J. Phys. Chem. B 2000, 104, 4463-4470
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Thermodynamic Properties of Poly(ethylene glycol)/Water Systems. 2. Critical Point Data Volker Fischer and Werner Borchard* Angewandte Physikalische Chemie, Gerhard-Mercator-UniVersita¨ t, Duisburg, Lotharstrasse 1, D-47048 Duisburg, Germany ReceiVed: April 22, 1999; In Final Form: February 14, 2000
The critical coordinates of seven poly(ethylene glycol)/water systems with different molar masses of the polymer have been determined by demixing experiments. From the critical point data, values of the interaction parameters of the Flory-Huggins-Staverman-van Santen theory and their extensions have been calculated and the Θ-temperature of the system has been extrapolated. The consistency of the results is discussed with regard to the known findings of earlier investigations. The averages of the molar masses of the polydisperse polymers have been deduced from data of membrane osmosis, light scattering, and time-of-flight mass spectroscopy. Although all polymer samples are polydisperse, it has been found that the scaling law φp,c(r) ) K1rx -j with j ≈ 0.38 is obeyed within the range of the experimental errors. The scaling amplitude K1 is nearly unity. By using the determined Θ-temperature, another predicted scaling law, Θ - Tc(r) ) K2rx-k, has been tested. Although it is very sensitive to all scattering of the experimental data the exponent k is found in the range of reasonable values.
Introduction If a system shows a liquid-liquid demixing when changing one of the independent variables temperature T or pressure P, the solvent has to become poorer for the solute during this change.1 Considering isobaric conditions the solution will show a cloudiness at a certain temperature, which indicates the first appearance of a new phase. This point is called the cloud point of the solution. Normally, two new coexisting liquid phases are observed that are extremely different in concentration and composition: a dilute and a concentrated one. But when examining the whole polymer concentration range of such a system at different temperatures, there are also coexisting phases that are quite similar in concentration and phase volumes. Consequently, in the concentration-temperature plot one or more points exist at which the coexisting phases are really identical. These points are called the critical points of the system. In the case of a binary system they even represent its upper or lower consolution points.1 It is obvious that critical points belong to the line of coexistence called binodal and to the line of stability, the spinodal. Thus, several thermodynamic conditions have to be fulfilled at a critical point and therefore critical points are very important for the determination of the thermodynamic characteristics of demixing systems.1-4 All principles mentioned above are of course also valid for systems consisting of linear, nonelectrolyte, polydisperse polymers, and a single low molecular solvent.1-5 In such systems, especially the different particle sizes of the solute and the solvent molecules and also the increased number of compounds have to be considered, they cause, among other reasons, the somewhat different features of polymer solutions.2 Nevertheless, for the characterization of polymer/solvent systems the critical point data of systems differing only in the number-average molar masses of the polymers are very useful.2,3,6-8 To determine the critical points of polymer/solvent systems, liquid-liquid demixing experiments are often applied to * To whom correspondence should be addressed.
polymer solutions,7,9-16 and, although it is already known that the common Flory-Huggins-Staverman-van Santen (FHSS) treatment cannot describe polymer/solvent systems quantitatively, the FHSS theory is frequently used for the analysis of the data.3,5,14,17 To reach some enhanced quantitative results, a number of refined theories and also by way of comparison the scaling laws of critical phenomena have been introduced in the past to explain the thermodynamic behavior of different polymer/solvent systems.18-36 In the case of the system poly(ethylene glycol)/water (PEG/ water), a large amount of work has been done to reach a convincing and even quantitative thermodynamic description.9-13,16,37-40 [Up to a molar mass of about 20 000 g mol-1 it is usual to denote these polymers as poly(ethylene glycols). They are believed to be true glycols. To synthesize higher molar masses, more involved reactions have to be applied and this may lead to an increased number of possible end groups. Consequently, such polymers are often called poly(ethylene oxides) then. For simplicity in our paper all of these polymers are summarized as poly(ethylene glycols) (PEGs).] For chain lengths above 2000 g mol-1 the systems undergoes a liquidliquid demixing showing a lower critical solution temperature (LCST).9-13 There is experimental evidence of a miscibility gap of the closed-loop type in all systems consisting of polymers with nominal molar masses below 10 000 g mol-1.9-12 For higher molar masses the experimental proof of an upper critical solution temperature (UCST) becomes quite difficult due to the increasing temperature and concentration range of the miscibility gap. Therefore, for “real” polymers only the LCSTs have been determined up to now. In spite of the extensive experimental work the understanding of the thermodynamic properties of the system is not complete yet. The discussed solvating mechanism of PEG in water strictly prohibits the application of the simple FHSS theory to the system.20,37,41,42 The formation of hydrogen bonds between the polymer segments and water molecules has been proposed. Hydrogen bonds represent a kind of interaction that is relatively
10.1021/jp9913214 CCC: $19.00 © 2000 American Chemical Society Published on Web 04/15/2000
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Fischer and Borchard
strong including also local orientation. This does not correspond to the mean field approximation of averaged interactions proposed by the common form of the FHSS theory.43 There are a number of modifications to the FHSS theory and some new theoretical approaches that reflect the nonrandomness character of such systems. They also include the formation of hydrogen bonds.18-23,44,45 All these refined or new models may lead to reasonable descriptions of many experimental data, but unfortunately almost all of them still have been tested only for a few binary mixtures. Some of these theories even lead to results with more theoretical than practical benefit yet and they mostly fail to describe the systems in the vicinity of their critical points. Therefore, at this time the simpler lattice theories remain useful tools to reach a first, but in a way schematic understanding of any polymeric system. Theory In general, the liquid-liquid demixing even of a polymer/ solvent system is characterized by the conditions of a heterogeneous equilibrium.1-3,15 Neglecting the influence of the chain length on polymer/solvent interactions and that of end group effects the Gibbs free enthalpy of mixing ∆G h M of a system taken per generalized base moles of the mixture is2,3
∆G hM RT
N
) φ1 ln φ1 +
φi
∑ i)2 r
ln φi + g(T,φp)φ1φp
(1)
φ,i
where φi is the volume fraction of a component i. In the following, the index 1 denotes the solvent and an i g 2 indicates a polymer component. rφ,i is the ratio of the molar volume of a pure polymer component i with i g 2 and of the pure solvent, R and T have their usual meanings, and g is an interaction function3,8,17 which is only dependent on the temperature T and N φi. The equilibthe polymer concentration φp, with φp ) ∑i)1 rium state of demixing of constant values of T and pressure P is in a certain distance of the critical region determined by
∆µ′i ) ∆µ′′i, i ) 1, 2, 3, ..., N
(2)
for all components. ∆µiγ is the difference of the chemical potential of the component i in the phase with index γ and that of the pure one. The interaction function g is closely related to the interaction parameters χ1 and χp of the FHSS theory.2,3,8,17 In the case of the simple FHSS theory and for a system that can be described adequately by a concentration independent interaction function g(T), they even coincide.3 This situation is represented by the ag interaction function introduced earlier in part one of this paper.16 The critical coordinates of such systems are2
φp,c ) (1 + jrφ,n1/2)-1 Tc-1 )
[ (
1 1 1 1 + jrφ,n-1/2 + Θ ψ 2rjφ,n
(3)
)]
(4)
where Θ denotes the Θ-temperature of the system and ψ Flory’s entropy of mixing parameter of dilute solutions.2,5 rφ,n is the number average of all rφ,i. If the cloud-point curve (CPC) of a polymer/solvent system is known, it is possible to determine its critical point by phase volume ratio measurements.8,14-16,46 In the concentration-temperature diagram of the system the curve of the concentrations of equal phase volumes (or its extension) crosses the CPC and this intersection marks the critical point (CP) of the system.8,15,16 The concentrations of
equal phase volumes of the demixed system at different equilibrium temperatures may be evaluated by means of the phase volume ratios of a concentration series. This method has been described elsewhere in detail.15,16 Shultz-Flory Plots Following Flory the Θ-temperature identifies the point where a system consisting of a polymer of infinite chain length and a low molecular solvent is about to demix.2 The concentrationindependent interaction parameter χ1 of the unmodified FHSS theory reaches the value 0.5 then. Generally, the Θ-temperature marks a pseudo-ideal state of the system which is of theoretical and also practical interest.2,4,6,7 For a binary polymer/solvent system, Shultz and Flory proposed a method to determine the Θ-temperature by means of eq 4, using experimentally determined critical temperatures of systems only different in the molar mass of the polymer.6,7 Although eq 4 has been deduced in the limits of the FHSS theory and is restricted to systems showing an UCST, it has been applied to systems with lower critical points and to systems consisting of polydisperse polymers with narrow molar mass distributions (MMDs).5 Later, Stockmayer derived a more common expression that unfortunately includes a parameter set which is experimentally difficult to determine.5,47 Therefore, ψ can only be calculated from the critical point data in the scope of the FHSS theory by means of eq 4. Then the FHSS interaction parameter χ1 can be resolved into its entropic and enthalpic components via5
χ1,H ) -T
( )
( )
∂(χT) ∂χ , χ1,S ) ∂T ∂T
(5)
which can be also written in terms of the parameters introduced by Flory for infinite dilution:
lim (χ1,H) ) κ,
φpf0
lim (χ1,S) ) 1/2 - ψ
(6)
φpf0
where κ is the enthalpy parameter of dilute solutions.2 For all polymers that possess a distinct MMD, the critical point is shifted to the right-hand side of the maximum or minimum of the CPC. The extreme of the CPC is called the precipitation threshold. For the threshold temperatures TT, Shultz has developed an equation similar to eq 4 by using the results of Tompa’s work on heterogeneous FHSS systems and supposing exponential distributions of the polymers studied:6,7
1 ≈ TT
[ {[
(1 - 0.184f 1 1 1+ 1Θ ψ 4f 2
-7/6
]( ) ( )}]
)
1
(rjφ,w)
1/2
+
1 (2rjφ,w)
(7)
where
f)
M hw (M hw-M h n)
(8)
h w the weight average, with M h n the number average molar mass, M and jrφ,w the weight average of all rφ,i. Koningsveld-Kleintjens Plot The theory presented above deals with two shortcomings which are the polydispersity of the polymer samples neglected
Thermodynamic Properties of PEG/Water Systems
J. Phys. Chem. B, Vol. 104, No. 18, 2000 4465
in the evaluation of the critical data, and the introduction of an interaction parameter which is independent of polymer concentration. In the earlier paper, it has been shown that a more reasonable description of the miscibility gap of the system PEG 6000/water can only be reached, if a concentration-dependent g function is used.16 The parameter set of the interaction function g may be determined in a more general way by a Koningsveld-Kleintjens plot.3,17,48 Detailed descriptions of this technique can be found in the quoted literature. From the spinodal conditions and the conditions of the critical state on the one hand, and from similar expressions of the parameter sets of a given function g(T,φp) on the other hand, linearized equations of known quantities may be obtained. They allow the determination of some parameter values and in a second step lacking values are calculated by means of the original expressions. Normally, for simplification, in these calculations the temperature dependence of g is limited to only one of the parameters used.48 Two of the approaches to the function g depending on temperature and polymer concentration that have been discussed in the first part may be cited here once again using the same notation. A simple linear concentration dependence of g has been found to increase the thermodynamic representation of the system eminently. It has been presented as the function bg:16 bg
) bg01 +
bg02
T
+ bg1φp
(9)
A closed form of the g function has been introduced by Koningsveld and Kleintjens and is given as function dg:16,48
dg
) dg0 +
(
dg11
+
)
dg12
T (1 - dg2φp)
(10)
Scaling Laws of the Critical Coordinates of Polymer/ Solvent Systems In the recent years, a growing number of investigations have been focused on the scaling behavior of polymer/solvent systems. Among the complete set of critical exponents in discussion, especially the scaling law of the critical concentrations and the value of its exponent have been confirmed by a number of evaluations.28-30,33,34,49 For a polymer/solvent system the critical concentration scales with the chain length of the polymer in terms of its monomer unit via
φp,c(r) ) K1rx-j
(11)
where rx is the degree of polymerization defined by rx ) Mp/ M0. Mp and M0 denote the molar masses of the polymer and of the monomer unit, respectively, and j is a critical exponent of about 0.38 for real systems.26,30,32,35,50,51 For the critical temperature Tc another power law is predicted: 30,35,36,52
|Θ - Tc(r)| ) K2rx-k
(12)
In the Flory theory the exponent k is 0.5 for large N. Cherayil concludes from theoretical considerations that k is 0.41 ( 0.05 or even 0.46.26,35 In experiments values about 0.47-0.50 are reported.33,34,49,53 Experimental Section Materials. The investigated samples are industrial PEGs from Merck-Schuchardt (PEG 6000, 10 000, 15 000, 20 000), Hoechst
TABLE 1: Results of the Membrane Osmosis Experiments for the Different PEG/Water Systems sample PEG 15 000 PEG 20 000 PEG 35 000 PEG 100 000 PEG 1 000 000
temp of the measurements/°C
M h n/g mol-1
30 35 35 25 30 35 25 30 35 25 30 35
13 641 ( 210 14 755 ( 486 17 423 ( 120 34 210 ( 480 32 708 ( 781 34 081 ( 1,003 40 611 ( 498 41 460 ( 690 40 485 ( 474 216 296 ( 60 966 204 097 ( 38 633 204 156 ( 78 307
(PEG 35 000), Union Carbide (WSR-10 PEG 100 000), and Serva (PEG 1 000 000). The nominal molar masses of the low molecular samples were detected by the manufacturer by titration of the polymer end groups. All polymers were used without further treatment. In the samples of high molecular masses of 100 000 and 1 000 000 g mol-1 a certain amount of a nonsoluble residue was noticed after mixing the polymers with water. This residue was removed by centrifugation at 15 000g for about 2 h in a preparative centrifuge which may cause a small concentration error of about 0.5 wt %. Tridistilled water was used as solvent. To prevent biological degradation of the samples during the measuring period in recent measurements, a small amount (4 mL in 1 L of solvent) of Raschit solution (4-chloro-5-methylphenol, 5 wt % in methanol) was added to the solvent. The resulting change of the polymer weight fraction of the samples is assumed to be below 4 ‰ and was neglected in the following. Measurements. The experimental technique and the data evaluation concerning the phase volume ratio measurements were performed as described earlier.11,15,16 Any measurement was executed with a concentration series of the polymer/solvent system at constant temperature after the equilibrium state of the demixed solutions was reached. The measurements of membrane osmosis were performed with a Gonotec membrane osmometer at 25, 30, and 35 °C. The membrane used was a two-layer membrane supplied by Gonotec with an exclusion limit of about 5000 g mol-1. The samples were measured without purification and without addition of Raschit solution. Because of certain amounts of low molecular parts in the samples after any sample concentration measurement, the cells were rinsed several times with solvent to recover the baseline and to free the membrane from any amounts of polymer which might have been adsorbed. For another PEG WSR-10 from Union Carbide, which may be similar but not identical to our sample due to a fractionation process applied to it, Amu has reported a number-average molar mass of 37 640 g mol-1 determined by membrane osmosis at 25 °C.54 The polymer chain distributions were tested by matrix-assisted laser desorption ionization-mass spectroscopy (MALDI-MS) for all PEGs up to a nominal molar mass of 20 000 g mol-1. The molar masses determined by means of the time-of-flight mass spectroscopy were detected in a matrix of dihydroxibenzoic acid and ethanol. Technical details concerning this method have been published by Karas et al. and other authors.55,56 In the case of the PEG 20 000 at least a trimodal distribution, and in the case of the PEG 15 000 a bimodal one, have been observed. The spectra have been used as a model to create a simpler theoretical description of the MMDs. Discrete distributions have
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TABLE 2: Averages of the Molar Masses of the Samples, Determined by Different Experimental Methods and Evaluation Techniquesa PEG sample PEG 6000 PEG 10000 PEG 15000 PEG 20000
PEG 35,000 PEG 100,000 PEG 1,000,000
method TOF-MS LS TOF-MS MO TOF-MS TOF-MS* LS MO TOF-MS TOF-MS* MO LK MO LS LK MO LK
M h n/g mol-1
M h w/g mol-1
6096
M h z/g mol-1
6 197 13 000 ( 2100 11 615
10457 14198 ( 348 12161 14518 17423 ( 120 15767 17 602 33 666 ( 755 33 666 40 852 ( 887
1.02
12 375
1.11
15 481 16 463 16 200 ( 1500
18 004 18 369
1.27 1.13
18 887 19 561
20 859 21 067
1.20 1.08
40 399
48 479
1.20
558 137
3.70
3 309 930
4.00
151 000 ( 25 000 151 000
40 852 208,183 ( 59,302 208 183
U
6293
832 732
MO ) membrane osmosis, values of the molar masses are the averaged results of measurements at different temperatures. Errors are from the 95% confidential range of the linear regression of the evaluation. LS ) static light scattering, evaluated according to Zimm. The molar masses presented are the averaged results of measurements at temperatures of 20 to 70 EC.58 TOF-MS, TOF-MS* ) values from normalized data of the spectra of the time-of-flight mass spectroscopy; they are calculated by evaluation of the whole data range orsdenoted by the asterisksexcluding the data of chains with molar masses
