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Thermodynamic Considerations of Microgel Swelling Behavior V. T. Pinkrah, A. E. Beezer,*,† B. Z. Chowdhry, L. H. Gracia, J. C. Mitchell, and M. J. Snowden* Medway Sciences, School of Sciences, University of Greenwich, Medway University Campus, Chatham Maritime, Kent ME4 4TB, U.K. Received September 22, 2003. In Final Form: July 6, 2004 A simple but novel thermodynamic model is presented, based upon van’t Hoff analysis, for the reversible swelling behavior of colloidal microgels. The swelling, as a function of temperature, of poly(N-isopropylacrylamide/N,N′-methylenebisacrylamide) as well as poly(N-isopropylacrylamide/vinylpyridine/N,N′methylenebisacrylamide) and poly(N-isopropylacrylamide/acrylic acid/N,N′-methylenebisacrylamide) microgel dispersions in H2O and D2O has been studied by photon correlation spectroscopy (PCS). PCS data was used to obtain the hydrodynamic diameter and hence the volume of the microgels (before and after reconstitution following freeze-drying) as a function of temperature. The choice of standard reference states, for analyzing the data attained, is discussed, and the one selected is that of the volume of the microgels at 333 K in H2O. For all microgels examined the volume, at this temperature, is shown to be independent of solvent (H2O, D2O). The derived data has allowed the exploration of a novel thermodynamic approach to the study of the swelling behavior of the microgels. The constant volume, at 333 K, for each of the polymer systems constituting the microgels is suggested to be an intrinsic property of the polymers themselves.
Introduction Poly(N-isopropylacrylamide/methylenebisacrylamide) [poly(NIPAM/BA)] microgels have attracted significant interest in the recent scientific literature1-11 due, in part, to their interesting physicochemical properties, in particular, their ability to undergo reversible volume phase transitions (VPT). The VPT properties of microgel systems can be manipulated and controlled by, e.g., co-polymerization with monomers possessing ionizable functional groups.12 Consequently the physicochemical properties of colloidal microgels based on poly(NIPAM/BA) and various copolymer derivatives have been widely investigated1-2,4,11 in relation to the VPT changes that occur in response to a number of external stimuli such as pH,13 temperature,1,3 and ionic strength.13-15 Poly(NIPAM/BA) * Corresponding author. Phone: + 44 (0) 208-331-9981. Fax: +44 (0) 163-488-3044. E-mail:
[email protected]. † Current address: School of Pharmacy, University of London, 29-39, Brunswick Square, London, WC1N 1AX, U.K. (1) Murray, M.; Snowden, M. J. Adv. Colloid Interface Sci. 1995, 54, 73. (2) Pelton, R. H.; Chibante, P. Colloid Surf. 1986, 20, 247. (3) Pelton, R. Adv. Colloid Interface Sci. 2000, 85, 1. (4) McPhee, W.; Tam, K. C.; Pelton, R. J. Colloid. Interface Sci. 1993, 24, 156. (5) Saunders, B. R.; Vincent, B. Adv. Colloid Interface Sci. 1999, 80, 1. (6) Pelton, R. H.; Pelton, H. M.; Morphesis, A.; Rowell, R. L. Langmuir 1989, 5, 816. (7) Snowden, M. J.; Vincent, B. J. Chem. Soc., Chem. Commun. 1992, 1103. (8) Kawaguchi, H.; Fujimoto, K.; Mizuhara, Y. Colloid Polym. Sci. 1992, 270, 53. (9) Wu, X.; Pelton, R. H.; Hamielec, A. E.; Woods, D. R.; McPhee, W. Colloid Polym. Sci. 1994, 272, 467. (10) Snowden, M. J.; Marston, N. J.; Vincent, B. Colloid Polym. Sci. 1994, 272, 1273. (11) Murray, M.; Chowdhry, B. Z.; Snowden, M. J. Chem. Br. 1995, 12, 943. (12) Hirotsu, S.; Hirokawa, Y.; Tanaka, T. J. Chem. Phys. 1987, 87, 1392. (13) Snowden, M. J.; Chowdhry, B. Z.; Vincent, B.; Morris, G. E. J. Chem. Soc., Faraday Trans. 1996, 92, 5013. (14) Daly, E.; Saunders, B. R. Langmuir 2000, 16, 5546.
microgel particles exhibit a temperature-induced, reversible VPT in H2O at 305 K.2,5 Water behaves as a good solvent at room temperature (microgel is in a swollen state) when polymer-solvent interactions are favored; however, water acts as a worse solvent as the temperature is increased (microgel in a shrunken state), leading to the collapse of the microgel when polymer-polymer attractions are favored over solvent-solvent interactions.5 At temperatures above the volume phase transition temperature (VPTT) of poly(NIPAM/BA), inter- and intrachain hydrogen bonding and hydrophobic interactions are said to be dominant.5 It has been proposed by Saunders et al.5 that the extent of poly(NIPAM/BA) microgel swelling is governed by the activity of water within the continuous phase. It is generally believed that local ordering of water molecules occurs in the vicinity of the poly(NIPAM/BA) chains and that the hydration process is exothermic.16 Microgel particle swelling is held to be entropically driven and is suggested to be dependent on hydrogen bonding between the water molecules and the amide (isopropylacrylamide and methylenebisacrylamide) and/or carbonyl groups of the microgel.17 Daly and Saunders14 reported that an increase in temperature increases the entropic contribution, which leads to particle deswelling and a collapse of the microgel particle as water becomes excluded from the particle interior. The classical theory of gel swelling, proposed many years ago by Flory and Rehner,18 assumes uniform distributions of polymer segments and crosslinking points throughout the polymer network. At swelling equilibrium, the chemical potential of water is equal inside and outside the microgel particles. Flory’s (15) Benee, L.; Snowden, M. J.; Chowdhry, B. Z. Encyclopedia of Advanced Materials; John Wiley and Sons Ltd.: New York, 2002. (16) Saunders, B. R.; Crowther, H. M.; Morris, G. E.; Mears, S. J.; Cosgrove, T.; Vincent, B. Colloid Surf., A 1999, 149, 57. (17) Daly, E.; Saunders, B. R. Phys. Chem. Chem. Phys. 2000, 2, 3187. (18) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.
10.1021/la035765+ CCC: $27.50 © 2004 American Chemical Society Published on Web 08/26/2004
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“network of swelling” has been discussed by both Pelton3 and Saunders5 in their reviews. There are a number of problems with Flory’s theory of gel swelling. First, it assumes a uniform cross-link density within the polymer network (there is evidence that this is not the case for many microgel systems). Second, the concentration dependence of χ (the Flory parameter) is not included (Napper et al.19 among others have considered this problem). Using PCS to obtain hydrodynamic diameter and volume data for microgel systems is commonplace. What really should be sought at constant gel number distribution is the amount of water in the microgel as a function of the temperature-induced volume change. Moreover, the majority of reports relating to microgel systems record the deswelling ratio (for temperatures above 298 K) as VT/ V298, and this (obviously) relates to a reference (i.e., standard) state at 298 K. This definition of a reference state is, presumably, selected on the basis of the usual thermodynamic conventions. However, as is evident from the scientific literature, the same microgel dispersed in different solvents (e.g., water20 or water/alcohol mixtures21-23) exhibits different values for hydrodynamic diameter at 298 K. The aforementioned definition also results in swelling ratios for temperatures 298 K. The foregoing procedures make comparisons of data between different microgel systems constituted of different polymers and/ or solvent systems awkward. The objective of this report is to explore the choice of other reference states and thereby discover what thermodynamic conclusions can be drawn with respect to the behavior of poly(NIPAM/BA) as well as poly(N-isopropylacrylamide/vinylpyridine/N,N′-methylenebisacrylamide) [poly(NIPAM/VP/BA)] and poly(N-isopropylacrylamide/acrylic acid/N,N′-methylenebisacrylamide) [poly(NIPAM/Acc/BA)] microgel dispersions. Unsurprisingly the selection of a different standard reference state will result in values for and hence conclusions from thermodynamic parameters (∆G, ∆H, and ∆S) that are different from those currently used. The solvents selected for this study were H2O and D2O. Temperature-based PCS observations have been undertaken with both freshly prepared and reconstituted freeze-dried microgel samples of identical composition. The approach reported has been developed to allow a more direct and revealing comparison of swelling as a function of temperature for these microgel systems than that attained to date. Experimental Section Materials. All chemicals were obtained from commercial suppliers and used without further purification. Microgel Preparation. Anionic poly(NIPAM/BA), cationic poly(NIPAM/VP/BA), and anionic poly(NIPAM/Acc/BA) microgel dispersions were prepared by surfactant-free emulsion polymerization, in H2O, at 343 K. The recipe for the synthesis of poly(NIPAM/BA) involved using 5 g of N-isopropylacrylamide (NIPAM) monomer (Aldrich), 0.5 g of potassium persulfate as the initiator (BDH Chemicals), and 0.5 g of the cross-linking agent N,N′-methylenebisacrylamide (19) Napper, D. H. Polymeric Stabilisation of Colloidal Dispersions; Academic Press: London, 1983. (20) Crowther, H. M.; Saunders, B. R.; Mears, S. J.; Cosgrove, T.; Vincent, B.; King, S. M.; Yu, Ga-Er. Colloid Surf., A 1999, 152, 327. (21) Saunders, B. R.; Crowther, H. M.; Morris, G. E.; Mears, S. J.; Cosgrove, T.; Vincent, B. Colloid Surf., A. 1999, 149, 57. (22) Crowther, H. M.; Vincent, B. Colloid Polym. Sci. 1998, 276, 46. (23) Zha, L.; Hu, J.; Wang, C.; Fu, S.; Elaissari, A.; Zhang, Y. Colloid Polym. Sci. 2002, 280, 1.
Pinkrah et al. (BA; Aldrich). In the case of poly(NIPAM/Acc/BA) microgel (0.0712 mole fraction Acc), 4.75 g of (NIPAM), 0.25 g of acrylic acid, 0.5 g of N,N′-methylenebisacrylamide, and 0.5 g of ammonium persulfate as the initiator were used. Similarly, for preparing the poly(NIPAM/VP/BA) microgel (0.246 mol fraction of 4-VP), 3.75 g of (NIPAM), 1.25 g of 4-vinylpyridine, 0.5 g of N,N′methylenebisacrylamide, and 0.5 g of 2,2′-azobis(2-methylpropionamidine) dihydrochloride (a cationic initiator; Wako Chemical Ltd.) were used. The methods used for the synthesis have been described previously in detail.13,24 All microgels were purified by extensive dialysis and centrifugation using deionized water. The final concentration of the colloidal microgel stock dispersions was ∼0.55 w/w %. Typical reaction yields were of the order of 95%. Freeze-drying was carried out via standard procedures utilizing an Edward’s freeze-dryer. Physical Measurements. PCS data were obtained using a Malvern Instruments Zetasizer 3000 instrument equipped with a 5 mW helium-neon laser (λ ) 633 nm). Scattered light from the samples was detected 90° to the sample. The hydrodynamic diameters of the microgel particle dispersions were calculated using the Stokes-Einstein equation. Stock microgel dispersions were diluted to a particle concentration of 0.1% (w/w). Freezedried microgel samples of a known mass (∼0.0140 g) were redispersed in a known volume (10 cm3) of deionized H2O or D2O to facilitate particle reswelling. Samples were continuously stirred (using a magnetic stirrer) for at least 12 h prior to measurements being undertaken. Samples were equilibrated at each temperature for a minimum of 10-15 min before data collection. Typically at each temperature (283-333 K at 5 K intervals) three measurements were made on all samples with each measurement consisting of 10 subruns to obtain an average hydrodynamic diameter. The data were analyzed using the Contin software provided by the PCS instrument manufacturer. The coefficient of variation (error) for the hydrodynamic diameter of the microgels was estimated to be less than 5%. It is to be noted that in the case of the volume data, relating to the microgels in D2O, a correction has to be applied to take into account the change in viscosity of D2O with temperature. To this end volumes of the microgel particles were corrected for the change in viscosity of the D2O (obtained from ref 25), relative to the viscosity of H2O, at each temperature, using the relationship
VD2O ) Vapp (ηH2O/ηD2O)3 where Vapp is the calculated volume and ηH2O and ηD2O are the viscosities of H2O and D2O respectively, at a given temperature.
Results and Discussion The experimentally derived data sets consist of hydrodynamic volume versus temperature data for the three microgel systems [poly(NIPAM/BA), poly(NIPAM/VP/BA), and poly(NIPAM/Acc/BA)] in the three solvent systems (stock H2O, freeze-dried/H2O, and freeze-dried/D2O). The thermally-induced volume change for the different microgel systems dispersed in H2O and D2O are shown in Figure 1a-c. The first notable observation is that the volume changes for the dispersions in the three systems are different. The volumes recorded at 298 K for the same microgel but in the different solvent systems are significantly different. The volume at 298 K is dependent upon the properties of the solvent and is subject to change from one microgel/solvent system to another. Therefore, the consequence of choosing 298 K as the standard state reference point for swelling/deswelling ratios will, necessarily, yield rather different values for each of the systems as they are dependent on the reference state. The second notable feature is that it appears that the volumes occupied (24) Pinkrah, V. T.; Snowden, M. J.; Mitchell, J. C.; Seidel, J.; Chowdhry, B. Z.; Fern, G. R. Langmuir 2003, 19, 585. (25) Cho, C. H.; Urquidi, J.; Singh, S.; Robinson, G. W. J. Phys. Chem. B 1999, 103 (11), 1991.
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Table 1. Volumes of the Different Microgel Systems Examined at 283 and 333 K
solvent
vol. (cm3) 283 K poly(NIPAM/ BA)
vol. (cm3) 283 K poly(NIPAM/ VP/BA)
vol. (cm3) 283 K poly(NIPAM/ Acc/BA)
vol. (cm3) 333 K poly(NIPAM/ BA)
vol. (cm3) 333 K poly(NIPAM/ VP/BA)
vol. (cm3) 333 K poly(NIPAM/ Acc/BA)
fresh stock H2O (ref) freeze-dried Re-H2O freeze-dried Re-D2O
1.8 × 10-13 ((2.3 × 10-15) 1.9 × 10-13 ((3.3 × 10-15) 1.9 × 10-13 ((9.7 × 10-14)
1.5 × 10-14 ((8.2 × 10-18) 1.6 × 10-14 ((1.4 × 10-18) 1.9 × 10-14 ((1.7 × 10-17)
1.8 × 10-12 ((8 × 10-13) 3.1 × 10-12 ((2.3 × 10-13) 3.8 × 10-12 ((1.8 × 10-13)
5.4 × 10-15 ((2.2 × 10-17) 3.8 × 10-15 ((1.6 × 10-17) 8.4 × 10-16 ((3.1 × 10-17)
2.9 × 10-17 ((5.2 × 10-19) 2.1 × 10-17 ((1.4 × 10-20) 3.1 × 10-17 ((5.2 × 10-22)
3.9 × 10-14 ((8.2 × 10-17) 3.4 × 10-15 ((7.2 × 10-18) 1.1 × 10-14 ((5.6 × 10-18)
Figure 2. Illustration of the equilibrium swelling of a crosslinked poly(NIPAM/BA) microgel system at constant temperatures.
Figure 1. Plots of the volume change as a function of temperature: (9) stock solution, (O) rehydrated in H2O, and (2) rehydrated in D2O for (a) poly(NIPAM/BA), (b) poly(NIPAM/ VP/BA), and (c) poly(NIPAM/Acc/BA) microgels, respectively.
by a given microgel, in the solvents examined in this work, at a temperature of 333 K are the same (see Table 1) within experimental error, irrespective and independent of the solvent environment. Moreover, all temperaturerelated volume ratios will, by definition, be swelling ratios. Hence, the volume at 333 K is taken to represent the standard state. The uniformity of the volume at 333 K for each polymer system, presumably, means that this volume represents an intrinsic property of the polymer(s) used to synthesize the microgel (and the poor quality of the solvent). This
further reinforces the notion that the volume at 333 K is a limiting volume for the microgels and may be independent of solvent. The swelling of the microgels can be represented in the form of an equilibrium-based equation, as shown in Figure 2. For example, consider placing a microgel, at the volume it would have at 333 K, into a solvent at some other given temperature, e.g., 298 K, to determine the equilibrium swelling at 298 K. Second, assume, in addition, that the observed change in volume (∆V) is free volume occupied only by solvent. Since the volume occupied by a microgel at 333 K for each of the microgel systems studied is, within experimental error, the same, then the volume data can be normalized (now strictly and explicitly the swelling ratio) to this limiting volume at 333 K. A limiting microgel size at 323 K has been reported in the literature previously. For example, Saunders et al. also made similar observations in which they observed that microgel particles were fully collapsed at 323 K.16,5 Wu et al. also concluded that above 323 K microgel particle size is constant.9 Figure 3a-c illustrates the temperature-related volume changes as a function of temperature for the three microgel systems in the different solvent systems normalized to the volumes at 333 K. Figure 3 therefore displays swelling ratios. As the volume at 333 K for each microgel/solvent system is constant, these plots illustrate the rate of volume change as a function of temperature. These data indicate that the poly(NIPAM/BA) microgel (stock or rehydrated freeze-dried/H2O) systems exhibit only moderate swelling ratio changes with temperature. In contrast, the rehydrated freeze-dried poly(NIPAM/BA) microgel in D2O shows a much more dramatic swelling ratio; it is 6 times more swollen than this microgel is in H2O systems (Figure 3a). For the other microgel systems, normalization of the volume data changes the volume-related ordering of the three solvent systems. In the case of poly(NIPAM/VP/BA) (Figure 3b), the rehydrated freeze-dried microgel in H2O behaves similarly to the microgel rehydrated in D2O; the swelling ratio is 1.5 times more swollen than that in the
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Figure 4. First-derivative plot as a function of temperature for the poly(NIPAM/BA) microgel: (9) stock solution, (O) rehydrated in H2O, and (2) rehydrated in D2O.
Figure 3. Graphical plots of normalized volume change as a function of temperature: (9) stock solution, (O) rehydrated in H2O, and (2) rehydrated in D2O for (a) poly(NIPAM/BA), (b) poly(NIPAM/VP/BA, and (c) poly(NIPAM/Acc/BA) microgels, respectively.
bulk. The relative position is dramatically changed in the case of the poly(NIPAM/Acc/BA) microgel (Figure 3c). This microgel rehydrated in D2O now takes the midposition, and the gel rehydrated in H2O shows the largest response to temperature. The swelling ratio in the rehydrated H2O system is 2 times greater than that for D2O but, more noticeably, 18 times more swollen than in the bulk. This could be a consequence of the presence of the acrylic acid group. These differences are not obvious in the volume data alone. Hence, normalization of the swelling data gives better discrimination between the different microgels, by grouping the behavior, and allows greater insight into the physicochemical properties of the microgels, which in turn allows quantification of the temperature-dependent response of the microgels to be explored. The firstderivative plot of the volume/temperature data for the poly(NIPAM/BA) microgel system (Figure 4) indicates the VPTT of the system in different solvent systems; the D2O system displays a VPTT ∼2 K different from that of the
H2O systems. The bulk system shows a VPTT of 302 K. This observation is in agreement with previous literature reports.26 The observed increase in the VPTT reported herein for the poly(NIPAM/BA) microgel dispersed in D2O, relative to the VPTT in H2O, has also been reported by a number of authors. Crowther et al. reported a 2 K increase in the VPTT of poly(NIPAM/BA) in D2O; the explanation given for this increase was that there was an increase in solvation of the polymer network by the D2O.20 Saunders et al. also report that the hydration of amides by water (D2O) can occur at the CdO and N-H sites via hydrogen (deuterium) bonding and suggest that the bonding of the amide groups of the poly(NIPAM/BA) with D2O is stronger than the bonding that occurs in H2O.21 Kobayashi et al. reported stronger hydration of amides in the case of D2O compared with H2O using infrared spectroscopy.27 A similar observation was made by Shibayama et al. for poly(NIPAM/ BA) macrogel suspended in D2O.28 Murray et al., using HSDSC, observed that the VPTT of poly(NIPAM/BA) microgel increases in D2O by approximately 1.5 K compared with the VPTT in H2O.29 Analysis of Thermodynamic Data. As discussed in the Introduction there are various thermodynamic theories18,19 involving complex events for the swelling of microgels that have been explored. However, there is an absence of a simple thermodynamic theory. Following the discussion already undertaken for the standard reference state, we select the volume occupied by the microgel at 333 K to represent the standard reference state. Accepting this, the following development is based on the assumptions that the volume occupied by the solvent at 333 K is representative of the microgel particles in their most collapsed state and that for each microgel the size is independent of the solvent environment. Using the equilibrium equation written in Figure 2, the following analysis can be undertaken. If ∆Vswell ) VT - V333 represents the change in volume on swelling (at temperature T) and this is assumed to be occupied only by water, consideration of Figure 2 (in terms of the equilibrium between swollen and deswelled microgel) (26) Saunders, B. R.; Cowther, H. M.; Vincent, B. Macromolecules 1997, 30, 482. (27) Kobayashi, M.; Yoshioka, T.; Kozasa, T.; Tashiro, K.; Suzuki, J. I.; Funahashi, S.; Izumi, Y. Macromolecules 1994, 27, 1349. (28) Shibayama, M.; Tanaka, T.; Han, C. C. J. Chem. Phys. 1992, 97, 6829. (29) Murray, M.; Rana, F.; Haq, I.; Cook, J.; Chowdhry, B. A.; Snowden, M. J. J. Chem. Soc., Chem. Commun. 1994, 15, 1803.
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Figure 5. Plot of ln K vs 1/T for stock dispersions of poly(NIPAM/BA) microgels. Plots for the other microgels show similar trends.
suggests that an equilibrium constant K can be expressed as
K ) [H2O]microgel/[H2O]bulk where [H2O]bulk is constant (a (activity) ) 1), and hence K ) [H2O]microgel. Describing the [H2O]bulk parameter as having an activity of 1, in this context, is valid because the PCS data is obtained from dilute microgel dispersions < 0.1 w/w%. Thus, from knowledge of ∆V as a function of temperature (and the appropriate molar volumes of water), it is possible to calculate [H2O]microgel, i.e., K, for each temperature. The van’t Hoff isochore30 relates K to T through a plot of ln K vs 1/T. The slope of such a plot, if linear, is equal to -∆H/R. Figure 5 shows a plot of ln K vs 1/T from which the slope has been determined and the enthalpy (∆H) calculated. Nonlinearity in van’t Hoff plots will arise if (1) there is a change in the mechanism of the process under study and/or (2) the enthalpy of the process(es) is temperature dependent. Figure 5 indicates that at low temperatures (below the VPTT) the plot shows a zero gradient. At higher temperatures (above the VPTT) the plot has a nonzero gradient. The data used to construct the plot therefore clearly falls into two different regimes. The reason, in part, for this is that the molar volumes of bulk water have been used to determine [H2O] microgel ≡ K; this may not be an entirely correct procedure (see below). The first observation that can be made regarding the van’t Hoff plots, for all three microgel systems, is that there are two main regions with the break point occurring, e.g., for the poly(NIPAM/BA) microgel in H 2O at the VPTT of ∼306 K (i.e., 1/T ) 0.00327 K-1). At temperatures below the inflection temperature the microgels are in their most expanded state (good solvent environment) and the concentration of water in the microgels at all temperatures VPPT region, eq 4 holds because linearity, i.e., a nonzero gradient, is observed in the Van’t Hoff plot shown in Figure 5. If eq 3 is written in the following way
∆Gswell ) ∆Hswell - T∆Sswell
(5)
where the symbols have their normal meaning in relation to the swelling of the microgel, then ∆Hswell and ∆Sswell would both be expected to have values of 306 K cannot be regarded as equivalent to bulk water, indicating that using appropriate molar volumes of water to calculate K introduces error. These conclusions are in contrast to the statement made by Daly and Saunders14 and quoted in the Introduction, stating that there is an increase in the entropic contributions with increase in temperatures note, however, the different standard state conventions
adopted. From our simple analysis we suggest that an increase in temperature increases the enthalpic and not entropic contributions. 4. Concluding Remarks The results of the volume analysis undertaken herein suggest that the volumes of each of the microgels examined, at 333 K, is a property of the microgel itself and that it may be independent of the solvent environment. For each of the three microgel systems examined, the final volume reaches the same value irrespective of the solvent system (stock/H2O, freeze-dried/H2O, freeze-dried/D2O) in which they are dispersed. The thermodynamic treatment of the data is justified by taking the volume at 333 K as the limiting size and hence the standard state. The results of the simple, van’t Hoff, thermodynamic analysis have also shown that the low-temperature region is dominated by entropic effects and furthermore indicate that at temperatures 306 K cannot be regarded as equivalent to bulk water. Acknowledgment. We thank one of the reviewers for bringing to our attention that the changes in viscosity for D2O, as a function of temperature, needed to be taken into consideration for the analysis of the PCS data as well as pertinent comments regarding the thermodynamic analysis. LA035765+
