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Understanding and Analyzing Freezing-Point Transitions of Confined Fluids within Nanopores Steven Shimizu, Kumar Varoon Agrawal, Marcus O’Mahony, Lee W. Drahushuk, Neha Manohar, Allan S. Myerson, and Michael S. Strano* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States

Downloaded by UNIV OF SUSSEX on September 10, 2015 | http://pubs.acs.org Publication Date (Web): September 9, 2015 | doi: 10.1021/acs.langmuir.5b02149

S Supporting Information *

ABSTRACT: Understanding phase transitions of fluids confined within nanopores is important for a wide variety of technological applications. It is well known that fluids confined in nanopores typically demonstrate freezing-point depressions, ΔTf, described by the Gibbs−Thomson (GT) equation. Herein, we highlight and correct several thermodynamic inconsistencies in the conventional use of the GT equation, including the fact that the enthalpy of melting, ΔHm, and the solid−liquid surface energy, γSL, are functions of pore diameter, complicating their prediction. We propose a theoretical analysis that employs the Turnbull coefficient, originally derived from metal nucleation theory, and show its consistency as a more reliable quantity for the prediction of ΔTf. This analysis provides a straightforward method to estimate ΔTf of nanoconfined organic fluids. As an example, we apply this technique to ibuprofen, an active pharmaceutical ingredient (API), and show that this theory fits well to the experimental ΔTf of nanoconfined ibuprofen.

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GT equation fails to accurately predict ΔTf for most fluids (water, chlorobenzene, trans-decalin, cis-decalin, benzene, heptane, naphthalene, and cyclohexane). To resolve this issue, we instead apply the energetic analysis from the metal nucleation theory. We show that the melting of nanoconfined fluids can be well described by the theoretically predicted Turnbull coefficient15,16 (γSL/ΔHm) which is experimentally shown to remain invariant with pore size but is variable for different molecules. We provide a workflow for predicting the freezing point of a confined fluid (for pore diameter >4 nm) by estimating the Turnbull coefficient using a theoretical correlation relating the Turnbull coefficient to the change in entropy of the bulk fluid upon freezing, ΔSm,bulk, as first reported by Digilov.17 We show that the Turnbull coefficient, when plotted against (3RGC/2ΔSm,bulk) exp(ΔSm,bulk/3RGC), leads to nearly straight line fitting for metals and semimetals, and the same was true for various organic fluids, albeit with a different slope. Finally, we test the accuracy of this method by applying it to the experimental18 ΔTf of nanoconfined ibuprofen, an active pharmaceutical ingredient (API).

INTRODUCTION Understanding phase transitions of nanoconfined fluids can provide crucial insights into adsorption and transport in nanopores, including those in carbon,1,2 silica,3,4 zeolites,5,6 and porous drug-delivery platforms.7,8 While freezing transitions inside sub-4-nm pores can be difficult to predict due to their extreme diameter dependence,9 transitions inside larger pores have been shown to vary linearly with the inverse pore radius as described by the GT equation. The most common form of the GT equation employed in the literature for freezing in cylindrical pores is ΔTf = Tf,bulk − Tf,pore =

2Tf,bulkVmγSL rporeΔHm

(1)

where Tf,pore and Tf,bulk are the freezing points of the fluid in the pore and in the bulk, respectively. Vm is the molar volume of the liquid or solid phase, and rpore is the radius of the pore. It should be noted that the right-hand side of the equation is always positive, meaning that the fluid experiences a freezingpoint depression when inside nanopores above 4 nm in size. This common form of the equation assumes that Vm,s = Vm,l = Vm and that the solid is surrounded by its own liquid melt inside the pore (γSL = γsolid−wall − γliquid−wall). Although many researchers use the GT equation assuming a constant values of ΔHm and γSL, thermocalorimetry data10 and theoretical models11−14 show that these two parameters can change by almost an order of magnitude with the size of the nanopores. However, as we show here, even after accounting for the dependence of ΔHm and γSL on the pore diameter, the © XXXX American Chemical Society

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EXPERIMENTAL DATA

We use available experimental data on nonpolar organic fluids (chlorobenzene, trans-decalin, cis-decalin, benzene, heptane, naphthalene, and cyclohexane) from the report of Jackson and McKenna10 and water from the report of Schreiber et al.19 All ΔTf data in these Received: June 15, 2015 Revised: September 1, 2015

A

DOI: 10.1021/acs.langmuir.5b02149 Langmuir XXXX, XXX, XXX−XXX

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Applying this radius-dependent ΔHm in the GT equation and assuming a constant γSL, the resulting plot of ΔTf vs 1/r should be extremely nonlinear. However, experimental data10,19 on ΔTf show linearity down to a 3 or 4 nm pore diameter (Figure 1b), indicating that the solid−liquid surface free energy (γSL) is also a function of diameter. Several researchers have employed the Tolman length correction13,14,21 to predict diameter-dependent γSL (eq 2)

reports were collected using differential scanning calorimetry (DSC), which has the added advantage of being able to simultaneously measure ΔHm. Nanoconfined ibuprofen crystals were generated using the process described by O’Mahony et al.18 The mass of ibuprofen in controlled pore glass (CPG) for each pore diameter was determined using thermogravimetric analysis (TGA, TA Instruments Q5000) by heating samples (10−15 mg) from 30 to 250 °C and removing the organic ibuprofen content with only inorganic CPG remaining. The decomposition temperature of ibuprofen is 130 °C, whereas CPG is reportedly stable up to 600 °C. ΔTf and ΔHm for ibuprofen at each pore diameter were determined using DSC (TA Instruments Q2000 DSC, calibrated using an indium standard −156.6 °C and 28.54 J/g) and are listed in Table S1. Samples of 2−5 mg were prepared and sealed in aluminum pans and scanned at a heating rate of 10 °C/min from 20 to 150 °C.

⎛ 2δ ⎞⎟ γSL(r ) = γSL, ∞⎜1 − ⎝ r ⎠

where γSL,∞ denotes the planar solid−liquid surface tension and δ denotes the Tolman length and is equal to (Re − Rs). Here Re is the radius of the equimolar dividing surface (a dividing surface such that the system on the either side of the surface would contain an equal number of molecules, while the density changes discontinuously at the surface itself). Rs is defined as the radius of the surface where the Laplace equation holds (surface of tension, i.e., ΔP = (2γ/Rs)). The magnitude and the sign of the Tolman length are still controversial,13 although most agree that it is on the order of a few tenths to a few hundredths of the Lennard-Jones collision diameter. Although the Tolman length described by eq 2 is commonly used, it is a first-order approximation of eq 3 (by expanding around 2δ/r = 0, a planar surface), originally derived by Tolman.12

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(2)

RESULTS AND DISCUSSION While the GT equation has been applied to the phase equilibrium of confined fluids,20 the equation assumes constant γSL and ΔHm. A few studies that have measured both freezing points and ΔHm as a function of the pore size have found very significant changes in ΔHm (data reproduced in Figure 1a).10,19

⎛ 1 γSL(r ) = γSL, ∞⎜⎜ + 1 ⎝

2δ r

⎞ ⎟ ⎟ ⎠

(3) 14

Recent theoretical work by Deinert and co-workers attempts to explain the changes in the enthalpy of fusion at small pore radii for the same data set used here.10,19 Their model (eq 4) assumes a spherical particle in equilibrium with its own melt and uses a simplified Tolman length (eq 2) to describe γSL(r). ΔHm = −

3Vm,sγSL r

+

12Vm,sδ dγSL, ∞ ⎞ ⎟ − dT ⎠ r2

Figure 1. Enthalpies of fusion (a) and the freezing-point depressions (b) plotted versus r and 1/r, respectively, for several organic molecules and water. Data reproduced from Jackson and McKenna10 and Findenegg.19

12Vm,sγSLδ r

2

⎛ 2γ dVm,s 3Vm,s dγSL, ∞ + T ⎜ SL + r dT ⎝ r dT

(4)

They fitted the data set of ΔHm as a function of pore radii using δ as a fitting parameter and demonstrated an excellent correlation with the experimental data. Here, we used this analysis and obtained δ and the diameter-dependent γSL(r). By

Figure 2. (a) Plot showing the discrepancy between experimental ΔTf and those predicted using the GT equation and fitted Tolman lengths from Shin et al.14 (b) Parity plot showing the freezing-point depressions for all fluids calculated by fitting the GT equation to the experimental ΔTf, using the experimental ΔHpore and the Tolman-length-corrected γSL (using eqs 2 and 3). (c) Plot of calculated (γSL/ΔHm) as a function of 1/r for water and organic molecules considered in Figure 1a,b. B


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Langmuir substituting the γSL(r) and the experimental ΔHm into the GT equation (eq 1), we show that the resulting ΔTf is much higher (by several hundred Kelvin) than those observed experimentally10,19 (Figure 2a). Thus, while the model of Deinert and coworkers successfully fits ΔHm as a function of pore diameter, it cannot accurately predict the freezing-point depression in nanopores. We then determined whether the Tolman length correction could accurately predict the experimentally observed ΔTf. To this end, we used experimental ΔHm to fit δ and γSL,∞ from eqs 2 and 3 to the experimental ΔTf using the GT equation (eq 1). Interestingly, the more precise Tolman length correction (eq 3) gives a worse fit to the data (Figure 2b) and yields unreasonably large values for δ (Table 1). The first-order approximation of

a few angstroms, contrary to the accepted range (a few tenths to a few hundredths of the Lennard-Jones collision diameter). This result calls into question the use of the Tolman length correction for solid−liquid systems. In fact, Tolman originally derived the equation for a vapor−liquid system,12 and since then, there have been no rigorous derivations for solid−liquid systems. Thus, using the Tolman length correction to predict γSL as a function of radius is not ideal. Moreover, the use of the Tolman length correction to predict ΔTf also requires a model for diameter-dependent ΔHm, when experimental data for ΔHmis not available. Instead, it can be reasoned that if ΔTf varies linearly with 1/r, then (γSL/ΔHm) should be a constant. Indeed, in plotting (γSL/ΔHm) (found using experimental ΔTf in the GT equation) vs 1/r, we show this quantity to actually be nearly invariant with radius (Figure 2c). We note that Turnbull16 was the first to introduce (γSL/ ΔHm) in the study of metal nucleation. We term this ratio the Turnbull coefficient and further nondimensionalize the quantity to (γSLVm,s2/3NA1/3/ΔHm), which physically represents the free energy of formation of a monolayer of a solid−liquid interface using one mole of material, divided by the enthalpy of fusion. Turnbull demonstrated that this ratio is important in describing nucleation in liquid metals,16 where many metals had the Turnbull coefficients of ∼0.45. To determine the Turnbull coefficient for nanoconfined fluids, we used the molecular dynamics (MD) simulations of the crystal-melt (solid−liquid) interface by Broughton and Gilmer,22 where it has been shown that the potential energy (e.g., interaction energy) varies smoothly from the solid to the liquid phase (section 1, Supporting Information). The enthalpies of the solid and liquid phases were estimated by the potential energy of the bulk solid and liquid atoms, and


Table 1. Fitted Values of γSL,∞ and δ by Fitting Experimental ΔTf to the GT Equation (Equation 1) Using Experimental ΔHm and the Tolman Length Correction Tolman correction (eq 2)

Tolman correction (eq 3)

fitted parameters

γSL,∞(mJ/m )

δ (Å)

γSL,∞ (mJ/m2)

δ (Å)

water chlorobenzene trans-decalin benzene heptane naphthalene cis-decalin cyclohexane

84.2 13.3 13.2 12.3 11.6 8.9 7.3 3.5

6.9 7.6 7.2 7.1 8.0 5.8 7.8 9.0

41 548 36.2 34.1 38.8 56.6 18.9 8.3 4.2

32 876 103 79.4 95.8 266 41.6 17.1 22.5

2

the Tolman length correction (eq 2) actually works better in all cases despite the prediction not being valid at such small sizes. Even in this case, the obtained Tolman length is on the order of

Figure 3. Schematic showing the three scenarios for the calculation of the Turnbull coefficient. The blue circles represent liquid, whereas those in black represent solid. C


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Figure 4. (a) Estimated Turnbull coefficients (γSLVm,s2/3NA1/3/ΔHm) for water and several organics. The red dashed line indicates the predicted Turnbull coefficient (1/2) when we consider only enthalpic changes and ignore entropic contributions during the phase transformation. (b) Plot of metals, semimetals, water, and organics showing that they fit to a unique straight line with zero intercept and a slope (d2) as described by eq 5. While metals and semimetals fit to a single line (d2 = 0.248), organics segregate in two lines (dashed lines with d2 = 0.143 and 0.058). The solid line in between the dashed line (d2 = 0.110) represents the average for organics.

possible. We can thus conclude that a first-order approximation to the Turnbull coefficient (while neglecting entropic contributions) is 1/2. Next, we estimated Turnbull coefficients for various confined fluids by plugging experimental ΔTf in the GT equation and found that the Turnbull coefficients range between 0.097 and 0.51 (Figure 4a). While Turnbull empirically found that the ratio (γSLVm,s2/3NA1/3/ΔHm) for metals is relatively constant16 (∼0.45 for metals and 0.32 for semimetals), Digilov17 later derived a model for the γSL (eq 5) which shows that the Turnbull coefficient can be related to the change in bulk entropy upon freezing (ΔSm,bulk) using a fitted constant d2,

ΔHm was calculated by subtracting the enthalpy of a bulk solid atom from that of a bulk liquid atom (Figure S1, Supporting Information). γSL was calculated by dividing the total energy change of the system (upon forming the interface) by the total number of molecules at the solid−liquid interface (while assuming the density of the interface to be equal to that of the bulk solid and ignoring the relatively small surface entropy contributions; section 1, Supporting Information). The ratio (γSL/ΔHm) ranged between 0.42 and 1.2, depending on which crystal face interacted with the liquid. While the liquid interfacial layer showed a small decrease in enthalpy, the first and second solid interfacial layers had a more significant increase in enthalpy (Figure S1, Supporting Information). Furthermore, for a simple approximation of the Turnbull coefficient, we first consider only the enthalpic contributions to the surface free energy (no entropic changes). Consider placing two blocks of bulk solid and bulk liquid with molar volumes Vm,s and Vm,l, respectively, to form a solid−liquid interface with an energy penalty of γSLVm,s2/3NA1/3. We assume that the two phases next to the interface retain their bulk densities. This leads to three potential outcomes for the change in enthalpy: (1) The solid interfacial layer undergoes disorder and has an enthalpy halfway between those of a solid and liquid, while the liquid interfacial layer has the same enthalpy as Hl,bulk. (2) The liquid interfacial layer rearranges itself and has an enthalpy halfway between those of a solid and liquid, while the solid interfacial layer does not undergo an enthalpy change. (3) The interfacial layers of both phases have enthalpies equal to half of ΔHm. These scenarios along with the estimated Turnbull coefficients (1/2, −1/2, and 0 for the three scenarios, respectively) are shown in Figure 3 (derivation in section 2, Supporting Information). Okui23 suggested a zeroth-order approximation of the Turnbull coefficient to be 1/2 since close to the melting point a molecule on the surface is essentially like a liquid. Our analysis above shows that this reasoning is valid only if one assumes that entropy plays no role and if the solid interfacial layer has an enthalpy midway between those of the solid and liquid phases (scenario 1). Physically, the surface energy cannot be negative (scenario 2), or else the bulk phases would not exist; it would be more favorable to form as much surface as

γSLVm,s 2/3NA1/3 ΔHm

⎡ 3R ⎛ ΔSm,bulk ⎞⎤ GC = d 2⎢ exp⎜ ⎟⎥ ⎢⎣ 2ΔSm,bulk ⎝ 3R GC ⎠⎥⎦

(5)

where d2 = dSL2/dL2 and dSL2 and dL2 are the mean-square amplitudes of thermal vibrations in the solid−liquid interface and the liquid phase, respectively. RGC is the gas constant. The model assumes the solid to be surrounded by its own melt with the interfacial region consisting of disordered planes of solid atoms next to several stratified layers of liquid atoms. The energy of interfacial atoms has been assumed to differ only slightly from that of the bulk phases. The derivation assumes that vibrational changes contribute most significantly to the energy changes at the interface. Although it is derived for monatomic species, the model can be extended to larger molecules, especially when the solid−liquid interface forms a layered structure with dominating vibrational energy as described above. Equation 5 implies that plotting (γSLVm,s2/3NA1/3/ΔHm) vs (3RGC/2ΔSm,bulk) exp(ΔSm,bulk/3RGC) for various molecules should result in a straight line with zero intercept and a slope with a specific d2. In fact, as Figure 4b demonstrates, metals, semimetals, and water fall roughly on a straight line intersecting the origin (d2 = 0.248, Table S2). All nanoconfined organics fall significantly below the metal Turnbull coefficients and can be fitted together with d2 = 0.110 (Table S2). Empirically, we observe that these organics can be grouped into the two separate regions, although our data is limited. While one group (d2 = 0.143) consists of benzene, chlorobenzene, cyclohexane, and trans-decalin, the second group (d2 = 0.058) consists of D


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Figure 5. Parity plots showing the predicted (using eq 5) vs the experimental ΔTf. The plot in (a) is based on the single d2 value (0.110) for the organics, while that in (b) is based on the two d2 groupings (0.143 and 0.058) for the organics. (c) Experimental ΔTf for ibuprofen (Table S1) in nanopores showing a linear relationship between ΔTf and 1/r.

thermodynamics typically fails below this size). The use of the Tolman length correction for determining γSL(r) was shown to be inadequate for predicting ΔTf. Finally, we show that combining the Turnbull coefficient with the Digilov model for metal nucleation can be used to predict the freezing behavior of nanoconfined organics including APIs.

naphthalene and heptane. Subsequently, we show that nanoconfined ibuprofen (Figure 5c) falls within this second group (Figure 4b). In either case, nanoconfined organic compounds have significantly lower Turnbull coefficients compared to nanoconfined metals and semimetals. Physically, a smaller d2 value of organics suggests that they undergo a much larger relative decrease in thermal vibrational amplitude upon freezing compared to metals, semimetals, and water. We could not find a distinguishing characteristic to account for the segmentation of organics into the two groups using the physicochemical properties (e.g., ΔVm, ΔHf,bulk, Tf,bulk, or dipole moment). It will be of interest to measure Turnbull coefficients of several more organic compounds to determine if the correlation and these groupings continue to hold. We tested the efficacy of the Turnbull coefficient in predicting ΔTf of confined fluids by generating parity plots (plots of experimental vs calculated data, Figure 5a,b) and found fairly good agreement especially when using the two groupings for organic molecules (Figure 5b). In summary, we provide a simple route to predicting the freezing-point depressions of nanoconfined organic fluids. Briefly, one can obtain the Turnbull coefficient using the appropriate group of d2 and the bulk ΔSm. ΔTf can then be obtained by incorporating the Turnbull coefficient in the GT equation. For example, using the bulk ΔSm for urea,24 we estimate the Turnbull coefficient for urea in the range of 0.086−0.211 (for a d2 range of 0.058−0.143). This technique can be applied to predict the size-dependent melting point of nanosized APIs25−28 inside a nanoporous matrix for drug-delivery applications, where the melting point is directly related to solubility, an important property for bioavailability. We analyzed the melting-point data (Table S1) on nanoconfined ibuprofen, which showed a linear relationship between ΔTf and 1/r, despite a drastic change in ΔHm with radius (Figure 5c). Then using eq 5, we estimated the Turnbull coefficient for ibuprofen and showed that ibuprofen falls within the second group of organic compounds (d2 = 0.058, Figure 4b).

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.5b02149. Tables S1 and S2 and Figures S1 and S2 as described in the text. Sections on the analysis of MD simulations and the derivation of the first-order model of the Turnbull coefficient. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported in part by the U.S. Army Research Laboratory and the U.S. Army Research Office through the Institute for Soldier Nanotechnologies under contract number W911NF-13-D-0001. We acknowledge support from the ShellMIT-EI Energy Research Fund.

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CONCLUSIONS The purpose of this paper is to provide clarification of the appropriate use of the GT equation for predicting ΔTf and interpreting γSL, namely, the fact that γSL and ΔHm are predicted to vary with pore size, although the ratio of these two values (Turnbull coefficient) is relatively invariant with pore size for pores greater than 4 nm in diameter (classical E


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