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Thermodynamic Model Fitting of the Calorimetric Output Obtained for Aqueous Solutions of OxyethyleneOxypropylene-Oxyethylene Triblock Copolymers Iain Paterson,† Jonathan Armstrong,‡ Babur Chowdhry,§ and Stephen Leharne*,† School of Environmental Sciences, University of Greenwich, Creek Road, Deptford, London SE8 3BW, U.K., Department of Physiology and Biophysics, School of Medicine, University of Southern California, Los Angeles, California 90033, and School of Chemical and Life Sciences, University of Greenwich, Wellington Street, Woolwich, London SE18 6PF, U.K. Received April 30, 1996. In Final Form: November 22, 1996X A high-sensitivity differential scanning calorimetry (HSDSC) study of aggregation transitions in dilute aqueous solutions of oxyethylene-oxypropylene-oxyethylene (EO-PO-EO) triblock copolymers (poloxamers) is reported. The data have been analyzed using a previously described thermodynamic model (Armstrong, J. K.; et al. J. Chem. Res. 1994, 364) based upon a mass action description of aggregation which has been further elaborated to include the effect of changes in the heat capacity of the initial and final states. As a consequence the model incorporates the underlying changes in the heat capacity of the system, thus obviating the need for baseline fitting and as such provides a useful mechanism for the analysis of the data. Model-fitting results are presented for aqueous solutions of various concentrations of the poloxamers P237 (EO62PO39EO62) and P333 (EO19PO56EO19). In addition model-derived results are presented for a number of other poloxamer solutions. The thermodynamic data obtained are further used to produce phase diagrams of the aggregation process as a function of concentration and temperature. Furthermore the calorimetric output is also used to compute critical micelle concentration and critical micelle temperature data. Data obtained for P333 complement spectroscopic data reported in the literature. The thermodynamic data obtained show a number of important trends. The heat capacity change values obtained are invariably negative, pointing toward the loss of solvating water structure on aggregation. Two measures of enthalpy are computed: the calorimetric enthalpysobtained from integration of the calorimetric outputsand the van’t Hoff enthalpysobtained from the change of the equilibrium constant characterizing aggregation with temperature. Both these measure of enthalpy are positive. The computed entropy changes are likewise positive, indicating that aggregation in these systems is an entropy-driven process. The van’t Hoff enthalpy/calorimetric enthalpy ratio further indicates the aggregation process to be cooperative. The temperature at which aggregation is half completed (T1/2) varies with copolymer concentration. The corresponding change in the van’t Hoff enthalpy results from the temperature dependence of the enthalpy. Data are also obtained for aqueous solutions of a further 12 EO-PO-EO block copolymers. Multiple linear regression analysis of the van’t Hoff enthalpy normalized to 298.15 K as a function of PO and EO block length points to the importance of the PO block in determining the size of the van’t Hoff enthalpy. Finally an enthalpy-entropy compensation plot indicates that the same solvent-solute interactions are responsible for the transitions in all the samples regardless of the copolymer composition and concentration.
Introduction In a number of previous publications we have reported results obtained from high-sensitivity differential scanning calorimetry (HSDSC) investigations of dilute poloxamer solutions.1-4 The poloxamers are a family of ABA block copolymers produced from oxyethylene (the A block) and oxypropylene (the B block). Two useful and comprehensive reviews of recent research dealing with these compounds have recently been published.23,33 A number of the reported investigations of these systems have shown that molecular aggregation or micellization occurs as the * Author for all correspondence. Telephone: +44 (0)181 331 8200. Fax: +44 (0)181 331 8205. E-mail:
[email protected]. † University of Greenwich, Deptford. ‡ University of Southern California, Los Angeles. § University of Greenwich, Woolrich. X Abstract published in Advance ACS Abstracts, March 15, 1997. (1) Armstrong, J. A.; Chowdhry, B. Z; Beezer, A. E.; Mitchell, J. C.; Leharne, S. J. Chem. Res. 1994, 364. (2) Mitchard, N. M.; Beezer, A. E.; Rees, N. H.; Mitchell, J. C.; Leharne, S.; Chowdhry, B. Z.; Buckton, G. J. Chem. Soc., Chem. Commun. 1990, 900. (3) Beezer, A. E.; Mitchard, N. M.; Mitchell, J. C.; Armstrong, J. K.; Chowdhry, B. Z.; Leharne, S.; Buckton, G. J. Chem. Res. 1992, 236. (4) Mitchard, N. M.; Beezer, A. E.; Mitchell, J. C.; Armstrong, J. K.; Chowdhry, B. Z.; Leharne, S.; Buckton, G. J. Phys. Chem. 1992, 96, 9507.
S0743-7463(96)00432-5 CCC: $14.00
result of increasing temperature.5-9 In our investigations1-4 we have shown that the aggregation process is sufficiently endothermic as to be followed by HSDSC. A more recent aspect of this work has been an attempt to capture as much information as is scientifically feasible and defensible from the HSDSC output. The verification that the aggregation process, as observed in the scanning calorimeter, is not subject to any kinetic limitation has provided an opportunity to attempt to fit the calorimetric output to a thermodynamic model of aggregation.1 From this model-fitting process estimates for the aggregation number and the thermodynamic parameters ∆HvH (the van’t Hoff enthalpy derived from the change in the equilibrium constant for aggregation with temperature) and T1/2 (the temperature at which half the total concentration of copolymer in solution has aggregated) have been made.1 As is common in DSC analysis the endothermic peaks had a suitable baseline subtracted and the area under the peak was then evaluated by integration. The area provides a measure of ∆Hcal (the calorimetric en(5) Zhou, Z.; Chu, B. Macromolecules 1988, 21, 2548. (6) Wanka, G.; Hoffmann, H.; Ulbricht, W. Colloid Polym. Sci. 1990, 268, 101. (7) Yu, G.-E.; Deng, Y.; Dalton, S.; Wang, Q.-G.; Attwood, D.; Price, C.; Booth, C. J. Chem. Soc., Faraday Trans. 1992, 88, 2537. (8) Alexandridis, P.; Holzwarth, J.; Hatton, T. Macromolecules 1994, 27, 2414. (9) Hecht, E.; Hoffmann, H. Langmuir 1994, 10, 86.
© 1997 American Chemical Society
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scanning calorimetry is that the endothermic peaks obtained are asymmetric with a steep leading edge. It can also be shown, however, that changes in ∆HvH with rising temperature will also produce a specific amount of peak asymmetry. It would therefore appear appropriate to devise some data analysis scheme in which optimum values for all the thermodynamic parameters mentioned are obtained and in which the optimum baseline is selected at the same time. A model-fitting software package, Scientist (Micromath Scientific Software, Salt Lake City, UT), which is capable of dealing with complex models has been used to a fit a thermodynamic model of aggregation to the HSDSC data. The purpose of this paper is to outline the construction of the mathematical model that was used for analyzing the HSDSC output for thermal aggregation. Data Analysis For an HSDSC system where the reference and sample cells are matched, the signal obtained on scanning is described by the following mathematical expression:
φCp,xs )
d (R(∆Hcal(T1/2) + ∆Cp(T - T1/2))) dT
(2)
where φCp,xs is the apparent excess heat capacity, ∆Hcal(T1/2) is the value for the calorimetric enthalpy at T1/2, and ∆Cp is the change in heat capacity between the inital and final states of the system. Differentiation of eq 2 provides the following formula for φCp,xs:
dR + R∆Cp φCp,xs ) ((∆Hcal(T1/2) + ∆Cp(T - T1/2))) dT
Figure 1. HSDSC output (a, top) for different concentrations of aqueous solutions of P237 and (b, bottom) for an up-scan and a down-scan of a 5 g dm-3 solution of P333.
thalpy). The ratio ∆HvH/∆Hcal provides a measure of the cooperativity of the aggregation process. A major limitation of the model-fitting process is the assumption that the heat capacity change between the initial and final states of the aggregation process is zero. Implicit in this assumption is the view that the enthalpy change for the process is independent of temperature. An inspection of the HSDSC output (see Figure 1), for aqueous poloxamer solutions however, reveals that the initial heat capacity of the system is larger than the final state heat capacity. The heat capacity change on aggregation, ∆Cp, for the process is therefore nonzero and negative. This has a number of implications for data analysis and model fitting, which are as follows. The underlying baseline for the calorimetric output contains a contribution from the changing composition of the system as the temperature is raised. This contribution can be formulated as
baseline ) RCp(2) + (1 - R)Cp(1)
(1)
where R is the extent of conversion to aggregates, Cp(2) is the molar heat capacity of the aggregate solution, and Cp(1) is the molar heat capacity of the surfactant monomers in solution. Any attempt at data analysis which seeks to identify and subtract an assumed baseline will result in some error if the selected baseline is inappropriate. Both the measured enthalpy values are temperature dependent, and it is possible that the change in both ∆Hcal and ∆HvH may be significant in the course of thermal aggregation. One characteristic of aggregation processes observed by
(3)
Values for R and the derivative dR/dT may be obtained by initially using the following mass action description of aggregation and mass balance:
K)
[Xn] [X]n
Ctotal ) n[Xn] + [X]
(4) (5)
Since R is defined as the extent of aggregation, it is reasonable to define it arithmetically as the fraction of monomer that is incorporated into micellar aggregates:
R)
n[Xn] Ctotal
(6)
It is readily shown using eqs 4-6, that the equilibrium constant is then equal to
K)
R n-1 n(1 - R)nCtotal
(7)
The temperature dependence of K is given by the van’t Hoff isochore, which is modified to include the change in heat capacity of the system so that
d ln K ∆HvH + β∆Cp(T - T1/2) ) dT RT2
(8)
In eq 8 ∆HvH is the van’t Hoff enthalpy and β is the ratio of the van’t Hoff enthalpy to the calorimetric enthalpy and is introduced such that the ratio of the heat capacity changes is the same as the ratio of the corresponding enthalpy changes. If eq 8 is integrated between the limits of T and T1/2, the result may then be combined with eq 7 to provide the following expression for R:
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ln(R) + (n - 1) ln(0.5) - n ln(1 - R) )
(
)
(( )
)
β∆Cp T1/2 ∆HvH 1 1 T -1 + ln + R T1/2 T R T1/2 T
(9)
It can be further shown that dR/dT is given by
(
)
1 dR ∆HvH + β∆Cp(T - T1/2) ) n dT 1 RT2 + R 1-R
(10)
Solving eqs 9 and 10 and substituting the results into eq 3 permits the evaluation of φCp,xs as a function of temperature. The transformation of the HSDSC outputswhich is a data set of power as a function of changing temperaturesto a data set of apparent excess heat capacity vs temperature and into a form that is readily utilized by the software is performed using the following steps. Firstly, the DA2 software was used to transform the power data to φCp,xs using the following equation:
φCp,xs )
4.18P σm
(11)
where P is power (mcal s-1), σ is the scan rate (K h-1), m is the amount of sample in the sample cell (mmol), and 4.18 is the constant used to convert calories to joules. Secondly, the DA2 software automatically uses an interpolation function to provide a data set where the temperature intervals are equally spaced. The software was further used to ‘level’ the initial portion of HSDSC output and appropriately alter the rest of the φCp,xs data. This had the effect of setting the initial heat capacity of the system, given in eq 1 [Cp(1)] to zero and clearly changing Cp(2). It does not, however, alter the difference between the two, ∆Cp. Finally the data were then used in the model-fitting process using the Scientist software. Scientist is capable of fitting experimental data to systems of equations using a hybrid minimization technique based upon a modification of the Powell algorithm. The software uses a combination of steepest descent with Gauss-Newton.30 It also features a powerful and robust root finder which is important for solving eq 9 for R. The modelfitting package will also provide estimates of the uncertainty in the optimized parameters, as well as provide values for the coefficient of determination (a measure of how much of the variance is explained by the model) and for the Model Selection Criterion (a measure of how much information is contained within the model). For all the model fitting undertaken in this study the coefficient of determination was better than 0.999, indicating that 99.9% of the variance is explained by the model, and the Model Selection Criterion was of the order of 6-8, indicating a high information content.
Experimental Details The poloxamer samples (obtained from ICI Chemical and Polymers, Cleveland, U.K.) were used as received. Surfactant solutions of several concentrations were prepared at room temperature using doubly distilled deionized water. The solutions were then injected into the sample cell of the MC2 scanning microcalorimeter (obtained from Microcal, Amherst, MA). Once in the instrument, the samples were kept under nitrogen at a pressure of 1 atm to suppress bubble formation during scanning. The instrument is capable of scanning from -10 to 105 °C. The instrument is extremely sensitive, which means that low scan rates may be used. In this study it was common for the solutions to be scanned at a rate of 1 K min-1. However the solutions were also scanned at other scan rates (0.16 and 0.5 K min-1). The lack of scan rate dependence of the signal demonstrates that the process is not subject to any kinetic limitation and may therefore be analyzed using thermodynamic formulations.
Results and Discussion HSDSC Output. Typical scanning calorimetric output is shown in Figure 1a. The plot exhibits data obtained for P237 solutions of several different concentrations. The peaks obtained are broad and asymmetric and clearly show the existence of a long post-transitional tail of lower heat
capacity than the pre-transitional portion of the calorimetric traces. The breadth of transition has been ascribed to the polydispersity of the samples.6 However more recently it has been suggested that the gradual nature of polyoxypropylene desolvation is possibly more important.23 The asymmetryscharacterized by a sharp leading edgesis indicative of an aggregation process. The steepness of this leading edge should provide qualitative clues as to the aggregation number for a monodisperse system. For very large aggregation numbers (>50) the calorimetric output will tend to be perpendicular to the baseline. This is also the expected output for a phase separation process.25 These traces are not as steep and reflect more modest aggregation numbers. It is, of course, possible that the polydispersity of the systems may alter the asymmetry of the signal. It is important however to note that Yu et al.,7 have established that removal of the diblock componentsswhich will make a contribution to polydispersitysfrom these commercial products has a negligible effect on the overall shape of the traces. The only effect is to change by 2 °C the onset of the aggregation process. One question that does however arise concerns the details by which these polydisperse systems aggregate. It must be anticipated that the range of molecular dimensions, especially of oxypropylene block sizes, gives rise to a range of hydrophobicities. It might therefore be expected that the more hydrophobic molecules would tend to aggregate first. However a model in which these molecules alone aggregate would demand a kinetically slow step as molecules of appropriate dimensions ‘seek’ similar molecules prior to aggregation. The absence of any kinetic limitations in the signalssas demonstrated by the lack of scan rate dependencesprecludes this. Figure 1b also demonstrates that the demicellization process as observed by down-scanning is the mirror image of the up-scan, thus suggesting that the one process is merely the reverse of the other. In the disaggregation process the oxypropylene blocks are rehydrated to subsequently form near neighborssthe reverse must then entail the dehydration and consequent aggregation of near neighbors. This therefore hints at a similar tendency for the molecules to aggregate regardless of their size. The large endothermic component of the aggregation transition arises from the loss of water of solvation, which has an icelike structure26 and which, as a consequence, carries with it a large negative entropy penalty. It might be expected, therefore, that any change in propylene oxide block size produces a change in both the enthalpy and entropy of solvation. Since the two changes make opposing contributions to the free energy of solvation, it might be anticipated that this remains broadly similar for the disperse range of molecules. This is not to suggest however that hydrophobicity is not important. It seems most likely that the more hydrophobic unmicellized molecules drive the aggregation process at any particular temperature; but it is strongly hinted at that they combine with near neighbors regardless of their molecular sizes. This implies that the micellar molecular size distribution may be different from that of unmicellized surfactant. Gao and Eisenberg34 have produced model calculations of the molecular weight distributions for the single chain and micellar fractions of two polydisperse block copolymer systems: poly(styrene-b-isoprene) in n-hexadecane and poly(styrene-b-sodium acrylate) in water. In both cases the distributions are Gaussian, the most notable difference being that the modal degree of polymerization for the single-chain fraction is lower than the corresponding modal value for the micellar fraction, though for the poly(styrene-b-sodium acrylate)/water system this difference is quite small. It therefore seems likely that, despite the polydispersity of the individual
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Table 1. Thermodynamic Parameters Obtained from the Model-Fitting Process for Various Aqueous Solution Concentrations of P237a
∆Hcal (kJ mol-1) ∆HvH (kJ mol-1) n T1/2 (K) ∆Cp (kJ mol-1 K-1) a
[P237] ) 1
[P237] ) 5
[P237] ) 7.5
[P237] ) 10
[P237] ) 15
[P237] ) 20
[P237] ) 25
[P237] ) 34.2
[P237] ) 50
281 (2.4) 514 (33.6) 14.0 (3.91) 332.5 (2.13) -18.2 (2.80)
268 (2.3) 848 (21.5) 8.8 (0.39) 319.6 (0.10) -11.3 (0.39)
266 (2.8) 834 (21.2) 9.0 (0.45) 318.7 (0.13) -13.4 (0.51)
266 (1.5) 877 (15.7) 8.7 (0.27) 317.2 (0.06) -11.8 (0.27)
278 (2.4) 999 (29.7) 11.7 (0.61) 316.9 (0.12) -18.4 (0.67)
272 (1.7) 960 (21.9) 10.8 (0.43) 315.9 (0.09) -18.0 (0.49)
253 (1.7) 937 (25.9) 12.0 (0.57) 316.4 (0.11) -20.9 (0.66)
290 (2.1) 936 (33.1) 15.0 (1.00) 316.5 (0.19) -29.2 (1.20)
265 (1.9) 966 (26.0) 11.7 (0.56) 313.2 (0.11) -23.0 (0.73)
Standard deviations are shown in parentheses. [P237] in g dm-3.
Figure 2. Indicative output of the model-fitting process. The calorimetric data were obtained for a 5 g dm-3 solution of P237. The solid circles are data points, and the solid line is the fitted curve. The dotted line represents the derived baseline.
poloxamer systems, thermal aggregation may be treated as a fairly uniform process. As for the thermodynamics of these systems it is clear that the free energy of the system along with its enthalpic content can be specified by a linear combination of partial molar free energies and enthalpies. Thus regardless of the polydisperse nature of the system the enthalpy and free energy of the system can be specified if the exact composition is known. The exact compositions of these commercial products are in fact unknown. This does not, however, prevent calorimetry from reporting on the differences between the inital and final states of the system. The results obtained however must be considered to be average values for the system. The phase behavior of these block copolymeric compounds is quite complex.23 For instance in dilute solution it has been concluded that as the temperature increases, the aggregation number also increases. However other processes such as rod formation as well as clouding will also occur.23 These processes may have an important impact upon the post-transitional tail in terms of its heat capacity value. It is possible that the above-mentioned processes may be responsible for the continuing decrease in the tail’s heat capacity. Scanning calorimetry is unable to give any precise details about this. In the analyses presented below the tail is seen as resulting from a residual unimer which is slowly incorporated into the micellar aggregates. There may, however, be other interpretations. Analysis of P237 Data. Table 1 contains thermodynamic data obtained from the model-fitting process for the HSDSC data of various aqueous solutions of P237. The results of model fitting must always be treated with caution. However the consistency of the data obtained and the good correspondence between fitted curves and datasa typical example is shown in Figure 2swould indicate that the model supplies a useful description of the thermodynamics of the system.
Several points need to be made about the information in Figure 2 and Table 1. The derived baseline shows that the thermal aggregation process takes a long time to go to completion, longer possibly than might be supposed by visual inspection of the thermal trace. This has important repercussions for baseline-fitting procedures which implicitly assume that the pre- and post-transitional portions of the peak can be delineated. In all our previous work it was assumed that the transition was complete at some point on the high-temperature tail. An analysis of the HSDSC tracesswhich provided the data for Table 1swas also carried out using the methods outlined in our previous publications. The results show that methods based on baseline fitting underestimate ∆Hcal, n, and T1/2 and provide roughly comparable values for ∆HvH. The differences in the values obtained for ∆HvH and ∆Hcal point to the aggregation process being a cooperative event. The ratio of ∆HvH to ∆Hcal (or as previously defined, β) provides an indication of the size of this cooperative unit. This is explained in the following way. For the calorimetric enthalpy the mole in the unit definition is identified as a mole of unimeric chains. From the equations outlined in the Data Analysis it can be shown that ∆HvH is given by the following formula:
∆HvH )
2 2(n + 1)Cp,1/2RT1/2 ∆Hcal
(12)
where Cp,1/2 is the apparent excess heat capacity at T1/2 the temperature at which the extension of aggregation, R, is equal to 0.5. Examination of the units used to described each of the quantities in the above expression reveals that the mole referred to in the van’t Hoff enthalpy units is defined by the universal gas constant, since the mole defined in the Cp,1/2 and ∆Hcal units cancels. From Table 1 it can be shown that the cooperative unit varies between three and four molecules. This substantiates our previous observations that the aggregation process involves the aggregation of aggregates.1,4 Further evidence for premicellization aggregation has been provide by Hecht and Hoffman using T-jump experiments.32 Light-scattering studies of dilute poloxamer solutions also confirm the view that, prior to the formation (which we follow by HSDSC) of the small aggregates (which are normally described as micelles), several extremely large aggregates are formed.5,10-12 Their formation is attributed to the presence of diblock impurities in the poloxamer compound. As the temperature rises, it has been shown that these aggregates break down to form smaller micellar aggregates.10,11 Since HSDSC can only measure ensemble averages, it is possible that our preaggregation aggregate size of between three and four represents the average of relatively few large aggregates and a large number of unassociated polymer molecules. (10) Brown, W.; Schille´n, K.; Almgren, M.; Hvidt, S.; Bahadur, P. J. Phys. Chem. 1991, 95, 1850. (11) Brown, W.; Schille´n, K.; Hvidt, S. J. Phys. Chem. 1992, 96, 6038. (12) Bahadur, P.; Pandya, K. Langmuir 1992, 8, 2666.
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Table 2. Effect of Inorganic Salt Type and Concentration on the Thermodynamic Parameters Obtained from the Model-Fitting Process for Various Aqueous Solutions of P237a ∆Hcal (kJ mol-1) ∆HvH (kJ mol-1) n T1/2 (K) ∆Cp (kJ mol-1 K-1) a
0.1 M PO43-
0.15 M PO43-
0.1 M ClO4-
0.2 M ClO4-
269 (1.3) 597 (7.8) 8.6 (0.23) 318.1 (0.09) -10.9 (0.17)
207 (4.0) 492 (25.6) 4.7 (0.49) 318.1 (0.25) -8.6 (0.56)
277 (1.7) 777 (14.3) 11.7 (0.44) 322.3 (0.13) -17.0 (0.41)
263 (2.8) 833 (28.1) 12.0 (0.77) 322.4 (0.21) -16.2 (0.68)
All P237 concentrations are 5 g dm-3. Standard deviations are shown in parentheses.
It is important to note that a number of studies of the thermal aggregation process in poloxamers suggestsusing a variety of techniquessthat the aggregation number becomes larger as a result of increasing temperature.6,13-16 Yet in our work the data are adequately described by assuming that the aggregation number is constant. It must therefore be supposed that the subsequent increase in aggregate size is very nearly an athermal event which is consequently unobservable by scanning calorimetry. The endothermic nature of the micellization transition arises from the desolvation of the POP block. Water of solvation, which adopts an icelike structure around the blocks, is lost as the temperature is raised.26 The proposed simultaneous changes in conformation of the POP block from a polar to a nonpolar configuration explain the reduction in polymer water interaction and the subsequent aggregation.14,17,18 This process occurs over the entire temperature range of the endothermic transition. Clearly the cooperative nature of the initial micelle-forming process must dominate the enthalpy changes observed in these systems, and either this masks out the further addition of surfactant monomer to the micelles or else the molecules that are added have already been desolvated in the initial micelle-forming process. The ∆Cp values in Table 1 further reinforce this point about water-polymer interactions. The ∆Cp values show that the aggregation process is accompanied by a negative change in heat capacity. In protein biochemistry it is commonly observed that the thermal denaturation of a protein is accompanied by an increase in heat capacity between the initial and final states of the system. This is customarily attributed to the exposure of hydrophobic residues to aqueous solvent along the polypeptide chain.19 A decrease in heat capacity, consequently, indicates a decrease in the exposure of hydrophobic residues to water, which would certainly be expected given the proposed aggregation mechanism. More strikingly the existence of the negative heat capacity change of aggregation indicates the parallel existence of an upper transition in which the aggregates dissociate. The effects of inorganic salts upon the aggregation transition of P237 are shown in Table 2. Perchlorate ions and phosphate ions occupy different positions in the Hoffmeister series and therefore have differing effects in terms of ‘salting-in’ and ‘salting-out’. It is common to explain these effects in terms of effects on water structure. The data indicate that perchlorate ions increase the temperature range over which aggregation occurs, which implies that perchlorate tends to salt-in the polymer. On the other hand phosphate ions reduce the aggregation temperatures, implying that they salt-out the polymer. In both cases the counter-cations were sodium ions and the pH of the phosphate solution was 7.2. (13) Rassing, J.; Attwood, D. Int. J. Pharm. 1983, 13, 47. (14) Linse, P.; Malmsten, M. Macromolecules 1992, 25, 5434. (15) Mortensen, K. J. Phys. 1 (Suppl.) 1993, 3, 157. (16) Glatter, O.; Scherf, G.; Schille´n, K.; Brown, W. Macromolecules 1994, 27, 6046. (17) Linse, P. Macromolecules 1993, 26, 4437. (18) Linse, P. J. Phys. Chem. 1993, 97, 13896. (19) Privalov, P. Crit. Rev. Biochem. Mol. Biol. 1992, 25, 281. (20) Clint, J. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1327.
Figure 3. Plots of ∆HvH vs T1/2 for (a, top) P237 and (b, bottom) P333.
The heat capacity change accompanying the aggregation transition denotes the temperature dependence of the enthalpy of aggregation, and it is this temperature dependence which should provide a reasonable explanation for the changes in ∆HvH with concentration outlined in Table 1. The dependence of T1/2 on concentration is readily explainable in thermodynamic terms. The mass action description of aggregation suggests that any increase in surfactant concentration must give rise to an increase in micelle concentration. Since micelle formation is an endothermic process, increasing monomer concentration has the effect of reducing temperature in an adiabatic system. It is therefore observed that increasing concentration reduces the temperature at which micellization occurs. Figure 3 provides a graphical indication of the functional relationship between ∆HvH and T1/2. The graph contains data obtained from Table 1 and the perchlorate data of Table 2. The plot is a reasonable straight line (R2 ) 0.9983) and suggests that ∆HvH changes because T1/2 is changing. The gradient of the slope, (d(∆HvH))/dT, should provide an estimate for ∆Cp. The computed value for ∆Cp from linear regression analysis is -25.5 kJ mol-1 K-1. This value is quite close to, though slightly larger than, -17.9 (σn-1 ) 5.2) kJ mol-1 K-1, the mean of the model-derived
2224 Langmuir, Vol. 13, No. 8, 1997
Figure 4. Plots showing the impact of changing copolymer concentration on the fraction of copolymer in micellar form at any particular temperature.
parameters. It is interesting to note that the perchlorate data actually link neatly with the concentration data and suggest that the main effect of perchlorate is to increase water activity and as a consequence reduce polymer activity. This reduced polymer activity would therefore give rise to an increase in the temperature range over which aggregation occurs. The phosphate data on the other hand do not give such easily explained results. It should be noted that the enthalpy of phosphate dissociation is very low (5.12 kJ mol-1) 31 and is therefore unlikely to account for the observations. The aggregation numbers obtained for P237 range from 8.7 to 15 (xj ) 11.3, σn-1 ) 2.2). These values are comparable to the value of 14 obtained by light scattering.11 However, given the cooperativity of the aggregation process, the average size of the cooperative unit being xj ) 3.2, σn-1 ) 0.57, it is possible that the number of molecules present in an aggregate is approximately 36. Phase Behavior of P237. The polydispersity of the samples must have important repercussions for our understanding of the micellization process and the exact meaning and value of the cmc.21 As the temperature of the aqueous surfactant system is raised, micellization will occur. The temperature at which this occurs is called the critical micelle temperature (cmt) and the surfactant concentration at this temperature is the cmc. The micelles formed are initially composed of the more hydrophobic materials, and as the temperature increases, the more hydrophilic molecules become gradually incorporated into the micelles. This has important implications for the composition distribution of the unmicellized surfactant monomer. The concentration of surfactant at the temperature at which micellization first occurs will be the cmc. However after this point the compositional differences between the solution and the micelles implies that the monomer concentration cannot be regarded as being the same as the cmc. It is possible, however, to investigate further the effects of polydispersity upon the micellization process. In Figure 4 a plot of the fraction of P237 in aggregated form as a function of temperature is shown for various total copolymer concentrations. The most noteworthy feature of the plot is the concentration dependence of the fraction of copolymer in the aggregated form. At any particular temperature, as the total copolymer concentration in(21) Linse, P. Macromolecules 1994, 27, 6404. (22) Karlstro¨m, G. J. Phys. Chem. 1985, 89, 4962. (23) Almgren, M.; Brown, W.; Hvidt, S. Colloid Polym. Sci. 1995, 273, 2.
Paterson et al.
creases, the aggregated fraction increases. Linse21 has shown, by modeling, that such a result would be anticipated for these systems and is related to the polydispersity of the copolymer. Analysis of P333 Data. Table 3 comprises modelfitting data obtained for P333. Comparison with the data acquired for P237 is invited. A number of key differences emerge. The range of ∆Hcal values obtained for P333 is larger than that observed for P237. This possibly reflects the greater size of the oxypropylene block in P333. It will be recalled that desolvation of this block and an accompanying conformational change are an integral part of the aggregation process. The range of ∆HvH values observed for P333 is smaller than that obtained for P237. The lower ∆HvH values for P333 probably reflect the smaller values for n, the aggregation number. It is interesting to note the differences in the aggregation numbers obtained for P333 and P237. The smaller values for P333 possibly mirror the greater thermodynamic stability associated with its aggregates (or more properly the greater free energy loss associated with aggregation), which must in some way reflect the greater hydrophobicity associated with the longer oxypropylene blocks. As with P237 the changes in ∆HvH seem to be functionally connected to the changes in T1/2, as shown in Figure 3b (r2 for the plot is 0.928). As previously, we would anticipate that the value of the gradient of the plot is equal to the heat capacity change for the aggregation process. In fact the gradient of -39.4 kJ mol-1 K-1 is much greater than the mean heat capacity value obtained from the model-fitting process (mean value is -8.75 kJ mol-1 K-1 , σn-1 ) 2.57). No explanation for this discrepancy can be offered. It is worthwhile at this stage to comment on the calorimetric enthalpies obtained in this study. By and large they do not appear to bear the same simple relationship with temperature as shown by the van’t Hoff enthalpy values. This may be due to changes in the cooperative unit as a function of temperature. Critical Micelle Concentration and Critical Micelle Temperature Data. It is possible to obtain both critical micelle temperature (cmt) and critical micelle concentration (cmc) data from the plots of φCp,xs vs temperature. The cmc is defined as the concentration at which micelles first appear in aqueous solution. The cmt is necessarily the temperature at which this occurs. Yu et al.7 have surmised that micelles first appear at that point at which the HSDSC signal first departs from the baseline, and as a consequence this temperature can be regarded as the cmt. On the other hand Hecht et al.24 have defined the cmt as the temperature of the intersection of the tangent drawn to the inflection point of the ascending DSC signal and the baseline. Alexandridis et al.8 have used a spectroscopic probe (1,6-diphenyl-1,3,5-hexatriene) to follow the micellization process. As the thermally induced micellization process develops, the absorbance of the probe increases because of the change in environment. A sigmoidal plot of the temperature dependence of absorbance is produced from which the temperature is identified at which a tangent drawn to the ascending linear portion of the plot intersects the extrapolated absorbance in the absence of aggregation. This latter method ought to be comparable with that of Hecht et al.24 For the sake of comparability we have measured the cmt in the same way as Hecht et al.24 It is worth noting that since the (24) Hecht, E.; Hoffmann, H. Colloids Surf. 1995, 96, 181. (25) Armstrong, J.; Chowdhry, B.; O’Brien, R.; Beezer, A.; Mitchell, J.; Leharne, S. J. Phys. Chem. 1995, 99, 4590. (26) Kjellander, R.; Florin, E. J. Chem. Soc., Faraday Trans. 1981, 77, 2053.
EO-PO-EO Triblock Copolymers
Langmuir, Vol. 13, No. 8, 1997 2225
Table 3. Thermodynamic Parameters Obtained from the Model-Fitting Process for Various Aqueous Solution Concentrations of P333a ∆Hcal (kJ mol-1) ∆HvH (kJ mol-1) n T1/2 (K) ∆Cp (kJ mol-1 K-1) a
[P333] ) 0.5
[P333] ) 1
[P333] ) 2.5
[P333] ) 5
[P333] ) 10
[P333] ) 15
[P333] ) 19.98
[P333] ) 20
423 636 3.0 303.4 -8.9
367 610 2.6 302.4 -7.0
366 652 2.5 300.8 -6.9
377 735 2.7 299.4 -4.0
379 791 3.0 298.5 -10.4
384 822 3.1 297.9 -10.8
389 844 3.2 297.3 -11.1
377 841 3.1 297.1 -10.9
Standard deviations are shown in parentheses. [P33] in g dm-3.
Table 4. Critical Micelle Concentration and Critical Micelle Temperature Information Obtained for P237 and P333 cmc (g dm-3) cmt (K) cmc (g dm-3) cmt (K)
1.00 313.9 0.50 297.3
5.00 310.1 1.00 296.5
7.50 309.2
P237 10.00 308.1
2.50 295.5
15.00 307.1
P333 5.00 294.5
20.00 307.5 10.00 293.6
25.00 306.3 15.00 293
34.22 305.3 19.98 292.7
50.00 303.8 20.00 292.8
ploying the adjusted data, a multiple linear regression analysis was performed using the data analysis tools of the spreadsheet Excel, and the following regression equation was obtained:
∆HvH ) -170.8 + 0.275 × POP + 0.006 × POE (13) where POP is the molecular mass of the polyoxypropylene block and POE the molecular mass of the polyoxyethylene blocks. The expression accounts for 57% of the variance (r ) 0.775). It is clear that the contribution made by POP to ∆HvH is nearly 50 times greater than that of POE. It is possible to calculate the free energy change of aggregation. If we take eq 7 and substitute the condition R ) 0.5 and combine this with the equation27
∆G ) -RT ln(K) Figure 5. Critical micelle concentration as a function of critical micelle temperature for (9) P237 and (b) P333. Data were obtained from the HSDSC output as outlined in the text. For comparison cmt (2) and cmc (1) data from ref 8 are shown.
amount of surfactant incorporated in micelles at this point is negligible, the cmc at this temperature must be equal to the initial concentration of surfactant in solution. Table 4 contains cmc and cmt data obtained for P237 and P333. The data obtained in this investigation for P333 are plotted with the data of Alexandridis et al.8 and are shown in Figure 5. It is clear that the two sets of data coincide with each other very neatly and point to the fact that both techniques provide comparable estimates for the cmt and cmc and that more importantly the process under investigation by HSDSC is the micellization process. Of interest is the fact that P333 will tend to micellize at ambient temperatures whilst even at high concentrations P237 is unable to micellize at similar temperatures. It will, however, micellize at higher temperatures. Analysis of the Other Poloxamer Solution Data. The data obtained for some 12 other poloxamer solutions are displayed in Table 5. It would be useful to examine what kind of relationship may draw the entire data set together. Indeed it would be more useful if the data presented in Table 5 can be drawn together with the data in Tables 1 and 3. To do this, the relationship between the amount of POP and POE and ∆HvH is explored. However to do this systematically, it is necessary to express all the ∆HvH values at some common reference temperature. In this case the reference temperature chosen is 298.15 and the values were adjusted using heat capacity data obtained from the model-fitting procedure.27 Em-
(14)
we obtain the following expression for ∆G: n-1 ∆G ) -RT1/2 ln(n(0.5n-1)Ctotal )
(15)
The values obtained for ∆G are shown in Table 6. From these values and the van’t Hoff enthalpy data it is possible to calculate the entropy changes, ∆S. These too are displayed in Table 6. The free energy changes are all negative, indicating that at T1/2 the aggregation process is spontaneous. Since the enthalpy values for the aggregation process are positive, it may be concluded that it is the entropy gain that is responsible for driving the aggregation process forward. A plot of entropy changes against the enthalpy changes is shown in Figure 6. The linearity of the plot (r2 ) 0.997) indicates that for the aggregation process changes in enthalpy are compensated by changes in entropy. The compensation temperatureswhich characterizes solvent-solute interactions28 sobtained is 281.9 K and is similar to that reported for aqueous solutions of polyoxypropylene oligomers of 275 K.29 Concluding Remarks A method has been presented which is capable of fitting calorimetric data obtained for surfactant aggregation in (27) Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press: Oxford, 1994. (28) Bedo¨, Z. S.; Berecez, E.; Lakatos, I. Colloid Polym. Sci. 1992, 270, 799. (29) Armstrong, J.; Chowdhry, B.; O’Brien, R.; Beezer, A.; Mitchell, J.; Leharne, S. J. Phys. Chem. 1995, 99, 4590. (30) Scientist Manual, Micromath, Salt Lake City, UT, 1995. (31) Fukada, H.; Takahashi, K. University of Osaka, personal communication. (32) Hecht, E.; Hoffmann, H. Colloids Surf. 1995, 96, 181. (33) Alexandridis, P.; Hatton, T. A. Colloids Surf. 1995, 96, 1-46. (34) Gao, Z.; Eisenberg, A. Macromolecules 1993, 26, 7353.
2226 Langmuir, Vol. 13, No. 8, 1997
Paterson et al.
Table 5. Thermodynamic Data Obtained from the Model-Fitting Process for Various Poloxamer Solutions of Concentration 5 g dm-3 a P122
P123
P124
P182
P184
P188
∆Hcal (kJ mol-1) ∆HvH (kJ mol-1) n T1/2 (K) ∆Cp (kJ mol-1 K-1)
144 (1.1) 161 (1.3) 2.7 (0.09) 336.4 (0.25) -2.1 (0.03)
153 (2.1) 303 (5.5) 8.5 (0.69) 340.6 (0.92) -5.6 (0.21)
170 (2.1) 422 (10.9) 5.0 (0.29) 319.9 (0.19) -5.7 (0.20)
215 (1.3) 433 (7.9) 5.2 (0.18) 317.6 (0.09) -6.6 (0.14)
222 (2.9) 423 (10.7) 4.7 (0.27) 319.5 (0.19) -5.4 (0.20)
158 (2.8) 227 (6.1) 1.8 (0.10) 332.7 (0.02) 0.8 (0.11)
P215
P217
P234
P235
P335
P407
∆Hcal (kJ mol-1) ∆HvH (kJ mol-1) n T1/2 (K) ∆Cp (kJ mol-1 K-1)
227 (1.5) 415 (4.8) 4.7 (0.14) 320.5 (0.11) -7.7 (0.13)
162 (1.5) 357 (5.7) 4.4 (0.17) 329.5 (0.17) -3.7 (0.11)
255 (1.7) 877 (10.5) 6.5 (0.17) 309.5 (0.05) -12.4 (0.35)
239 (2.1) 562 (10.6) 4.0 (0.15) 312.3 (0.07) -6.1 (0.24)
354 (1.5) 686 (9.7) 2.7 (0.07) 302.0 (0.02) -7.3 (0.17)
199 (1.5) 980 (17.3) 4.3 (0.14) 301.9 (0.03) -10.6 (0.57)
a
Standard deviations are shown in parentheses.
Table 6. Composition and Thermodynamic Data for All Systems Investigated ∆G (kJ ∆Hcal (kJ mol-1) mol-1)
∆S (kJ mol-1 K-1)
poloxamer
PO
EO
concn (g dm-3)
P122 P123 P124 P182 P184 P188 P215 P217 P234 P235 P335 P407
1200 1200 1200 1750 1750 1750 2050 2150 2250 2250 3250 4000
450 725 1000 650 1150 6600 2100 4450 1500 2400 3250 8000
5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0
-2.76 -5.77 -4.21 -4.29 -4.07 -1.63 -4.09 -4.04 -4.74 -3.58 -2.49 -3.65
144 153 170 215 222 158 227 162 255 239 354 199
0.44 0.47 0.54 0.69 0.71 0.48 0.72 0.50 0.84 0.78 1.18 0.67
P333 P333 P333 P333 P333 P333 P333 P333
3250 3250 3250 3250 3250 3250 3250 3250
1700 1700 1700 1700 1700 1700 1700 1700
0.5 1.0 2.5 5.0 10.0 15.0 19.98 20.0
-2.77 -2.40 -2.29 -2.47 -2.71 -2.78 -2.84 -2.76
423 367 366 377 379 384 389 377
1.40 1.22 1.22 1.27 1.28 1.30 1.32 1.28
P237 P237 P237 P237 P237 P237 P237 P237 P237
2250 2250 2250 2250 2250 2250 2250 2250 2250
5450 5450 5450 5450 5450 5450 5450 5450 5450
1.0 5.0 7.5 10.0 15.0 20.0 25 34.2 50
-7.25 -5.71 -5.71 -5.56 -6.07 -5.80 -5.82 -5.54 -5.06
281 268 266 266 278 272 253 290 265
0.87 0.86 0.85 0.86 0.90 0.88 0.82 0.93 0.86
aqueous solution to a thermodynamic mass action model. The developed model provides estimates of the aggregation number which correspond with values obtained by light scattering; it suggests the presence of premicellar aggregates which have been observed using other techniques; it provides values for the heat capacity change of aggregation, whose sign corresponds with that expected given the important role of dehydration in the aggregation process; and it is capable of fitting the data over the entire
Figure 6. Enthalpy-entropy compensation plot for the data obtained in this study and shown in Table 6.
temperature range using a minimal number of parameters and assumptions about the underlying aggregation process. Since the model incorporates the underlying heat capacity changes which give rise to the calorimetric baseline, using the model to fit the calorimetric data obviates the necessity of baseline fitting. Finally, because the model provides a good representation of the process as ‘viewed’ by scanning calorimetry, it seems quite probable that the model can be used for the deconvolution of aggregation transitions for several polymeric surfactants simultaneously present in aqueous solution. Preliminary results have borne this out, and this will be reported in a subsequent communication. Acknowledgment. S.L. and B.C. wish to gratefully acknowledge the receipt of an EPSRC grant (GR/H95174) which was used to partially support the work reported in this paper. LA960432G
