Pressure-Modulated Differential Scanning Calorimetry. An Approach

We refer to it as pressure-modulated differential scanning calorimetry (PMDSC), ... Analytical Parameters for Amplitude-Modulated Multiplexed Flow Ana...
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Anal. Chem. 2006, 78, 984-990

Pressure-Modulated Differential Scanning Calorimetry. An Approach to the Continuous, Simultaneous Determination of Heat Capacities and Expansion Coefficients# K. Boehm, J. Ro 1 sgen,† and H.-J. Hinz*

Institut fu¨r Physikalische Chemie, Westfa¨lische Wilhelms-Universita¨t Mu¨nster, Corrensstrasse 30, 48149 Mu¨nster, Germany

A new method is described that permits the continuous and synchronous determination of heat capacity and expansibility data. We refer to it as pressure-modulated differential scanning calorimetry (PMDSC), as it involves a standard DSC temperature scan and superimposes on it a pressure modulation of preselected format. The power of the method is demonstrated using salt solutions for which the most accurate heat capacity and expansibility data exist in the literature. As the PMDSC measurements could reproduce the parameters with high accuracy and precision, we applied the method also to an aqueous suspension of multilamellar DSPC vesicles for which no expansibility data had been reported previously for the transition region. Excellent agreement was obtained between data from PMDSC and values from independent direct differential scanning densimetry measurements. The basic theoretical background of the method when using sawtooth-like pressure ramps is given under Supporting Information, and a complete statistical thermodynamic derivation of the general equations is presented in the accompanying paper. Until recently, measurements of partial molar volumes and expansion coefficients relied almost exclusively on densimetric and pycnometric techniques.1-4 These methodssalthough very accurateshave one pronounced disadvantage particularly in view of the application to rare biological compounds. They require relatively large amounts of material for each measurement. Therefore, such fundamental properties as volume changes and expansion coefficients, which are frequently associated with phenomena such as conformational changes, ligand binding, aggregation, and dissociation reactions, received less attention * To whom correspondence should be addressed. E-mail: [email protected]. † Current address: Department of Biochemistry and Molecular Biology, Sealy Center for Structural Biology, University of Texas Medical Branch, 5.154 Medical Research Building, Galveston, TX 77555-1052. E-mail: [email protected]. # This publication is dedicated to Serge N. Timasheff on the occasion of his 80th birthday in admiration of his fundamental contributions to science and with gratitude for a long-lasting personal friendship. (1) Picker, P.; Tremblay, E.; Jolicoeur, C. J. Solution Chem. 1974, 3, 377-384. (2) Kratky, O.; Leopold, H. Methods Enzymol 1973, 27, 98-110. (3) Hall, R. E. J. Wash. Acad. Sci. 1924, 14, 172. (4) Geffken, W.; Kruis, A.; Beckmann, Ch. Z. Phys. Chem. 1933, 20B, 398419.

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than they deserved. Therefore, the introduction of pressure perturbation calorimetry (PPC) by Brandts5 constituted a significant progress in the determination of volumetric properties of molecules of biological interest. PPC involves pressure jumps during the quasi-isothermal mode of the DSC instrument and observes the resulting heat signal changes as a function of time. For two-component solutions, eq 1 permits the calculation of values for the coefficient of thermal expansion, R j ) (1/vj)‚ (∂vj/∂T))p, of the partial volume vj of the solute.

R j ) R0 -

∆Qrev Tgsvj∆p

(1)

T is the absolute temperature, gs ) c (g/mL)Vcell (mL) is the total weight of solute in the sample cell, ∆Qrev is the reversible heat change resulting from the pressure jump ∆p; R0 is the expansibility of the solvent. It should be noted that calculation of R j requires an independent measurement of the partial specific volume vj of the solute, for example, by densimetry. There is a certain drawback to the PPC method. For the measurement of the coefficient of thermal expansion Rp of a one-component system having the volume V, Rp ) (1/V)(∂V/∂T)p, according to eq 2, the DSC scan is interrupted at temperature T to provide quasiisothermal conditions for the pressure jumps.

(∂Qrev/∂p)T ) - TVRp

(2)

To allow for continuous measurements, we describe in the following a modification of the method that permits the continuous, synchronous determination of expansion coefficients and heat capacity. The theoretical basis pertaining to the sawtooth pressure format applied in this study is briefly developed in the Supporting Information together with a block diagram of the pressure control unit. A detailed derivation of the theory of PMDSC is given in the accompanying paper (Ro¨sgen and Hinz). We refer to the method as pressure-modulated differential scanning calorimetry (PMDSC), since it can involve arbitrary formats of pressure modulation superimposed on the continuous temperature scan of a DSC instrument. Theory and practice of DSC measurements at constant pressure have been described previously in detail by (5) Lin, L.; Brandts, J. F. Anal. Biochem. 2002, 302, 144-160. 10.1021/ac0509760 CCC: $33.50

© 2006 American Chemical Society Published on Web 01/21/2006

Ro¨sgen and Hinz.6 The fundamental equation describing the temperature and pressure dependence of the experimental heat capacity Cf, when an arbitrary experimental pressure variation f is applied to a DSC scan, is the following:

Cf ) Cp - [T(∂V/∂T)p(∂p/∂T)f]

(3)

Cp is the heat capacity at constant pressure, T is the absolute temperature, and R/p ) (∂V/∂T)p is the expansibility,7 (∂p/∂T)f is the slope of the pressure temperature curve which depends on the particular pressure variation f chosen. It can be expressed as the rate of pressure change, (∂p/∂t)f, chosen in the particular experiment times the reciprocal heating rate, (∂t/∂T)f, of the calorimeter. As shown in the appendix, the resulting heat capacity of the solute, Csolute , in a two-component solution consisting of p solute and solvent, can be calculated using the standard equation for the analysis of the apparent heat capacity difference, ∆Capp, between sample and reference cell in a DSC scan, where the reference cell contains the solvent or buffer.

{

}

∆Capp vjsolute Csolute ) Csolvent + p p vjsolvent msolute

[( T

)

]

∂p ‚(R/,solute - vjsoluteRsolvent ) (4) p ∂T f p

According to eq 4, Csolute is determined by two terms. The first p term in braces is the standard expression describing the heat capacity of the solute at constant pressure as a function of temperature. The second term in brackets shows the modification of the heat capacity signal by the experimental rate of pressure change (∂p/∂T)f ) (∂p/∂t)f(∂t/∂T)f and the difference in expansibilities of solute and solvent, respectively. To demonstrate the accuracy and precision of the PMDSC method for simultaneous determination of heat capacity and expansion coefficients, we present in this study the results of PMDSC measurements and of differential scanning densimetry (DSD) studies on aqueous salt solutions of NaCl and Na2SO4. We chose these simple systems, because both their heat capacity and expansion data reported in the literature are of higher accuracy than those existing for biological samples. We demonstrate in the present study on the salt solutions that heat capacity and expansibility data, which so far required separate measurements on different instruments, can be obtained with the same accuracy and precision in a single PMDSC experiment. This convincing result prompted us to apply the method also to a much less well characterized biological system. We determined for the first time the expansion properties of distearoylphosphatidylcholine (DSPC) in the temperature range of the main transition by PMDSC. MATERIALS AND METHODS PMDSC Studies. The heat capacity was determined using a nanodifferential scanning microcalorimeter (N-DSC), model 6100 from Calorimetry Science Corp. The volume of both sample and reference cell is 0.29 mL. The pressure control unit was constructed in collaboration with Ju¨rgen Kro¨ninger of the electronic (6) Ro ¨sgen, J.; Hinz, H.-J. In Handbook of Thermal Analysis and Calorimetry; Kemp, R. B., Ed.; Elsevier: Amsterdam, 1999; Vol. 4, pp 63-108. (7) Harned, H. S.; Owen, B. B. ACS Monogr. Ser. 1958.

workshop of the Institute of Physical Chemistry of the University of Mu¨nster. A block diagram of the control unit is given as Supporting Information. The noise level of a scan with constant pressure is less than 15 nW, and the reproducibility with or without refilling is in the order of 0.4 µcal/K. The maximum pressure is limited to 6.5 bar to prevent damaging the calorimetric cells. The pressure control program provides several formats of pressure change as well as possibilities for waiting times at any pressure. In the present study, we applied a sequence of continuous sawtoothlike ramps between 0.3 and 5.3 bar excess pressure. A pressure rate of 0.077 bar/s was routinely applied at a heating rate of 1 K/min for Na2SO4 and 2 K/min for NaCl solutions. The sampling rate of pressure and Cp values was 1 s-1 for all measurements with Na2SO4 solutions and 6 s-1 for NaCl solutions. In case of the DSPC solution, we used a sequence of continuous sawtoothlike ramps between 2 and 5 bar excess pressure with a pressure rate of 0.017 bar/s and a heating rate of 0.01 K/min. DSD Measurements. Density measurements were performed in differential mode using two coupled high-temperature DMA 602 HT cells and a DMA 60 control unit of Anton Paar (Graz, Austria). Temperature scanning was controlled by a Haake PG 20 temperature controller. This arrangement reduces the influence of temperature fluctuations, as both cells are affected simultaneously. To collect data at isothermal conditions, the heating was stopped every 0.4 °C, and after temperature equilibration, a defined number of oscillations were counted. To obtain very high precision around 25 °C, separate measurements were performed at temperature intervals of 0.1 °C. The standard deviation of the measured signal for the setup chosen was ∼10-5 s for each scan. The accuracy of the measurements is excellent, as shown by the good agreement with literature values. RESULTS AND DISCUSSION Density Studies on Aqueous Na2SO4 and NaCl Solutions. Following Harned and Owen,7 the dependence on concentration of the apparent molal volume φV of a salt can be described by eq 5. φ0V of Na2SO4 refers to the volume at infinite dilution and can be equated to the partial molal quantity. c is the molal concentration of Na2SO4, c° the standard concentration 1 mol kg-1.

φV ) φ0V + SVxc/c°

(5)

Apparent molal volumes of Na2SO4 were determined in the present study by differential density measurements in the concentration range 0.05-2.74 mol kg-1, and the results are given in Figure 1A. Filled red triangles represent the present results, and the line shows the fit of the data to eq 5. The fit parameters for sodium sulfate are φ0V ) 11.64 cm3 mol-1 and SV ) 11.82 cm3 mol-1. The other symbols indicate data points from the literature. Obviously there is excellent agreement between the present data and the literature values. The data of Geffken and Price8 (open squares) and of Gibson9 (cross-hairs) are based on pycnometric studies. In the more recent references of Surdo et al.10 and Phutela and (8) Geffcken, W.; Price, D. Z. Phys. Chemie, 1934, 26B, 81-98. (9) Gibson, R. E. J. Phys. Chem. 1927, 496-510. (10) Surdo, A. L.; Alzola, E. M.; Millero, F. J. J. Chem. Thermodyn. 1982, 14, 649-662.

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Figure 1. Apparent molal volume as function of the square root of molality at 25 °C. (A) Apparent molal volume of Na2SO4 (aq): red 2, present DSD measurements (20000 counts/measurement); 0, Geffken and Price;8 green b, Surdo et al.;10 +, Gibson;9 ", Phutela and Pitzer;11 s, fit curve of DSD data only. (B) Apparent molal volume of NaCl (aq): red 2, present DSD measurements (20000 counts/ measurement); O, Allred et al.;12 0, Oloffson;13 3, Millero;14 blue diamond Picker et al.;15 pink diamond, Gibson et al.;16 + in black square, Geffken;17 +, Baxter and Wallace;18 + in hexagon, Vaslow;19 green b, Perron et al.;20 orange \,Mironenko et al.;21 s, fit curve of DSD data only.

Pitzer,11 various densimetric techniques were used. The fit parameters φ0V ) 11.15 cm3 mol-1 and SV ) 12.32 cm3 mol-1 when using all literature data are within error limits identical to our results. In general, 1:1 electrolytes show volume and expansibility behavior that differs significantly from that of 1:2 electrolytes. This difference is manifest especially in the values of SV and SE (eqs 6 and 7). Therefore, we also determined for NaCl solutions the volumetric parameters to have a second independent parameter set for demonstration of the accuracy and precision of the expansibility values obtained from the PMDSC studies. Figure 1B shows the results of DSD measurements on NaCl solutions in the concentration range from 0.021 to 1.37 mol kg-1 NaCl. Apparent molal volumes are plotted against the square root of molal concentration of NaCl. The filled triangles show the present data, and the fit to eq 5 is represented by the solid line. The fit parameters are φ0V ) 16.54 cm3 mol-1 and SV ) 2.04 cm3 mol-1. The graph shows also a variety of published data, and the agreement between the present values and those from literature is good. (11) Phutela, R. C.; Pitzer, K. S. J. Chem. Eng. Data 1986, 31, 320-327.

986 Analytical Chemistry, Vol. 78, No. 4, February 15, 2006

Figure 2. PMDSC measurements on Na2SO4 and NaCl solutions in water (A) PMDSC measurement on a 0.05 mol/kg Na2SO4 solution in water. Experimental conditions: heating rate, 1 K/min, dp/dt ) 0.077 bar/s; pressure sequence, 0.3-5.3 -0.3 bar; sampling rate, 1 pt/s. Red line: fit of the data to eq S-9; inset, magnification of the PMDSC signal, Top: Residuals as function of temperature. (B) PMDSC measurement on a 1.37 mol/kg NaCl solution in water. Experimental conditions: heating rate, 2 K/min, dp/dt ) 0.077 bar/s; pressure sequence, 0.3 -5.3-0.3 bar; sampling rate, 1 pt/6 s; red line; fit of the data to eq S-9, Top: residuals as function of temperature.

PMDSC Studies. The density studies on the Na2SO4 and NaCl solutions provided the volumetric data that are required as reference data for the evaluation of the accuracy and precision of the expansibility results obtained with the PMDSC method. Figure 2A shows a representative PMDSC measurement on a 0.05 m Na2SO4 solution. The pressure modulation followed a sawtooth pattern between 0.3 and 5.3 bar with a linear rate of pressure change of ∆p/∆t ) 0.077 bar/s (5 bar/65 s). As shown under Supporting Information (Figure S-1A), the ideal output signal (µW) of the DSC instrument corresponding to a sawtooth pressure change is a step function pattern. Due to the nonezero relaxation time τ of the DSC instrument, the real response function of the calorimeter is the result of a convolution of the ideal step function having the amplitude (TR/p(∆p/∆T) with an exponential function having a time constant of τ ) 1/k ) 9 s (Figure S-1A). The effect of the relaxation time on the Cp signal of the DSC instrument is negligible in the absence of pressure changes.6 The effect of pressure modulation on the heat capacity signal is magnified in the inset of Figure 2A. The deviations between

experimental and fitted curve are plotted on top of Figure 2A. It is evident that the fit represents the PMDSC curve very well. Figure 2B shows an analogous PMDSC measurement on a 1.37 m NaCl solution, obtained with a lower sampling rate of 1pt/6 s and with a heating rate of 2 K/min. Obviously neither the quality of the measurements nor that of the fit is reduced by these changes as is shown by the plot of the residuals at the top of Figure 2B. The absolute positive and negative deviations are again small ((4 µW). One observes in both residual plots a “spike” pattern that tends to show larger deviations in the turning points of the sawtoothlike pressure changes. This is particularly evident in Figure 2A. The finding can be rationalized in the following manner. As a result of the higher sampling rate, one obtains more frequently data points at the edge of the sawtoothlike pressure changes. These are obviously responsible for the spikes seen in the plot of the residuals. Randzio22 pointed out that in the case of Brandts’ pressure perturbation calorimetry involving isothermal pressure jumps, little spikes occur in the signal at the beginning of each pressure jump. These spikes are apparently caused by the sensitivity of the heat sensor to discontinuities in the pressure p or pressure slope dp/dT. Our PMDSC method allows us to check this hypothesis. We also observe such spikes, and they depend strongly on temperature. At low temperature they are insignificant, but they become more pronounced at high temperature. Based on their characteristic pattern (sudden, instantaneous signal change at pressure slope reversal, followed by an exponential decay), it is obvious that these spikes are due to an isolated event that takes place exactly at the reversal of the pressure slope. Accordingly, we took these effects into account in data analysis. The excellent agreement between our PMDSC results and literature values demonstrates both the correctness of the hypothesis concerning the origin of the spikes and the successful elimination of their perturbing effects. To check the quality of both the Cp and expansibility data obtained from fits of the PMDSC data, we also measured the heat capacity of the salt solutions at constant mean pressure of 2.8 bar as a function of concentration and temperature and compared our values with data from literature. This is shown in Figure 3A,B for Na2SO4 and NaCl solutions, respectively. Inspection of the graphs shows the excellent agreement between the heat capacity data resulting from the present PMDSC and DSC measurements and the literature values. A more detailed analysis of the PMDSC-Cp,φ data is given in the following section. Determination of Apparent Molal Heat Capacity Cp,φ. Variation with Concentration. The values of the apparent molal heat capacity Cp,φ of sodium sulfate (aq) and sodium chloride as function of the square root of the molality at 25 °C are displayed (12) Allred, G.; Wooley, E. J. Chem. Thermodyn. 1981, 13, 147-154 (13) Olofsson, I. J. Chem. Thermodyn. 1979, 11, 1005-1014. (14) Millero, F. J. J. Phys. Chem. 1970, 74, 356-362. (15) Picker, P.; Tremblay, E.; Jolicoeur, J. J. Solution Chem. 1974, 5, 377-384. (16) Gibson, R. E.; Loeffler, O. H. J. Am. Chem. Soc. 1941, 63, 443-449. (17) Geffken, W. Z. Phys. Chem. 1931, 155A, 1-28. (18) Baxter, G. P.; Wallace, C. C. J. Am. Chem. Soc. 1916, 38, 70. (19) Vaslow, F. J. Phys. Chem. 1966, 7, 2286-2294. (20) Perron G.; Fortier; J. L.; Desnoyers J. E. J. Chem. Thermodyn. 1975, 7, 1177-1184. (21) Mironenko, M. V.; Boitnott, G. E.; Grant, S. A.; Sletten, R. S. J. Phys. Chem. B 2001, 41, 9909-9912. (22) Randzio, S. L. Thermochim. Acta 2003, 398, 75-80.

Figure 3. Cp,φ as function of square root of molality at 25 °C. (A) Apparent isobaric heat capacity Cp,φ of Na2SO4 (aq): red 2, data from PMDSC; 3, data from DSC; s, fit of both data from PMDSC and DSC; green b, Magalhaes et al.;25 blue 9, Randall et al.;23 red dash line, Perron et al.;26 +, Saluja et al.27 (B) Apparent isobaric heat capacity of NaCl (aq): red 2, data from PMDSC; s, regression curve of PMDSC data; green 1, Allred et al.;12 blue 9, Olofsson et al.;13 O, Clark et al.24

in Figure 3A,B. The solid lines represent the fit curves to the Cp,φ data from PMDSC measurements. In the case of sodium sulfate, a linear fit using eq 6a represents the data well. The analogy to eq 5 is obvious. However, the Cp,φ versus c1/2 plot of sodium chloride is evidently not linear. Therefore, we used eq 6b to fit the experimental Cp,φ values in analogy to Perron et al.,20 Allred and Wooley,12 and others.13,23 0 Cp,φ ) Cp,φ + SCp,φ‚

c 0 Cp,φ ) Cp,φ + A C F0 c°

( )

1/2

xc°c + BC‚

(6a)

xc°c

(6b)

c is the molality of the salt, c° ) 1 mol/kg is standard concentra0 tion, Cp,φ is the heat capacity at infinite dilution, and F0 is the density of the pure solvent (water). The factor AC(F0)1/2 is the limiting slope derived from the Debye-Hu¨ckel theory. Its value 0 is 28.95 J mol-3/2 kg1/2 K-1 at 25 °C.20 The values of SCp,φ and Cp,φ obtained for sodium sulfate from fitting of our Cp,φ data are 245.5 and -201.9 J mol-1 K-1, respectively. (23) Randall, M.; Rossini, F. D. J. Am. Chem. Soc. 1929, 51, 323-344.

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The literature data included in Figure 3A are from the studies of Magalhaes25 Perron,26 and Saluja27 and were all determined using a Picker flow calorimeter.28 Randall23 used a self-constructed calorimetric unit. Evidently the agreement with these data from the literature is, in general, good over the total concentration range. The error bars shown in Figure 3A belong to the values of Randall.23 For our Cp,φ values, the calculated maximal standard deviation is smaller than 2 J mol-1 K-1 and therefore not visible in the graph. The variation with salt concentration of Cp,φ of aqueous sodium chloride solutions at 25 °C is presented in Figure 3B. The solid curve represents our data and the various symbols refer to the Cp,φ values from the literature. Included are values determined with Picker flow microcalorimeters by Allred12 and Olofsson13 as well as calculated values from the review article of Clarke.24 The 0 parameters obtained from fitting our data set to eq 6b are Cp,φ ) -1 -1 -1 -1 -81.5 J mol K and BC ) 14.1 J mol K . They are nearly 0 identical to the parameters reported by Perron et al.:20 Cp,φ ) -1 -1 -1 -1 -81.4 J mol K and BC ) 14.9 J mol K . Variation with Temperature. Figure 4A shows the temperature dependence of Cp,φ at three selected sodium sulfate concentrations. Literature data by Rogers and Pitzer29 and Saluja et al.27 shown in the graph were determined using a Picker flow calorimeter, and those by Holmes and Messmer30 were based on a mathematical fit to data from different references. The solid lines represent the Cp,φ values derived from our PMDSC studies. The values of Roger and Pitzer were calculated using a constant partial molar volume, whereas Saluja was taking the volume change into account. In general, these values coincide with our corresponding curves. The dashed green curves were obtained from DSC measurements at a constant excess pressure of 2.8 bar, and the fit was carried out under the assumption of a constant volume (V(25 °C)). To prove, furthermore, the agreement between the Cp,φ values from DSC and PMDSC for the same concentration, we calculated the DSCCp,φ curves under the assumption of variable volumes. These curves are shown as pink dash-dotted lines. For a concentration of 0.5 mol/kg sodium sulfate, this curve is identical with values derived from the corresponding PMDSC measurement. 0 The variation with temperature of Cp,φ is shown for five NaCl concentrations in Figure 4B. In general, there is very good agreement with the literature data. Only at low concentrations is a small deviation for temperatures of >50 °C observed when compared with the calculated data of Clarke and Glew.24 However, in summary, we can conclude that PMDSC studies result in heat capacity data of high accuracy and precision. Determination of Apparent Molal Expansibility. Variation of Expansibility with Concentration. The apparent molal expansibility R/p ) (∂V/∂T)p has been determined by various methods. Here we concentrate on results obtained using the new PMDSC method. Figure 5A shows R/p values for the solute Na2SO4 at 25 (24) Clarke, E. C. W.; Glew. D. N. J. Phys. Chem. Ref. Data 1985, 14 (2), 489610. (25) Magalhaes, M. C. F.; Ko ¨nigberger, E.; May, P. M.; Hefter, G. J. Chem. Eng. Data 2002, 47, 590-598. (26) Perron, G.; Desnoyers, J. Can. J. Chem. 1975, 53, 1134-1138. (27) Saluja, P. P.; Lemire, R. J.; LeBlanc, J. C. J. Chem. Thermodyn. 1992, 24, 181-203. (28) Picker, P.; Leduc, P.-A., Philip, P. R.; Desnoyers, J. E. J. Chem. Thermodyn. 1971, 3, 631-642. (29) Rogers, P. S. Z.; Pitzer, K. S. J. Phys. Chem. 1981, 85, 2886-2895. (30) Holmes, H. F.; Messmer, R. E. J. Solution Chem. 1986, 15 (6), 495-518.

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Figure 4. Cp,φ as function of temperature at various salt concentrations. (A) Apparent isobaric heat capacity of Na2SO4 solutions at three concentrations: s, data from PMDSC; green dash line data from DSC fitted with V ) const ) V(25 °C); red dash-dot-dot-dash line data from DSC fitted with temperature-dependent volume V )V(T); blue [, Rogers and Pitzer;29 +, Saluja et al.;27 red b, Holmes and Messmer30 (B) Apparent isobaric heat capacity of NaCl solutions at five concentrations: s, data from PMDSC; +, Perron al.;16 blue 9, Allred et al.;12 ref b, Clarke al.24

°C as a function of concentration, Figure 5B shows the corresponding data for NaCl. Values from the literature are included in both figures. The expansibility φΕ according to the nomenclature of Harned and Owen is equivalent to the term R/,solute that p has been plotted in Figure 5 to conform with recent literature. In the concentration range studied, we found that the expansibility as a function of the square root of molality is best fitted by the linear eq 7a. The fit parameters for Na2SO4 are as follows: φ0E ) 22.51 × 10-2 cm3 mol-1 K-1 and SE ) -8.98 × 10-2 cm3 mol-1 K-1.

φE ) φ0E + SE‚xc/c°

(7a)

c φE ) φ0E + SE‚xc/c° + BE‚ c°

(7b)

Gucker31 also reported a linear dependence of expansibility on the square root of concentration. His values were based on studies by Gibson and referred to molar concentration. We calculated the corresponding molalities, and the result is shown as the dark-red (31) Gucker, F. T. J. Am. Chem. Soc. 1934, 56, 1017.

Figure 5. Apparent molal expansibility as function of square root of molality at 25 °C. (A) Apparent molal expansibility of Na2SO4 solutions: red 2, data from PMDSC; s, fit of PMDSC data; ", Phutela and Pitzer;11 red dash line, Gucker.31 (B) Apparent molal expansibility of NaCl solutions: red 2, data from PMDSC; s, fit of PMDSC data; +, calculated points from volume data by Perron et al.;20 blue dashdot-dash-dot line, values presented by Millero;14 red dash line, Gucker;31 red dash-dot-dash-dot line, Fortier et al.;32 blue 9, Alary et al.33 The gray area represents the confidence interval of the PMDSC data.

long-dashed line shown in Figure 5A. It is evident that Gucker‘s analysis is in very good agreement with our results from PMDSC measurements. The values shown as open blue hexagons were calculated from volume data reported by Phutela and Pitzer.11 They are slightly different from our and Gucker’s data. This difference can be rationalized when considering the method used by Phutela and Pitzer for the calculation of the expansibilities. They measured the volume at different temperatures under excess pressures between 20 and 100 bar and fitted these data together with values from the literature to V(p,T) functions. Values for the apparent volume were derived from this function and listed for a pressure of 1 bar for various concentrations. We used their data to calculate the expansibilities at 25 °C shown in Figure 5A. The background marked gray displays the 95% confidence interval of the fit curve of the expansibilities derived from the PMDSC studies. In Figure 5B, the expansibility of sodium chloride at 25°C is plotted as a function of the square root of the molality. In view of the confidence interval displayed again in gray, the agreement with the linear curve reported by Gucker31 for values of c1/2 > 0.5 is obviously good. The comparison of our fit parameterssφE° ) 9.35 × 10-2 cm3 mol-1 K-1 and SE ) -1.83 × 10-2 cm3 mol-1 K-1,

with those of Gucker, -φE° ) 9.3 × 10-2 cm3 mol-1 K-1 and SE ) -2.15 × 10-2 cm3 mol-1 K-1ssupports this quantitatively. Other investigators who measured also in the low-concentration range like Millero14 and Fortier32 have postulated a nonlinear dependence according to eq 7b. Millero determined his data with a magnetic flow densimeter in the temperature range of 0-55 °C with salt concentrations between 0.1 and 1 mol/kg. His calculated expansibility function (from the temperature derivative of the fit parameters of a V versus c1/2 plot) is shown in Figure 5B as a dashed-dotted curve. The expansibility values are much lower over the total concentration range. In contrast, the curve representing data from Fortier32 is in very good agreement with our data with the exception that Fortier et al. also assumed a nonlinear relationship for the dependence on concentration as did Millero. Fortier measured the expansibility directly with a flow calorimeter that allows also for dilatometric measurements. He measured from 10 to 40 °C using five different NaCl concentrations from 0.2 to 1.9 mol/kg. Based on the same method, Alary provided data for 25 °C employing NaCl solutions with concentrations between 0.28 and 5 mol/ kg.33 The expansibility values plotted as crosses were calculated by us from volume data reported by Perron20 for NaCl concentrations up to 1 mol/kg and six different temperatures between 1.5 and 45 °C. In summary, the agreement of the PMDSC expansibilitiy data as a function of concentration with all literature values except for those by Millero is good to fair, as the 95% confidence interval shown in gray demonstrates. We do not have an explanation why the one data set of Millero deviates from all other sets. Variation of Expansibility with Temperature. The temperature dependence of the expansibility of a 0.5 mol/kg sodium sulfate solution derived from PMDSC studies is shown as a solid black line in Figure 6A. We plotted also the data of Phutela and Pitzer as a green dash-dotted curve. The curves practically coincide with only negligible deviations. Figure 6B shows the temperature dependence of the expansibility of an aqueous 1.37 mol/kg sodium chloride solution derived from PMDSC measurements as a solid black curve and data of Fortier32 as red squares. The excellent agreement is obvious. We show only the curve for one concentration, because the dependence of (∂φE/∂T) on concentration is not very pronounced and curves at different concentrations would overlap. PMDSC Measurements on DSPC. The excellent precision and accuracy of the PMDSC studies on salt solutions encouraged us to apply the new technique to biological systems. Here we present PMDSC measurements of the expansibility of aqueous suspensions of (L-R-distearoylphosphatidylcholine (C18:0), DSPC) in the temperature range of 54-56 °C. To avoid problems with kinetics,34 we used a slow heating rate of r ) 0.01 K/min and a pressure modulation between 2 and 5 bar with dp/dt ) 0.017 bar/ s. The PMDSC measurements were accompanied by accurate DSD measurements. The volumetric parameters from these DSD measurements are in excellent agreement with values from the literature35 cited in parentheses. (32) Fortier, J. L.; Simard, M.-A.; Picker, P.; Jolicoeur, C. Rev. Sci. Instrum. 1979, 50 (11), 1474-1480. (33) Alary, J. F.; Simard, M. A.; Dumont, J.; Jolicoeur, C. J. Solution Chem. 1982, 11 (11), 755-776. (34) Grabitz, P.; Ivanova, V. P.; Heimburg, T. Biophys. J. 2002, 82, 299-309. (35) Nagle, J. F.; Wilkinson, D. Biophys. J. 1978, 23, 159-167.

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Figure 6. Apparent molal expansibility as function of temperature. A: Apparent molal expansibility of an aqueous 0.5 mol/kg sodium sulfate solution. s, data from PMDSC; green dash-dot-dot-dash line, Phutela and Pitzer11 (B) Apparent molal expansibility of an aqueous 1.37 mol/kg sodium chloride solution s,data from PMDSC; red 9, Fortier et al.32

The measured maximum apparent volume value for the Pβgel-Phase is 768.9 cm3/mol (768.8). The minimum volume for the LR liquid crystal phase is 806 cm3/mol (807.5). The volume change ∆V for this transition is 37.1 cm3/mol (35.6) corresponding to a change of 4.8%. The PMDSC curve is presented in Figure 7A together with the fit of the data to eq S-9 (Supporting Information). The residual plot is also given in Figure 7A, indicating a good fit. The expansibility data derived from this fit are presented in Figure 7B together with those obtained from the DSD measurements. As the DSPC suspensions are under atmospheric pressure in the DSD studies, whereas in PMDSC measurements they are exposed to an average excess pressure of 3.5 bar, the transition temperatures Tm observed by PMDSC are shifted and higher than those from DSD. To allow for an easy comparison of the magnitude and shape of the transitional expansibilities resulting from the new method the DSD graph was normalized to the Tm value of the PMDSC curve. Inspection of Figure 7B shows that the expansibility of DSPC at Tm obtained by DSD is 114.6 cm3/mol K. The value resulting from PMDSC is by 8% smaller, i.e., 106.1 cm3/ mol K. The only value found in the literature is 102.6 cm3/mol K.35 It was reported by Nagle and Wilkinson35 and was derived from density studies. CONCLUSION Continuous pressure modulation of DSC scans (PMDSC method) has been shown here to provide simultaneously heat 990 Analytical Chemistry, Vol. 78, No. 4, February 15, 2006

Figure 7. PMDSC measurement on a 10.2 mg/mL (12.9 mM) suspension of DSPC multilamellar vesicles in water. (A) Experimental conditions: heating rate, 0.01 K/min, dp/dt ) 0.017 bar/s; pressure sequence, 2.0-5.0-2.0 bar; sampling rate, 1 pt/s red line; fit of the data to eq 16. Top: Residuals as function of temperature. (B) Apparent molar expansibility of a 10.2 mg/mL (12.9 mM) DSPC suspension of multilamellar vesicles in water as function of temperature. s, data from PMDSC. Data from DSD (red dashed curve) were shifted by +0.25 K to standardize the transition temperatures.

capacity and expansibility data of high accuracy and precision. The method has been successfully tested on aqueous salt solutions of NaCl and Na2SO4. They were selected as test solutions, because for these solutions the largest number of accurate data is available in the literature. The applicability to biologically relevant systems has been demonstrated by obtaining for the first time expansibility data for the main transition of an aqueous suspension of multilamellar vesicles of DSPC. These lipid systems are particularly suitable for PMDSC studies due to the relatively large volume changes associated with the lamellar-to-liquid crystalline transition of their alkyl chains. SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

Received for review June 3, 2005. Accepted December 6, 2005. AC0509760