crystal size in the polyisoprene/benzene system may account for the deviations from the FH prediction, it appears that such an explanation is not consistent with the n-hexadecane results. It appears that another explanation for the breakdown of the FH model of diluent melting in uncrosslinked polymers needs to be found.