15992
J. Phys. Chem. 1996, 100, 15992-15999
Thermodynamic Properties of Poly(ethylene glycol)/Water Systems. 1. A Polymer Sample with a Narrow Molar Mass Distribution Volker Fischer, Werner Borchard,* and Michael Karas‡ Angewandte Physikalische Chemie, Gerhard-Mercator-UniVersita¨ t, Duisburg, Lotharstrasse 1, D-47057 Duisburg, Germany ReceiVed: March 14, 1996; In Final Form: July 15, 1996X
Liquid-liquid demixing experiments of a poly(ethylene glycol)/water system have been performed. The polymer is characterized by a nominal molar mass of 6000 g mol-1 and a narrow molar mass distribution. The cloud-point curve of the system and a large number of phase volume ratios at different polymer concentrations and temperatures have been measured. With these results the critical coordinates and the shadow curve of the system have been determined using the volume conservation equation. Generally, the system behaves like a binary one. The miscibility gap is well described using the scaling theory up to temperatures of about 12 K above the lower critical point. The cloud-point and the shadow curve of the system have been computed using the Flory-Huggins theory. For the first time thermodynamic parameters have been used that are calculated directly from the concentrations of coexisting phases obtained by evaluation of phase volume ratio measurements. The examined forms of the interaction function g of the base molar free enthalpy of mixing ∆G h M do not describe the system satisfactorily.
Introduction Poly(ethylene glycol) (PEG) is one of the simplest technical polymers that has been investigated in aqueous solution. In particular, the liquid-liquid demixing in the system PEG/water has been of interest.1-6 Nevertheless, a complete understanding of the thermodynamic properties of the system and even a quantitative description of the demixing are not yet reached. To obtain this, the polymer chain distribution of the synthetic polymer has to be considered. In general the liquid-liquid demixing of polydisperse polymers in a low molecular solvent may be described by the known conditions of a heterogeneous equilibrium.7-15 But in contrast to a strict binary mixture the molar mass distribution (mmd) of the polymer in the coexisting phases is different from that in the polymer bulk phase, which leads to a dependence of the coexisting polymer concentrations upon the initial concentration.9-11,13-16 The second and more specific feature of the PEG/water system is the fact that polymer/solvent systems with interaction due to specific intermolecular forces could not be described satisfactorily by theories developed for nonpolar polymer/solvent systems such as the Flory-Huggins theory. It has been argued that in fact hydrogen bonds between parts of the polymer molecules and the solvent cause the almost unique solubility of the polymer in water.17 They represent a kind of interaction that is relatively strong, including also local orientations. The system does not obey the rules of simple combinatorial mixing statistics owing to random distributions of different particles. The occurrence of the lower critical solution temperature (LCST) in the PEG/water system was explained exclusively by the breakdown of the hydrogen bonds at higher temperatures.3,4,17,18 Therefore, as already proposed in 1961 by Freeman and Rowlinson, it is reasonable to distinguish between polymer/ solvent systems like PEG/water or poly(propylene glycol)/water and other systems with nonpolar solvents.19 * To whom correspondence should be addressed. ‡ Present address: Professor Dr. M. Karas, Abt. fu ¨ r Instrumentelle Analytische Chemie, Johann Wolfgang Goethe-Universita¨t, Fankfurt am Main, Theodor-Stern-Kai, D-60590 Frankfurt am Main, Germany. X Abstract published in AdVance ACS Abstracts, September 1, 1996.
S0022-3654(96)00791-5 CCC: $12.00
Cloud-Points and Shadow Curve. The cloud-point of demixing polymer solution with a miscibility gap is defined by the first occurrence of a liquid-liquid phase demixing, which is normally combined with the appearance of the first cloudiness in the solution.16,20,21 All the cloud-points of solutions with different initial polymer concentrations build up the cloud-point curve (CPC). The points on the CPC coexist with those on the shadow curve.9,14,16 In an earlier paper it has been shown that the cloud-point and shadow curve of a system consisting of a polymer and a solvent could be obtained using the linearized volume conservation equation, which may be summarized as follows:16
φ)
( )
φp′ 1 Ψφp′′ φp′′
(1a)
with
Ψ ) [(φ + 1)φp]
(1b)
where φp′ and φp′′ represent the volume fractions of the polymer in the coexisting phases and φ the ratio of the volumes of the phases ′′ and ′ at constant temperature and pressure. The initial concentration φp is close to φp′ near the cloud-point temperature. By this definition of the phase volume ratio φ, it is clear that φ e 1 for all concentrations. Usually, plots of φ(Ψ) for initial polymer concentrations below the concentration of equal phase volumes are evaluated separately from those above it, leading to results for either the dilute or the concentrated branch of the CPC. An extrapolation of the φ(Ψ) plots to φ ) 1 yields the concentration φp,φ)1, which is the critical concentration at T ) TC. At a cloud-point the initial phase becomes nearly identical with the single prime phase. The conjugated coexisting phase is only present in an extremely small quantity. The phase volume ratio φ is close to zero as the second phase is beginning to appear. The plots of φ(Ψ) extrapolated toward Ψ ) 0 lead therefore to the shadow concentration φp′′ and the ratio φp′/ φp′′, which is of theoretical interest. In the case of systems containing polydisperse polymers it is dependent on temperature T, pressure P, and the concentration of all components in the © 1996 American Chemical Society
Poly(ethylene glycol)/Water Systems
J. Phys. Chem., Vol. 100, No. 39, 1996 15993
coexisting phases. In polydisperse polymer/solvent systems, which are in reality multicomponent systems, the plots of φ(Ψ) will not be linear, which is generally the situation with solutions of synthetic polymers. Nevertheless, for a single initial concentration φp eq 1a is always valid and is the tangent to the nonlinear φ(Ψ) curve. By use of a concentration series of a polydisperse polymer in a solvent, the initial mmd remains the same in all solutions as long as no phase separation occurs. Since the concentration of the cloud-point phase is nearly identical to the initial phase, we may expect also that the mmd in the cloud-point phase is almost the same as in the initial phase.11,22 At the critical point the mmd will be the same in the cloud-point and in the shadow phase. In all other cases there are coexisting phases that do not belong to the quasi-binary section and therefore have unknown chain distributions. Thus, the CPC is a starting point for calculations. Diameter Curve and Determination of the Critical Point. Extrapolation of the φ(Ψ) curves to the value φp,φ ) 1 for different temperatures from phase volume ratio data leads to a set of φp,φ ) 1 and T values that can be plotted in the concentration/temperature diagram of the system. These points mark the line of equal volumes of coexisting phases, which is equivalent to the diameter curve as in case of a binary system. In a binary system every coexistence point is the mirror image of its conjugated or correlated coexistence point with respect to the diameter concentration.16 For a polydisperse polymer with a narrow mmd and a linear relationship between φ and Ψ the situation might be similar. The CPC and the shadow curve nearly coincide, and the CPC is almost symmetrical with regard to the diameter curve. If not, the assumption of a nearly binary system is not valid. Therefore, the polymer concentrations of the coexisting phases along the diameter curve can, to a good approximation, be calculated according to eq 2.
φp′′ ) 2φp,φ ) 1 - φp′
∆G hM RT
φi
i)2 ri
ln φi + g(T,φp)φ1φp
(3)
where φ1 is the volume fraction of the solvent, φi that of a polymer component i, and ri the ratio of the molar volume of the pure polymer component i to that of the pure solvent. R and T have their usual meanings, and g is an interaction function9,11,15 that depends on the temperature T and the polymer N
concentration φp, with φp ) ∑ φi. The equilibrium state of i)2 demixing is determined by
∆µi′ ) ∆µi′′
(4)
where i ) 1, 2, 3 ..., N with i ) 1 for the solvent. ∆µ1γ is the difference of the chemical potential of the solvent in the phase with index γ and that of the pure solvent. ∆µiγ with i g 2 represents the same for a polymer homologue i. For a Gibbs free enthalpy of mixing according to eq 3 it follows that
(
)
∆µ1γ 1 ) ln(φ1γ) + φpγ 1 - γ + (φpγ)2 χ1γ RT jr n
(5a)
and
(
)
∆µiγ 1 ) ln(φiγ) - (ri - 1) + φpγri 1 - γ + (φ1γ)2 ri χpγ RT jrn (5b) where γ represents the phase indices ′ or ′′ and i ) 2, 3, ..., N and jrn is the number average of ri. The parameter χ1γ represents the interaction parameter of the solvent in the phase γ, and the parameter χpγ represents the same of the polymer.8,9,11,15 They are defined by
( ) ∂g ∂φp
T,P
( )
T,P
χ1γ ) g(T,φpγ) - φ1γ
(2)
with
N
) φ1 ln φ1 + ∑
(6a)
and
T, P ) const. By means of the φp,φ)1(T) curve, it is possible to determine the critical coordinates of the system. The φp,φ)1(T) curve or possibly its extension intersects with the CPC, yielding the critical demixing point of the system.9,23,24 The precision of this determination is restricted by an often observed, distinct change of the curvature of the φp,φ)1(T) curve near the critical point, which affects especially the accuracy of the determination of the critical concentration.24 Its influence on the critical temperature is normally small owing to the fact that the CPC is usually flat in the critical region. Near the critical temperature the light scattering in the system is increased by the critical opalescence, which may complicate the precise determination of the temperature of the first clouding of the measured samples in the vicinity of the critical point.25 Besides the method described, there are also other intrinsically nonlinear extrapolations that have been used to determine the critical coordinates, but they are at least as laborious and they do not lead to more precise results.26 Thermodynamic Properties. The liquid-liquid demixing of a polymer/solvent system is characterized by the conditions of a heterogeneous equilibrium as already mentioned. If endgroup effects are ignored, the Gibbs free enthalpy of mixing per generalized base mole of the mixture ∆G h M of the system is 8,9 described by
χpγ ) g(T,φpγ) + φpγ
∂g ∂φp
(6b)
By the use of the distribution coefficient σ, it follows that8,14,27
( )
φi′ ) eriσ φi′′
(7a)
with
σ ) ln
( )
φ1′ + χ1′(φp′)2 - χ1′′(φp′′)2 - χp′(φ1′)2 + χp′′(φ1′′)2 φ1′′ (7b)
which reflects the fractionation of a polymer component i enriched either in the dilute or in the concentrated coexisting phase. By a summation over all polymer species i, eq 7a leads to the polymer concentration φp′′: N
φp′′ ) ∑φi′ e-riσ
(8)
i)2
Assuming a certain mmd in the original polymer sample and assuming an appropriate thermodynamic framework, a stepwise iteration may be performed that should give the CPC and the shadow curve of the system. Such iteration procedures of
15994 J. Phys. Chem., Vol. 100, No. 39, 1996
Fischer et al.
Figure 1. Result of the MALDI time of flight mass spectroscopy for the PEG 6000 sample used in the experiments.
varying kinds are established since 1968 by Koningsveld and Kleintjens, Soˆlc, and other authors.11,28,29 Nevertheless, the quality of the resulting description of the phase diagram of the system is dependent on the considered form of the interaction h n) and the general restrictions of the applied function g(T,φp,M theory. Critical Scaling. Even in binary polymer solutions, the CPC shows a very flat characteristic around the critical point.20,30 It is known that the not extremely extended Flory-Huggins theory cannot describe this curvature of the CPC correctly, tending to give a parabolic representation, although the experimental data suggest a cubic one.20,30-34 A scaling fit near the critical point of the system is helpful for classifying the qualitative behavior of the system and for refining the description of the phase diagram close to the critical point. The following equations represent the scaling laws, which seem to be valid both for binary fluids and for binary polymer solutions at least close to the critical point:30-35
|φp′ - φp′′| ) A1τ
(9)
|φp′ + φp′′ - 2φp,c| ) 2A2(1-ν)
(10)
and
is the reduced temperature difference
)
(T - Tc) Tc
(11)
A1 and A2 are system dependent constants, and τ and V are the univeral critical exponents. For a mean-field approach the values of the critical exponents τ and ν are 0.5 and 0, respectively.31,36,37 In reality values around τ ) 0.34 up to 0.45 and ν ≈ 0.1 have been found.32,34 Equation 10 has been deduced by Widom from thermodynamic considerations.38 Nevertheless, there are a number of experimental findings suggesting a simpler description of this diameter without singularity.31-33 In these cases the small anomaly in the diameter curve that is predicted by eq 10 may be hidden by the experimental errors occurring within the critical region.31,32 Experimental Section Materials. A PEG 6000 from Merck-Schuchardt was used without further purification. The polymer chain distribution was tested by MALDI-MS (matrix-assisted laser desorption ionization mass spectroscopy). The plot of the distribution of the polymer so determined is presented in Figure 1. It is remarkable that the mass differences between single peaks coincide with the mass of one monomer unit of polymer (44.053 au). Tridistilled water was used in earlier series of measurements
without any additives. Considering a possible bacteriological degradation of PEG in water, in later experiments a small amount (4 mL in 1 L of solvent) of raschit solution (4-chloro5-methylphenol, 5% by wt in methanol) was added. The resulting change of the polymer weight fraction of the samples is assumed to be below 0.4% and has been neglected in the following. Measurements. Solutions are prepared by weighing. After 2 days the solutions become clear and homogeneous. Cylindrical glass tubes are filled with the sample solutions and are sealed by melting off the glass tubes. The tubes of one measurement series are brought into a LAUDA UD 20 thermostat filled with oil for heating. After a fixed temperature of the oil bath is reached, a constancy better than 0.2 °C above 100 °C is maintained. The temperature is measured in the bath using a PT100 resistance thermometer. The pressure increment in the sealed tubes due to heating is neglected with regard to the small influence of pressure on demixing found by other authors for the PEG/water system.3,4 The small amount of water that evaporates into the upper gas-filled volume of the tube is also disregarded. It causes a change of the volume fraction of the samples lower than 0.3%. The cloud-points have been determined by visual observation of the first cloudiness occurring on slow heating. In a few cases, including only solutions with low viscosity, a very small drop of a second phase has been observed at the bottom of the glass tube after a period of constant heating, although a pronounced cloudiness could not be found. The temperature of this measurement is then interpreted as the cloud-point temperature. Some samples were measured 2 or 3 times to prove the experimental findings. To ensure comparability, all these samples were carefully homogenized again at lower temperatures before a new measurement was started. In general the phase equilibria are reached after 48 h at constant temperature. Phase volume ratio measurements were executed using a cathetometer for the determination of the relative heights of the meniscuses of the phases, assuming constancy of the diameter inside the glass tubes. The bottom phase volume has been corrected for the spherical segment at the lowest part of the tube.16 The measurements of the interfaces of the liquid phases have not been corrected because these phase boundaries are not or only slightly curved. In the case of the upper liquid-gas meniscuses, the surface tension of the liquid causes a more distinct curvature. The measurement is performed at the lowest part of the curved meniscuses, and the results are not corrected, thus ignoring a very small amount of liquid above the measured height. The accuracy of the measurements of the length of the cylindrical columns of the coexisting phases is better than (3 × 10-3 cm. The polymer volume fractions of the solutions were calculated using their mass fractions and assuming a temperature independent ratio of the densities of PEG and water. The density values were taken from the literature.39 Results and Discussion The CPC of the PEG 6000/water system is presented in Figure 2. In a first approximation a polynomial curve of 6th degree is fitted to the experimental data, disregarding some data points that show large deviations. The circles belong to the first series of measurements, the crosses to the later, when raschit solution was added to the solvent. Obviously, all results are consistent and will not be treated separately in the following. The heterogeneous region is detected in the concentration range between φp ) 0.002 and 0.4 at roughly 143 °C. Between φp ) 0.1 and 0.2 the curve is remarkably flat with a threshold near
Poly(ethylene glycol)/Water Systems
Figure 2. Experimental CPC of the PEG 6000/water system and φp,φ)1 values calculated from phase volume ratio data: (O) experimental cloud-points of the first measurement series; (×) experimental cloudpoints of the second measurement series with raschit solution added to the solvent; (s) approximated polynomial curve of 6th degree; (4) concentrations φp,φ)1 calculated according to eq 1a; (- - -) regression curve for the φp,φ)1 values. The threshold of the CPC is found at φp,thres ) 0.1138 and ϑthres ) 131.57 °C calculated by means of the polynomial approximation of the CPC.
Figure 3. Plots of φ vs ψ for different temperatures ϑ: (s) linear regression curve of the values at 132.5 °C; (‚‚‚) same at 139.7 °C; (- ‚-) same at 144.75 °C.
φp,thres ) 0.11 at 131.6 °C. In binary systems the critical point coincides with this threshold. In multicomponent polymer solutions the threshold is expected on the left-hand side of the critical point, which is confirmed in the following.10 The multiple measurements of the cloud-points indicate that there is a limiting precision of 0.2 °C to reproduce cloud-point temperatures. Deviations of more than that are due to practical errors like a delayed detection of the cloudiness of the sample, extremely low values of turbidity, a missing constancy of the bath temperature, etc. The plots φ and Ψ for different temperatures are all straight lines according to eq 1a. In Figure 3 three plots φ(Ψ) out of many others are shown for the temperatures ϑ ) 132.5, 139.7, and 144.75 °C (ϑ denotes the temperature in degrees Centrigrade). The lines represent the linear regression curve fitted to the experimental data. The phase volume ratios used have been corrected as mentioned. By transfer of the interpolated concentration values φp,φ)1 into a concentration-temperature plot, an extension of the resulting curve toward the CPC is possible, illustrated also in Figure 2. Since the results of the different plots φ(Ψ) for dilute and concentrated initial polymer concentrations are identical
J. Phys. Chem., Vol. 100, No. 39, 1996 15995
Figure 4. Experimental CPC, extrapolated cloud-points, and corresponding coexisting concentrations calculated by means of the phase volume ratios. The results of the different evaluations for polymer concentrations below and above φp,φ)1 are drawn into one figure with a shift of the temperature scale. The temperature scale on the lefthand side is valid for the results of the φ(ψ) representation of samples with φp e φp,φ)1 (curve A). The temperature scale on the right-hand side is valid for the results of samples with φp g φp,φ)1 (curve B). The symbol O represents experimental cloud-points, b represents cloudpoints extrapolated from phase volume ratios, 9 represents corresponding coexisting concentrations, and solid and dashed lines represent regression curves.
within the margins of the experimental errors, in Figure 2 only the former results are presented. The φp,φ)1(ϑ) curve is slightly nonlinear. By use of the extrapolation of the φp,φ)1(ϑ) curve in Figure 2, the critical point of the PEG 6000/water system is determined to be at φp,c ) 0.148 and ϑc ) 131.64 °C. Similar results are obtained using the findings of the φ(Ψ) plots for large initial polymer concentrations. The critical concentration is then slightly shifted to φp,c ) 0.151. Both values are clearly different from the threshold concentration mentioned above. Near the critical temperature the scattering of the φp,φ)1(ϑ) values becomes more distinct, but there is not reliable hint for the existence of a bend in the φp,φ)1(ϑ) curve. The scattering of the points is due to the larger experimental uncertainties at temperatures near the LCST where only a small number of samples are demixed and the phase volume ratios are very sensitive to any deviations of the experimental conditions. Consequently, the lower critical point of the system is supposed to be at φp,c ≈ 0.15 and ϑc ) 131.64 °C. By means of a φ vs ϑ plot for different polymer concentrations that was used earlier to confirm critical concentrations,20,23 it could be shown that the critical concentration of the PEG 6000/water system is located slightly above φp ) 0.1488 and certainly below φp ) 0.1661. This finding, not presented here, confirms our above-mentioned result with φp,c ≈ 0.15. As already mentioned, the extrapolation of the function φ(Ψ) toward vanishing values of the phase volume ratio represents a general way to determine the gross polymer concentration of coexisting phases. The results of these calculations for different temperatures in the PEG 6000/water system are presented in Figure 4. By comparison of the presented data, it may be stated that the data based on phase volume ratio measurements and the experimental cloud-point data are in good agreement within only small margins of error. The calculated coexisting concentrations confirm that the system behaves like a binary one in the investigated temperature range with regard to the loci of the CPC and the shadow curve. This is also proved by an examination of the symmetry of the CPC with respect to the φp,φ)1(ϑ) curve. A diameter construction of the phase diagram shows that the calculated curves of coexistence on the left- and right-hand sides of the critical point
15996 J. Phys. Chem., Vol. 100, No. 39, 1996
Fischer et al.
Figure 5. Example of a double logarithmic scaling plot for the PEG 6000/water system. The calculated optimum critical coordinates are φp,c ) 0.1492 and ϑc ) 131.49 °C. The experimental cloud-points have been used that are described by different curves on the left- and right-hand side of the critical concentration. The symbol × represents log(φp′′ - φp′) vs log , 4 represents log(φp′′ + φp′ - 2.0φp,c) vs log , and solid lines represent linear regression curves; see text.
coincide with the CPC within the margins of the experimental errors. Consequently, the diameter curve is indeed the mirror axis of the CPC of the system as expected for a binary system. If the system can be treated like a binary one, the basic scaling laws introduced earlier could be used to calculate scaling exponents. The CPC of the system is then regarded to be the binodal. Calculations are performed using both the experimental cloud-point data and the data from measurements of the phase volume ratios. In a first attempt the coordinates of the critical point determined by means of the φp,φ)1(ϑ) curve in Figure 2 have been used. In a second step the critical temperature ϑc has been allowed to float toward the best value at which the data fit a straight line when using a double logarithmic representation due to eq 9. An analogous procedure has been performed with regard to the double logarithmic plot following eq 10 using the newly adjusted values of the critical temperature. In these calculations the left- and right-hand sides of the CPC with regard to the critical point are described separately. In the case of the experimental cloud-point data, polynomial curves are chosen to determine mean values free of experimental scattering. The data extrapolated from phase volume ratios are fitted by curves following an empirical relationship according to
ϑ ) B1(φpγ)B2 + B3 with γ ) ′ or ′′
(12)
In Figure 5 an example of these double logarithmic plots is presented. It is based on the experimental cloud-point data when the critical coordinates are readjusted as mentioned above. In Table 1 the resulting scaling exponents, the parameters A1 and A2, the ranges of the fits, and the assumed critical coordinates are presented. The crosses denote the curve following eq 9, whereas the triangles mark the diameter law. The evaluated data start at a distance of about 0.3 K away from the presumably real critical temperature in the case of the experimental cloudpoint data. The cloud-points extrapolated from the phase volume ratio data commence at about 0.6 K above the assumed critical temperature. The critical coordinates, which are calculated by approaching a straight line fit in the double logarithmic representation of the scaling laws, differ only slightly from the values calculated by extrapolation of the experimental data. Therefore, the adjusted values seem to be located within the margin of errors of the experimental data. The scaling exponent τ reaches values of about 0.34 in all different calculations, which is the expected value for binary low- and
high-molecular liquid mixtures.20,31,32,40 The corresponding parameter A1 equals values of about 1.24 with only small deviations regarding the different evaluation methods. All plots according to eq 10 do not fit the data to a straight line near the critical region. The calculated parameter values A2 and also the scaling exponent ν show distinct changes when using different critical coordinates. In the same way slight differences between the data sets, which are used for evaluation, cause large changes in the resulting parameters and exponents. In the case of the experimental cloud-point data after the readjustment of the critical temperature, the exponent ν reaches values close to zero, whereas the phase volume ratio data are described by an exponent value of about 0.16 and 0.14, respectively. The diameter law seems to be very sensitive to any shift of the critical concentration, so the accuracy of the available data may not be high enough to obtain consistent exponent values. Even the distance to the critical temperature, and therefore the lack of data very close to the critical concentration, is a possible reason for the uncertainties found. However, in all calculations the exponent (1 - ν) reaches values in the range ∼0.8-1.0 and therefore does not exceed the limits reported in the literature.31,32,34,40 Iteration Procedures. By use of the findings that are summarized above, iteration procedures could be performed to test some theoretical approaches. In all calculations a Poisson distribution for the PEG 6000 has been assumed based on the results of time of flight mass spectroscopy. The Poisson distribution is typical for short-chain poly(ethylene glycols) owing to their method of preparation.6,41 From the results of the time of flight mass spectroscopy, φi and ri data pairs are calculated that describe the discrete polymer chain distribution in the bulk polymer. With these mmd data and the results of the phase volume ratio measurements it should be possible to achieve thermodynamic parameters of the system by a modified least-squares fit. Equations 4-8 are valid for every set of φp′, φp′′, and ϑ values from phase volume ratio measurements. The left- and righthand sides of eqs 4 and 8 should be equal when introducing the experimental results. Only the function g(T,φp) is unknown and has to be given. Accordingly, the parameters of an assumed function g could be determined by minimizing the differences between the left- and right-hand sides of these equations for every set of φp′, φp′′, and ϑ values from phase volume ratio measurements. The quality of the fit is determined by the squared sum of these differences calculated for the whole data set. This sum has been minimized during the fit process. For the PEG/water system in the first step four slightly different forms of the interaction function g(T,φp) are tested. They are presented in Table 2 with the parameter values determined by the fit procedure. In this first approach a possible dependence of g on the polymer chain length is not considered but the equation of the critical state has been used as a further condition to adjust the results by means of the known critical coordinates. For the critical state the following equation has to be fulfilled:15
[
-
]
∂3{φ1φpg(T,φp)} ∂φp
3
)c.p.
jrz (rjw) φp,c 2
2
+
1 (1 - φp,c)2
(13)
Assuming an only temperature dependent g function, it reduces to that derived by Stockmayer,42 which is presented in the first row of Table 2. An adjustment of the miscibility gap cannot be achieved then because the critical concentration φp,c is only dependent on the weight and centrifugal average ratios of the molar volumes jrw and jrz. In all other cases one parameter of
Poly(ethylene glycol)/Water Systems
J. Phys. Chem., Vol. 100, No. 39, 1996 15997
TABLE 1: Parameter Values and Critical Exponents Derived from the Double Logarithmic Plots of the Scaling Laws range of the linear regression
origin of the data experimental cloud-point data experimental cloud-point data cloud-point data from phase volume ratio measurements cloud-point data from phase volume ratio measurements
used critical concentration φp,c
3.1623 × 10-3 < < 0.033 3.1623 × 10-3 < < 0.033 8.3541 × 10-4 < < 0.033 3.3555 × 10-3 < < 0.033 3.3555 × 10-3 < < 0.033 1.1487 × 10-3 < < 0.033
0.1483 0.1483 0.1492 (calculated value)
used critical temperature Tc/°C
parameter value
scaling exponent
131.64 131.64 131.49 (calculated value) 131.49 (calculated value) 131.49 (calculated value) 131.64
A1 ) 1.241 A2 ) 1.387 A1 ) 1.291 A2 ) 1.524 A2 ) 1.632 A1 ) 1.248
τ ) 0.334 ν ) 0.106 τ ) 0.346 ν ) 0.077 ν ) 0.056 τ ) 0.34
3.3705 × 10-3 < < 0.033 1.476 × 10-3 < < 0.033
0.1483
131.64 131.64 (calculated value)
A2 ) 1.113 A1 ) 1.248
ν ) 0.166 τ ) 0.34
3.3713 × 10-3 < < 0.033 3.3713 × 10-3 < < 0.033
0.1483 0.149 (calculated value)
131.64 (calculated value) 131.64 (calculated value)
A2 ) 1.114 A2 ) 1.207
ν ) 0.166 ν ) 0.14
TABLE 2: Different Expressions for the Function g(T,Op) and Their Parameter Values Determined by a Fit to the Experimental Data from Phase Volume Ratio Measurements
function g(T,φp)
ag ) ag01 +
bg ) bg01 +
cg ) cg01 +
dg
) dg0 +
φp,c )
ag02
T
(
rw (rz)0.5
)
rz
ag01 ag02
6bg1 ) -
+ cg1φp + cg2φp2
6cg1 - 6cg2 + 24cg2φp,c ) -
dg11
+
)
dg12
T (1 - dg2φp)
{(
6
dg11
+
(rw)2φp,c2
+
)
dg12
Tc
) 1.39 ) -272.35 K bg1 ) 0.21 bg02
rz
+
(rw)2φp,c2
g2(1 - dg2)(1 - dg2φp,c) d
the set has been calculated using the condition of the critical state during the fit process. In Figure 6 the experimental cloud-points and the calculated CPCs are summarized. The corresponding shadow curves do not differ significantly from the corresponding CPC. The curves that are calculated using the ag function of Table 2 are not presented. They do not approximate the experimental CPC even if parameter values are used that are taken from light scattering measurements of dilute solutions of a PEG 10 000 sample.43 The parameter values g01 and g02 determined by the fit of the data of the phase volume ratio measurements also do not lead to a reasonable description of the experimental CPC. Nevertheless, both calculated CPCs run through their minima at a volume fraction of about φp ) 0.05, which corresponds to a third of the experimentally determined critical concentration. By use of the concentration dependent g functions bg and cg of Table 2 and the parameter values listed there, the description of the
) 2.41 ) -730.53 K
bg01
1 (1 - φp,c)2
+ bg1φp
T
cg02
(
1
1+
T
bg02
parameter values determined
condition of the critical state12,15,23,24,27
) 1.39 ) -272.14 K cg1 ) 0.21 cg2 ) 0.000 97 cg01
1 (1 - φp,c)2
}
-4
cg02
rz
1 )+ 2 (rw)2φp,c2 (1 - φp,c)
) 0.19 ) 1.06 dg12 ) -177.10 K dg2 ) 0.46 dg0
dg11
miscibility gap becomes more reasonable because the fitted parameter values are adjusted by the critical condition. Since their CPCs coincide, they are both described by the dasheddotted curve in Figure 6. The loci of the calculated CPCs are near the locus of the experimental CPC, but the shape of the curves does not coincide with that of the experimental CPC. A distinct improvement of the quality of the calculated CPC is not reached by changing the concentration dependence of the g function from the linear type of bg to the squared type of cg. The closed form of the g function dg, which has been introduced by Koningsveld and Kleintjens, leads to a reasonable locus of the calculated CPC.27 But the shape of the calculated curve, which is the dotted curve in Figure 6, also deviates from the very flat one of the experimental CPC. Nevertheless, all g functions presented are restricted by the simple linear temperature dependence introduced and the relatively small temperature range observed. More complicated
15998 J. Phys. Chem., Vol. 100, No. 39, 1996
Fischer et al. and
δ2′ ) δ2′′ ) (1 - ν); C2′ ≈ C2′′ ≈
A2 φp,c
(14c)
With these relations the coexistence curve of the PEO/water system has been calculated. The parameter and exponent values based on the experimental cloud-point data and the readjusted critical coordinates are used, namely, A1 ) 1.291, τ ) 0.346, A2 ) 1.632, and ν ) 0.056. This leads to the solid curve in Figure 6, which describes the coexistence curve of the system up to about 12 K above the LCST very well and which is in this case the total investigated range. Conclusions Figure 6. Calculated CPCs on the basis of the Flory-Huggins theory in comparison with the experimental cloud-points and a scaling representation of the miscibility gap: (O) experimental cloud-points; (- ‚ -) CPC calculated using the interaction function bg and the parameter values presented in Table 2, where the interaction function cg yields an analogous curve; (‚‚‚) CPC calculated using the closed function dg by Koningsveld and Kleintjens with parameter values presented in Table 2; (s) scaling representation according to eqs 14a-c with parameter and exponent values from the double logarithmic plots of the scaling laws given by eqs 9 and 10. Assumed critical coordinates are the same as in Figure 5.
expressions are not considered yet because of the growing number of parameters that then have to be fitted. It seems evident that some extensions have to be introduced into the theory if the PEO/water system is examined. The association of the polymer chain segments with water molecules may especially complicate its thermodynamic properties.17 Therefore, the basic thermodynamic approaches examined above do not lead to a suitable description of this system. The coexistence curve of a binary fluid mixture can be described by a scaling representation around the critical point of the system.33,34,40 The coexisting concentrations φpγ are determined by γ
φpγ ) φp,c(1 + Cjγδj + ...) with j ) 1, 2, ... (14a) where γ denotes the phase indices′ and ′′, Cj and δj represent scaling parameters and exponents, and j is a running index starting at unity. It has been recognized that the symmetry of the classic scaling theory for fluids is lost in such systems.33,34 In the classic case generally the following features of symmetry have been found:
δj′ ) δj′′; Cj′ ) (Cj′′
(15a)
According to Stein and Allen who have investigated binary fluid mixtures, it may be supposed that at least
δ1′ ) δ1′′; C1′ ) -C1′′
(15b)
δ2′ ) δ2′′; C2′ ) C2′′
(15c)
In macroscopic terms the PEG 6000/water system with a narrow mmd of the polymer behaves like a binary one. This is proved for a temperature region close to the lower critical demixing point of the system by cloud-points and phase volume ratio measurements. The description of the phase diagram is based on the volume conservation equation. The extrapolation of the phase volume ratio data toward vanishing values of φ yields calculated cloud-points and their corresponding coexisting concentrations. In the case of the PEG 6000/water system the results confirm its nearly binary character. This is also indicated by a simple diameter construction, which yields a shadow curve coinciding with the measured CPC within the range of accuracy of the measurements. However, it becomes evident that the precision of the experimental results is still limited and has to be refined if possible. Considering the PEG/water system as a binary one, it obeys the scaling laws close to the critical coordinates with a scaling exponent τ of about 0.34, which is identical to those found in other low- and high-molecular binary fluids. Contrary to this, the law of the rectilinear diameters is only obeyed at a certain distance from a critical temperature larger than 1.4 K. Close to the critical point the double logarithmic plots of the diameter law are highly nonlinear. The value of the scaling exponent ν in this evaluation is strongly dependent on the supposed critical coordinates and cannot be fixed accurately. The PEG 6000/water system cannot be described satisfactorily by a simple Flory-Huggings treatment, even if the parameters of the assumed interaction function g are fitted to the available data of the phase volume ratio measurements. Assuming a Poisson distribution for the mmd, which is based on accurate results by the MALDI time of flight mass spectroscopy, four slightly different expressions of the g function are tested, all assuming an only linear dependence of the interaction function on the reciprocal temperature. The calculated CPCs do not coincide at all with the experimental data. Only a scaling representation using the parameter and exponent values determined by the scaling laws leads to a reasonable description of the miscibility gap of the system up to about 12 K above the LCST found.
and
are valid for both binary fluid mixtures and pure fluids.44 According to this, the scaling representation of the PEG/water system is restricted to j ) 1, 2 and the symmetry of eqs 15b and 15c is assumed. In combination with eqs 9 and 10 this leads to
δ1′ ) δ1′′ ) τ; C1′ ) -C1′′ )
A1 2φp,c
Acknowledgment. Thanks are given to the Deutsche Forschungsgemeinschaft (DFG) for financial support of the PEG/ water project, to Dipl.-Phys. A. Deppe for the preparation of the MALDI time of flight spectra, and to Dipl.-Chem. Dr. Ralf Ju¨schke for giving us the permission to use his regression program. We thank Professor Dr. R. Koningsveld for discussions and a critical review of the manuscript. References and Notes
(14b)
(1) Bailey, F. E., Jr.; Callard, R. W. J. Appl. Polym. Sci. 1959, 1 (1) 56.
Poly(ethylene glycol)/Water Systems (2) Nakayama, H. Nihon Kagakkai 1970, 43, 1683. (3) Saeki, S.; Kuwahara, N.; Nakata, M.; Kaneko, M. Polymer 1976, 17, 685. (4) Saeki, S.; Kuwahara, N.; Nakata, M.; Kaneko, M. Polymer 1977, 18, 1027. (5) Malcolm, G. N.; Rowlinson, J.-S. Trans. Faraday Soc. 1957, 53, 921. (6) Bekturov, E. A. Synthetic Water-soluble Polymers in Solution; Huethig & Wepf Verlag: Basel, Germany, 1986; p 135ff. (7) Haase, R. Thermodynamik der Mischphasen; Springer: Berlin, 1956. (8) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1978. (9) Koningsveld, R. Thesis, University of Leiden, Germany, 1967. (10) Rehage, G.; Mo¨ller, D. J. Polym. Sci., Part C 1967, 16, 1787. (11) Koningsveld, R.; Staverman, A. J. J. Polym. Sci., Part A-2 1968, 6, 305. (12) Gordon, M.; Chermin, H. A. G.; Koningsveld, R. Macromolecules 1969, 2, 207. (13) Koningsveld, R.; Staverman, A. J. J. Polym. Sci., Part A-2 1968, 6, 367. (14) Koningsveld, R.; Staverman, A. J. J. Polym. Sci., Part A-2 1968, 6, 383. (15) Koningsveld, R.; Kleintjens, L. A. Makromol. Chem. 1973, 8, 197. (16) Borchard, W.; Frahn, S.; Fischer, V. Macromol. Chem. Phys. 1994, 195, 3311. (17) Kjellander, R.; Florin, E. J. Chem. Soc., Faraday Trans. 1 1981, 77, 2053. (18) Casassa, E. F. Fractionation of Synthetic Polymers: Principles and Practices; Tung, L. H., Ed.; Marcel Dekker: New York, Basel, 1977; Chapter 1. (19) Freeman, P. I.; Rowlinson, J. S. Polymer 1960, 1, 20. (20) Borchard, W.; Rosskopf, G.; Rehage, G. Z. Phys. Chem. Neue Folge 1980, 119, 165. (21) Koningsveld, R.; Staverman, A. J. J. Polym. Sci., Part A-2 1968, 6, 349. (22) Koningsveld, R.; Staverman, A. J. Kolloid Z. Z. Polym. 1967, 218 (2), 114.
J. Phys. Chem., Vol. 100, No. 39, 1996 15999 (23) Koningsveld, R.; Staverman, A. J. J. Polym. Sci., Part A-2 1968, 6, 325. (24) Koningsveld, R.; Staverman, A. J. J. Polym. Sci., Part C 1967, 16, 1775. (25) Borchard, W. Ber. Bunsen-Ges. Phys. Chem. 1972, 76 (3/4), 224. (26) Wolf, B. A. Makromol. Chem. 1969, 128, 284. (27) Vanhee, S.; Kiepen, F.; Brinkmann, D.; Borchard, W.; Koningsveld, R.; Berghmans, H. Macromol. Chem. Phys. 1994, 195, 759. (28) Soˆlc, K. Macromolecules 1970, 3, 665. (29) Bohdanecky, M. J. Polym. Sci., Part C 1968, 23, 257. (30) Stanley, H. E. Phase Transitions and Critical Phenomena; Oxford University Press: Oxford, 1971. (31) Scott, R. L. Ber. Bunsen-Ges. Phys. Chem. 1972, 74 (3/4), 296. (32) Greer, S. C. Phys. ReV. A 1976, 14 (5), 1770. (33) Levelt-Sengers, J. M. H.; Straub, J.; Vicentini-Missoni, M. J. Chem. Phys. 1971, 54 (12), 5034. (34) Nakata, M.; Kuwahara, N.; Kaneko, M. J. Chem. Phys. 1975, 62 (11), 4278. (35) Estler, W. T.; Hocken, R.; Charlton, T.; Wilcox, L. R. Phys. ReV. A 1975, 12 (5), 2118. (36) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979; p 121ff. (37) Widom, B. Physica A 1993, 194, 532. (38) Widom, B.; Rowlinson, J. S. J. Chem. Phys. 1970, 52 (4), 1670. (39) Vennemann, N.; Lechner, M. D.; Oberthu¨r, R. C. Polymer 1987, 28, 1738. (40) Heller, P. Rep. Prog. Phys. 1967, 30, 731. (41) Brochure of the Chemische Werke Hu¨ ls AG Polydiole & Polywachse, 6th ed.; Hu¨ls AG: Marl, Germany, 1981; p 3. (42) Stockmayer, W. H. J. Chem. Phys. 1949, 17, 588. (43) Fraaije, V. Diplomarbeit, Gerhard-Mercator-Universita¨t GH Duisburg, Germany, 1991. (44) Stein, A.; Allen, C. F. J. Phys. Chem. Ref. Data 1973, 2, 443.
JP960791K
