In the Laboratory
W
Achieving Absolute Negative Pressures in Liquids: Precipitation Phenomena in Solution Z. P. Visak, L. P. N. Rebelo,* and J. Szydlowski Instituto de Tecnologia Quimica e Biologica, ITQB 2, Universidade Nova de Lisboa, Av. da Republica, Apartado 127, 2780-901 Oeiras, Portugal; *
[email protected] Does negative pressure exist? There is more then one reason for doubting its existence. First of all, pressure is normally studied in the gas phase, thus neglecting liquids and solids. Second, the kinetic theory of gases defines pressure as the average force of impact per unit area that particles (molecules) apply to the walls of a container. Consequently, pressure must be positive. Furthermore, the common assumption that pressure is proportional to density makes it hard to imagine that it could be negative. For most people, the term “negative pressure” is reserved solely for pressures between zero and one atmosphere; that is, relative pressures with respect to the atmosphere. Nevertheless, negative pressures— more specifically, absolute negative pressures—do exist. Background Detailed descriptions of negative pressures have already appeared (1–6 ). In one of them (5), experimental as well as theoretical achievements with respect to reaching negative pressures in the liquid phase are reviewed. As for theory, the authors have shown that in the case of systems that cannot expand infinitely (condensed phases such as liquids and solids) negative pressure regimes are not forbidden. A simple van der Waals equation of state (vdW EOS) is used to establish and illustrate the range of temperatures and volumes within which a van der Waals liquid can exist under negative pressures. When considering negative pressures, the following must be taken into account. Negative pressures can only be achieved in condensed phases (liquids, solids, solutions of liquids in liquids and solids in liquids); gas phases cannot be put under tension. Small amounts of samples must be used; in principle, the smaller the volume, the higher the possible negative pressure. The liquids must be highly pure (free of solid impurities) as well as thoroughly degassed, so that only negligible amounts of air or other gases are present. The walls of the container where the liquid is confined should be well polished and dust free. Good adhesion (wettability) between the liquid and the walls of the container must be present. The condensed phase under negative pressure is a metastable one (superheated) and can thus easily “break” or “collapse” when subjected to any external disturbance such as vibrations or slight mechanical shocks. In the material below we explain the concept of absolute negative pressure in a very simple but accurate fashion. In contrast to quantities such as density, mass, and absolute temperature, there is no zero-limit value for pressure provided that the sample under study is a condensed phase (solid or liquid). Absolute pressure, as defined by thermodynamics, can achieve any value, positive or negative. The concept is more intuitively acceptable when one focuses attention on the solid phase. Imagine a metal rod, initially at atmospheric pressure,
which one starts to pull from either end. If the force applied is strong enough, it will inevitably induce absolute negative pressures in the bulk of the metallic rod, which will eventually lead to the fracture of the material. The same technique can be applied to liquids, although the strategy for achieving and maintaining that state is experimentally more complicated. If a liquid that fully occupies the volume of its container adheres well to its walls, then a slight expansion of the container’s internal volume will induce a state of static and isotropic negative pressure in the liquid. This could, conceptually, be performed using a cylinder with a movable piston. From a microscopic point of view, the liquid is in an expanded-volume state where the average molecular pair distance is slightly longer than that found at identical temperature but at some positive pressure. Alternatively, one can envision an experiment where the total volume of a closed system is kept constant (and thus its molar volume is also constant—isochoric line), but the temperature is significantly decreased to a value below the equilibrium temperature for that isochore at some normal positive pressure. This is called the Berthelot method. Experimentally, the latter methodology has been much more successful than the former. At some sufficiently low temperature one will observe the collapse of the pure liquid state. The system has relaxed to its totally stable state of liquid–gas equilibrium at that temperature (vapor pressure). Absolute negative pressures are thus metastable states. They constitute a specific subclass of superheating, one in which P < 0. The inherent thermodynamic limits of these states in a (P,T,V ) space depend upon the specific substance considered and are bounded by the so-called mechanical spinodal. The mechanical spinodal is the locus where (∂P/∂V )T = 0 and thus where compressibility diverges to infinity. In any pure substance there are two spinodal loci, the limit of liquid superheating and that of gas supercooling, which meet as a P–T cusp at a singular point1—the critical point. Here one additional thermodynamic constraint emerges, ( ∂ 2 P/ ∂ V 2)T = 0. Unfortunately, in practice, the experimental limit in pulling a liquid to negative pressures is more severe (lower tension) than its thermodynamic limit given by the negative pressure of the spinodal curve. Either homogeneous nucleation of gas bubbles in the bulk of the liquid or heterogeneous nucleation at the walls of the container (or at suspended solid impurities) will provoke premature collapse of the liquid. In the first case, as one “dives” deeper and deeper into the metastable state at negative pressures, density fluctuations can be large enough to trigger the formation and rapid increase of gas nuclei. This practical limit of stretching a liquid to its collapse is also commonly referred to as its tensile strength. Usually the weakness of the system lies at the liquid–solid interface (walls or suspended impurities) and the experiment ends well before the liquid’s tensile strength is reached.
JChemEd.chem.wisc.edu • Vol. 79 No. 7 July 2002 • Journal of Chemical Education
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transition curve
p
Once the concept of absolute negative pressure is accepted and understood, an additional question remains to be answered. How can it be measured? A relatively direct determination of negative pressure values can be achieved by combining the Berthelot method (explained above) with the Bourdon method for measuring normal pressure. If the container is a capillary glass helix of very small volume (typically, 0.001 cm3) sealed at both ends, then, as pressure is varied, the helix deforms in a fashion that depends on the sign of the pressure. Any measure of this helix deformation constitutes a pressure sensor. The device is calibrated against standard manometers in the positive pressure region, and it is assumed that its response in the negative region is a smooth extrapolation of the positive pressure behavior. This methodology has been successfully applied in the determination of P–T conditions of occurrence of loci of maximum density and negatively sloped melting lines (3, 7 ). In spite of the merit of permitting an almost direct determination of negative pressures, the small volumes do not permit one to visualize the sample. Here, the best and simplest alternative is the estimation of pressure through the use of the thermal pressure coefficient of the liquid (5, 6 ). All these concepts apply equally well to liquid mixtures and solutions. In these cases, there are additional phenomena that can be studied. For well-tuned (P,T,x) conditions of potential phase separation at negative pressures, these states of stretched liquids might provoke demixing phenomena. The current work illustrates that. Phase-separating solutions demonstrate negative pressure better than pure substances do, for in the latter case one cannot visually detect anything. In particular, polymer solutions scatter light strongly mainly upon phase separation. This is usually referred to as turbidity or cloudiness. Polymer solutions have an additional advantage. One can fine-tune the molecular weight of polymers to obtain the one-phase–two-phase transition line at experimentally convenient (moderate) negative pressures and temperatures. Several recent articles (6, 8, 9) have investigated liquid– liquid (L–L) equilibria in binary systems under negative pressure regimes. For polymer-plus-solvent mixtures, one can detect the occurrence of that regime by observing transitions at temperatures corresponding to shifts in a direction opposite to that in the positive-pressure regime. From a phenomenological point of view, liquid–liquid immiscibility is governed by the attainability of a critical value for the excess molar Gibbs energy, G E. Among binary mixtures of weakly interacting species, the most common situation is one in which immiscibility occurs as temperature is lowered (low-temperature demixing). This is because of the interplay between the normally endothermic (positive) excess molar enthalpy, H E, and the (typical) increase of entropy upon mixing (G E = H E – TS E). Therefore the enthalpic factor favors phase separation, while the entropic factor favors mixing. But the latter decreases significantly in importance as temperature decreases (TS E), and consequently, lowtemperature phase separation can occur. Pressure also determines the location of the phase transition, as it may either increase or decrease the G E value of the mixture. The key values are the magnitude and sign of the excess volume, V E = (∂G E/∂P)T . If a mixture expands upon mixing (V E > 0), pressurization might provoke phase separation (10). This is a very interesting phenomenon that
isochore
D
+ vapor pressure A
C
B G
T
F
– E
Figure 1. Graph of the course of a Berthelot-type experiment with a binary solution. ABC: heating process along the vapor-pressure curve; owing to expansion, at C (Tc = Tfill, the filling temperature) the liquid occupies the entire capillary. CD: isochoric heating, pressure rises at a much higher rate; on reversing the process by cooling (DCE), the liquid comes under tension when crossing C. At E a phase transition occurs (TE = Tcp, the cloud point temperature). At F the system has already collapsed and is returning to the stable condition (G). The shaded area corresponds to a two-phase (heterogeneous) region.
may even, theoretically, provoke limited solubility of binary gas mixtures (11). In contrast, for mixtures that contract upon mixing, pressurization helps miscibility; thus, phase separation can only potentially occur under conditions of stretched liquids at absolute negative pressure. Polymer solutions of weakly interacting species usually present a negative excess volume, for there is enough free volume for the solvent molecules to occupy their own space with no net volume cost. More about the basic types of liquid–liquid phase transition diagrams can be found in the work of Rebelo (10). In this work, we used a 1.80% (w/w) solution of polystyrene (Mw = 25,000) in acetone. Overview of a Berthelot-type of Experiment Our research group has undertaken a program of reaching absolute negative pressures and their direct determination by a Berthelot–Bourdon methodology (7 ). In the current set of experiments we performed only a Berthelot method. Negative pressures can be maintained for a considerable period of time. The method is based on the thermal contraction of the liquid and its adhesion to the walls of the container. Figure 1 presents the basic idea of the Berthelot experiment with the solution under negative pressures. As can be seen in the figure, part of the transition curve (dividing the one- and two- phase regions) lies within the region of negative pressures. This part of the curve is assumed to correspond to a smooth extrapolation of the portion in the positive pressure region. Therefore, when a system is under tension, it is possible to provoke a liquid–liquid phase transition. To achieve a phase transition at negative pressure in a polymer solution, one must carefully choose the system with regard to parameters such as polymer molecular weight, polymer concentration, and solvent quality. In this work we present original data for a 1.80% (w/w)
Journal of Chemical Education • Vol. 79 No. 7 July 2002 • JChemEd.chem.wisc.edu
In the Laboratory
solution of polystyrene (PS) of Mw = 25,000 in acetone. The experiment described here is relatively simple and inexpensive, as no special measuring equipment is required. A thermometer (accuracy ca. ± 0.1 K) and a heating plate equipped with a magnetic stirrer and a thermostat suffice. Detailed descriptions of materials and of preparation and precautions can be found as supplemental material.W Experimental Procedure and Results Figure 2 presents a series of six photographs and their corresponding stage points in a phase diagram (see also Fig. 1). The capillary is immersed in a water bath (beaker) at a temperature of about 23 °C. The solution is in the two-phase region of the phase diagram, under its vapor pressure (Fig. 2A). Phase separation, which is indicated by cloudiness or a milky appearance of the solution, can be observed very clearly. Next, the capillary is heated and the process follows the line ABC of Figure 1 (actually the vapor pressure of the solution, which
transition curve
is close to that of pure acetone for this low concentration). At 24 °C (the transition line), the cloudiness disappears. The system has just entered the one-phase region, restoring the homogeneous solution. Figure 2B shows a transparent homogeneous solution at 28 °C (point B in the plot). As heating continues, the liquid expands and, at a certain point, fills the entire capillary; the bubble finally disappears. The temperature at which this occurs is called the filling temperature, Tfill (Fig. 2C). In this experiment, the gas phase disappeared at 35 °C. This temperature is related to the size of the gas bubble trapped after the capillary is sealed. Sealing the capillary with a small gas bubble might prevent the capillary from achieving overly high temperatures (high Tfill). From point C, if one continues to warm the capillary, the liquid will evolve along an isochore (CD) where the pressure starts to rise rapidly (isochoric heating, prepressurization stage). For the solution, one can, to a first approximation, estimate the thermal pressure coefficient
p
p
isochore
+ A
transition curve D
+
vapor pressure
vapor pressure
D
A
T
T –
–
transition curve
transition curve
isochore
p
isochore
p + B
isochore
+
vapor pressure
vapor pressure
E
B
T
T
–
–
E
transition curve
transition curve
p
+ C
+
vapor pressure
vapor pressure
F
C
G
T –
isochore
p
isochore
–
T
F
Figure 2. Stage-presentation of the Berthelot’s experiment using a 1.8% (w/w) solution of 25,000 polystyrene in acetone. A: Two-phase region at positive pressure (vapor pressure of the solution); cloudiness appears from the bottom, but the bubble is present at the top. B: Transparent, homogeneous solution, recovered by heating. C: Capillary at the filling temperature (T = Tfill); bubble disappears. D: Final point of the prepressurization stage; filled capillary at moderately high positive pressure. E: Phase transition (cloud point) at negative pressure; cloudiness rises from the bottom, no bubble is present (T = Tcp < Tfill). F: Collapse of the metastable liquid at negative pressure, where a small bubble starts to form at the top, while the system is on its way to a totally stable state, at positive pressure (vapor pressure).
JChemEd.chem.wisc.edu • Vol. 79 No. 7 July 2002 • Journal of Chemical Education
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In the Laboratory
γV = ∂P ∂T
= ∂V ∂T V
P
∂V ∂P
1 T
=
αP KT
(1)
to be equal to that of pure acetone (low concentration of polymer). Using empirical equations for the expansion and compressibility coefficients, αP and KT, respectively (see 12 and 13), one finds, for the narrow temperature interval of interest for the present measurements, that V = (10.0 ± 0.2) bar K1, under orthobaric conditions. Along isochores, V is constant within experimental accuracy (14); therefore we used this value to estimate the pressure in both the positive and negative regimes. After a value for V is known, pressure can be calculated for any temperature T along the isochore:
Table 1. Data for 1.80% (w/w) Solution of 25,000 Mw Polystyrene in Acetone Tfill /K
Tcp /K a
P/bar b
300.7
298.7
20
302.6
300.0
26
303.7
301.2
25
305.0
302.1
29
306.9
303.4
35
308.2
304.2
40
313.0
308.2
48
aEstimated bEstimated
uncertainty is ± 0.2 K. uncertainty is ± 5 bar.
T
P = P fill +
T fill
γV dT ≈ γV T – T fill
(2)
Figure 2D shows the ultimate point of the prepressurization stage. The image of the capillary is similar to that in Figure 2C, but the temperature is higher, about 42 °C. According to the estimate that the average V of the diluted polymer solution is 10 bar K1, and assuming rigid walls of the glass capillary (neglecting deformation effects due to nonzero values for αP and KT of glass) the pressure in the capillary at this stage of the experiment is +70 bar. After reaching point D, one should keep the capillary under the achieved moderately high pressure for at least two hours. This is done to promote the disappearance of trapped gas bubbles in the crevices of the glass walls and to allow gases dissolved in the liquid to diffuse out. It also promotes good adhesion of the liquid to the walls of the container. From this point onward, the temperature is constantly decreased (path DCE, Fig. 1). Thus, one eventually comes back to the temperature of filling (point C, T = Tfill). This is an important stage of the experiment. If no gas bubble appears in the liquid after crossing below the filling temperature, then the system has just entered the superheated region, and soon afterwards it will enter the region of negative pressures. In other words, the liquid is stretched or under tension. As temperature continues to drop, at point E, when the temperature is equal to cloud point, Tcp, a phase transition occurs again, but now at absolute negative pressure. The term “cloud point” is used because of the cloudiness induced by the growth of the second phase in formation. A cloud point at about 31 °C was observed (four degrees below Tfill, therefore at an estimated negative pressure of about 40 bar). Soon the cloudiness fills most of the capillary. At this point, the system is even more metastable, owing to compositional heterogeneity. In Figure 2E one can easily see the second phase appearing from the bottom of the capillary and spreading to its top. As cooling continues, the system enters deeper and deeper into the two-phase region of this metastable regime. In practice, this stage is very difficult to maintain. At a temperature slightly lower than Tcp the liquid usually collapses and tension is released. The system returns to the totally stable one-phase region at vapor pressure condition (point G in Fig. 1). Figure 2F captures the instant (point F) between the collapse and relaxation stages (point G) and shows a small bubble starting to form at the top (T ≈ 30.5 °C).
872
Hazards While we have never witnessed any capillary fracture in the set of the current experimental runs, it is advisable to use a strong glass container serving as a water bath, together with a protective shield. Laboratory goggles and rubber gloves are recommended. Discussion This experiment managed to achieve several points in the negative pressure region (cloud-point temperatures, Tcp, with corresponding negative pressures of transition, P). These points are reported in Table 1 and plotted in Figure 3 along with data obtained at positive pressure. No comparison T–P data are available in the negative pressure region. Therefore, to certify the reliability of the observed data we included data determined at positive pressures, using the much more accurate light-scattering technique (15, 16 ). Here, positive pressures were not estimated but measured (accuracy ± 0.1 bar), and temperature was controlled (within ±0.005 K) using a bath of excellent accuracy. Thus, the data in the positive pressure region serve as reference data. The good match (Fig. 3) between the points at negative pressures and a smooth extrapolation of the positive pressure points is reassuring. Summary and Conclusions This paper describes an inexpensive and relatively easy way to demonstrate that absolute negative pressures exist in liquids and how they can be generated. It also reports phase equilibrium in polymer solutions under negative pressure. It presents a few results of the phase transition in the negative pressure region for a solution of 25,000 Mw polystyrene in acetone. The results are in good agreement with experiments carried out in the positivepressure region, which were performed using a much more precise and accurate technique (15, 16 ). The experimental demonstration of absolute negative pressures constitutes by itself a challenging topic within the broad general area of physical chemistry, even more so when this can be combined with measurements of properties in these regimes.
Journal of Chemical Education • Vol. 79 No. 7 July 2002 • JChemEd.chem.wisc.edu
In the Laboratory der Waals. The metastable region is located inside the binodal curve and outside the envelope defined by the spinodals. At sufficiently low temperatures the liquid isotherms cross the P = 0 line and their minima are located at P < 0, whereas those on the gas side of the diagram only approach the P = 0 value asymptotically at large molar volumes.
200
p / bar
150
100
50
Literature Cited
0
-50 270
280
290
300
310
320
330
T/K Figure 3. Experimental data in the (䊉) positive and (䊊) negative pressure regimes; 䊝 is the transition point described in this work.
Acknowledgments ZPV and JS are grateful to Fundação Ciência e Tecnologia (BD/1176/2000 and BCC/16424/98, respectively). This work was financially supported by PRAXIS under contracts 2/2.1/ QUI/178/94 and POCTI 34955/EQU/2000. W
Supplemental Material
Descriptions of materials and of preparation and precautions are available in this issue of JCE Online. Note 1. In the well-known (P,T ) projection of the phase diagram of a pure fluid, the gas spinodal lies above the vapor pressure curve and that for the liquid is below the liquid–gas equilibrium line. In the similarly well-known (P,V ) projection, the gas spinodal is the locus of all maxima of the subcritical isotherms and the liquid spinodal is determined by the minima of those isotherms. Again, the two spinodals meet at the critical point and define a dome-shaped envelope within which the system is in a totally mechanically unstable (negative compressibility) state. These subcritical S-shaped isotherms are predicted by any cubic equation of state, such as that of van
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