1. Introduction
The role of pressure on the phase diagram of polymer liquids and also polymer mixtures has been intensely studied in the past decades, and there has been increased interest in the effects of pressure on the miscibility of polymers(An et al. 1997; An & wolf, 1998; Blaum & Wolf, 1976; Geerissen et al. 1985; Hammouda & Bauer, 1995; Hosokawa et al. 1993; Lefebvre et al. 2000; Maderek et al. 1983; Rabeony et al. 1998; Wolf & Blaum, 1976, 1977; Wolf & Jend, 1977,1978). One reason is the need for such data to more fully understand polymer miscibility in relation to the various proposed theories and equations of state. Another is the realization that such pressure effects could be important in many situations where such blends are used, e.g., when mixing a blend in an extruder or in forming articles from a blend by injection molding. These needs have led to the development of pressure cells that can be used with both light and neutron scattering such that the phase behavior and interaction strengths of blends can be measured. In past work, a wide range of phase behavior at chosen composition or near critical point of polymer solutions and polymer blends was found(Beiner et al. 1998, 2002; Blaum & Wolf, 1976; Hammouda et al. 1997; Janssen et al. 1993; Lefebvre etal 1999; Schwahn et al. 2001; Wolf & Blaum, 1977; Wolf & Jend, 1977; Zeman &Patterson, 1972; Zeman et al. 1972). There are also many works on the theories about the pressure effects on the thermodynamics of polymer liquid and blends(An et al. 1997; An & wolf, 1998; Dudowicz & Freed, 1995, 2006; Kumar, 2000; Patterson & Robard, 1978; Walsh & Rostami, 1985). As several outstanding problems remain unexplained in these blends, we decided to investigate the dependence on pressure, an independent thermodynamic variable.
The phase behavior of polymer liquids is commonly described in the terms of the lattice model of Flory and Huggins (FH), and the thermodynamics of typical polymer containing systems are understood in the framework of the incompressible random phase approximation. According to original FH theory, the rigorous incompressible system should be unaffected by pressure. In contrast to rigid lattice theories, equationofstate (EOS) theories are capable of predicting the thermodynamics of polymer containing systems.
Alternatively, the influence of pressure also can be deduced by calculating the pressure coefficient
Here x is composition (mole fraction), v _{M}, S _{M} and h _{M} are solution molar volume, entropy and enthalpy, and v ^{E} and h ^{E} (in equation 2) are the excess volume and enthalpy, respectively. Subscript c denote the critical quantities. The strict equalities in Eq. 1 simplify should v ^{E} and h ^{E} have identical functional forms describing their T and x dependences.
Eq. 2 is valid only at the critical composition. If, as is commonly so, h
^{E}>0, the sign of (dT / dP)
_{c} is governed by the sign of
The ClausiusClapeyron equation(Ma, 1982) describes the coexistence line of two phases in a TP plane according to
In this study we carried out cloudpoint measurements for binary system transdecahydronaphthalene(TD)/ polystyrene (PS) in a pressure range up to 800bar to determine coexistence curves and critical lines. The purpose of this paper is to test whether the SanchezLacombe lattice fluid theory (SLLFT) can describe the equilibrium behavior and pressure effects of polymer containing systems. The spinodals, the binodals, the FH interaction parameter, the enthalpy of the mixing, and volume change of the mixing for TD/PS system were calculated as a function of pressure, temperature and composition on the basis of the SLLFT.
2. Theoretical background
In the lattice fluid theory, as formulated by Sanchez and Lacombe(Lacombe & Sanchez, 1976; Sanchez & Lacombe, 1976), the energy of mixing for binary polymer containing systems is related to the Gibbs energy per mer (indicated by the double bar) of the mixture (index M) and that of the pure components (index1or2) by
According to SLLFT(Lacombe & Sanchez, 1976; Sanchez & Lacombe, 1976, 1978 ), the thermal equation of state and Gibbs energy per mer for the pure components and the binary mixtures are expressed in eqs 25, respectively(An et al. 1997; An & wolf, 1998),
where r represents the average number of lattice sites occupied by a molecule in the mixtures, rN is the total number of lattice sites occupied in the mixtures, and
In eqs 47, the different symbols have the following meaning:
According to the combining rules(Lacombe & Sanchez, 1976; Rodgers et al. 1991), the closepacked volume of a mer, υ ^{ *}, and the average number of lattice sites occupied, r, in the mixture are written by:
Therefore, total volume of binary mixtures becomes
where the symbols
k is the Boltzmann constant. The parameters
where
σ _{i} and σ ^{ 0} _{i} represent the symmetry numbers of the r _{i} mer and the r _{i} ^{0} mer and σ _{i} =σ ^{ 0} _{i} =1. z is the coordination number of the lattice which equals 12 in the present work.
For the pure component i, the total interaction energy per mer is
If there are specific interactions as described by Sanchez and Balazs(Sanchez & Balazs, 1989), the total interaction energy per mer becomes a function of temperature and is derived as following:
where
According to our previous calculations(An et al. 1997; An & wolf, 1998), the total interaction energy per mer should be linearly dependent on pressure besides as a function of temperature, i.e.,
where
where P
_{0} is the reference pressure (normally P
_{0} is chosen as 1 bar or atmosphere) and
On the basis of the FH lattice model(Flory, 1953), the FH interaction parameter,
where
According to our previous assumptions(An et al. 1997; Sun et al. 1999), eq. 24 can be rewritten by means of the SanchezLacombe (SL )theory,
3. Pressure induced phase separation of TD/PS polymer solutions
The cloudpoint temperatures (T_{cl}’s) of transdecahydronaphthalene (TD)/ polystyrene (PS, M_{w}=270 000) solutions were determined by light scattering measurements over a range of temperatures(1 16 ^{}C), pressures(100 to 900 bar), and compositions (4.221.6 v% polymer). The system phase separates upon cooling and T_{cl} was found to increase with rising pressure for constant composition. In the absence of special effects this finding indicates positive excess volume for the mixing. Special attention was paid to the demixing temperatures as function of pressure for different polymer solutions and the plots in Tϕ plane (where ϕ signifies volume fractions). The cloudpoint curves of polymer solution under pressures were observed for different compositions, which demonstrates that pressure has more effect on the TD/PS solutions that far from the critical point than that near the critical point. Figure 1 shows isopleths (i.e., lines of constant composition of the solution) cloudpoint curves observed for TD/PS polymer solutions on a temperaturepressure plane. The open circles indicate experimental points and the solid curves describe the behavior of the data points. Curves were measured at ϕ = 4.2, 8.4, 12.8, 17.2, and 21.6(v% at P=1bar), respectively. The clout point curves at each composition behave similarly to one another. The region above the curve is the onephase homogeneous region, while the region below represents the twophase region. As shown, the phase separation pressures increase with raising temperature.
With the preselected polymer concentrations, isobaric critical lines can be constructed from the isopleths in Figure 1. The thus obtained critical lines from 100 bar to 800 bar, respectively, show in Figure 2 on a Tϕ plane. In Figure 2 the coexistence boundaries are described on a Tϕ plane at the indicated pressures (bar)(the boundary under 1 bar was extrapolated from Figure 1). The shape of the coexistence curve depends only slightly on pressure, which can be recognized by comparing the curve at 100bar with the curve at 800 bar, and this system shows that it is an upper critical solution temperature (UCST) behavior system.
PVT Diagram
The PVT property of PS was determined with a PVT100 dilatometer. The experimental data was determined as a series of isotherms. Figure 3 presents the selected isobars with the interval of pressure 200bar and shows the specific volume,
Scaling Parameters
Fitting the PVT data that in the range between AB and CD (extrapolated to the glassy state temperature in Figure 3) and in the range between EF and GH (equilibrium melt state in Figure 3) to the SL equations of state, i.e., eqs 4 and 6, we obtained two series scaling parameters, each includes three scaling parameters
obtained from the glassy state temperature (the temperature range for the measurements of TD/PS phase behavior under pressure) of PS fit the experimental results better than that obtained from the equilibrium melt state (Figure 4a). The reason is that the phase separation behavior of the polymer solutions was measured in the temperature range that bulk PS is at glassy state temperature but in equilibrium state. The further calculations were based on the series scaling parameters obtained from the data of PS that extrapolated to the glassy temperature in Figure 3.
Pressure dependence of phase separation behaviors
From eqs 20, 21 and 23, the interaction Gibbs energy without specific interactions can be reexpressed into(An & Wolf, 1998)
where
T_{i}*/K  P_{i}*/bar 
 
TD  621  3110  0.935 
PS  714  4432  1.068 
PS*  725  4041  1.025 
Spinodals and binodals under pressure
Like the spinodals and binodals of ternary and quaternary polymer blends were calculated with the method(Horst 1995; Horst & Wolf, 1992) (the knowledge of the first and second derivatives of
Figure 5 shows the calculated results of the spinodals and the binodals compared with the experimental data as showed in Figure 2. The dashed and the solid lines represent the spinodals and the binodals calculated with the SL theory under indicated pressures, respectively; the solid circles represent the experimental data, respectively, which were obtained from Figure 2. In Figure 5 the qualitative agreement between the spinodals and the binodals calculated and the experimental cloud points is acceptable at different pressure. From Figure 5, it can be seen that the critical temperatures (
Interaction energy parameters under pressure
In previous paper (An et al, 1997; An & Wolf, 1998), the only one adjustable interaction energy parameter,
calculated by eq. 25, and then used for the phase diagram calculation under pressure. In this work, however,
FH interaction parameter
In the FloryHuggins theory model (Koningsveld & Staverman, 1968), the interaction parameter (γ) was considered as a function of composition and temperature. The FloryHuggins interaction parameter was dependent on the concentration of the components, temperature and pressure in the meanfield latticegas (MFLG) model and semiphenomenological treatment(van Opstal & Koningsveld, 1992). According to the results of our calculation, the FloryHuggins interaction parameter should be as a function of P, T, and ϕ for the system of TD/PS.
The FH interaction parameter,
Figure 7 shows the dependence of FloryHuggins interaction parameter on pressure and temperature against concentration. We can see that the FloryHuggins interaction parameter is linear increasing with the composition of PS increases. It also can be observed that the FH interaction parameter reduces with the increasing of temperature at the fixed composition and pressure, which agrees with the experiment result that this system exhibits UCST behavior. The effects of pressure and temperature on γ are identical with that derived by de Loos et al. for the systems polyethylene/ethylene(de Loos et al. 1983).
Figure 8 shows the plots of the FloryHuggins interaction parameters against temperature at fixed PS concentration (
From these results calculated, the relations of temperature, pressure, and concentration of TD/PS polymer solutions are presented. The influence of pressure on calculated FloryHuggins interaction parameter of TD/PS polymer solution shows that FloryHuggins interaction parameter increases on raising the pressure, i.e., miscibility is reduced.
Enthalpy of mixing
It is known that the Gibbs energy can be split into its enthalpic and entropic parts in the SL theory(Lacombe & Sanchez, 1976; Sanchez & Lacombe, 1978). The enthalpy of mixing per mer,
where
Volume change of mixing
As the ClausiusClapeyron indicates that the phase behavior of a mixture under pressure is governed by the sign of the excess volume of mixing; for polymer solution is the same. Therefore we will analyse the reduced excess volume or the volume change of mixing. In the SL theory (Lacombe & Sanchez, 1976; Sanchez & Lacombe, 1978), volume changes upon mixing are calculable. The volume change of mixing,
where
Figure 10 shows the temperature and polymer solution composition dependences of
In order to observe the pressure effects on the volume change of mixing for TD/PS polymer solution, we calculated the excess volume of TD/PS polymer solutions at 280K for various pressures. Figure 11 shows the volume change of TD/PS polymer solutions vs composition calculated with eq. 28 at 280K for the indicated pressures. We can know more clearly from Figure 11 that for low pressure the volume change is negative but becomes positive on raising the pressure. The change of sign occurs at about 100bar. At this pressure
4. Pressure induced miscibility in PEO/(P (EObDMS) mixtures
The cloudpoint temperatures (T_{cl}’s) of poly (ethylene oxide) (PEO) and poly (ethylene oxidebdimethylsiloxane) (P (EObDMS)) homopolymer and blockoligomer mixtures were determined by turbidity measurements over a range of temperatures (105 to 130 ^{}C), pressures (1 to 800 bar), and compositions (1040 wt% PEO). The system phase separates upon cooling and T_{cl} was found to decrease with rising pressure for constant composition. In the absence of special effects this finding indicates negative excess volumes. Special attention was paid to the demixing temperatures as function of pressure for different polymer mixtures and the plots in TФ plane (where Ф signifies volume fractions). The cloudpoint curves of polymer mixture under pressures were observed for different compositions. Figure 12 shows isopleths (i.e., curves of constant composition of the mixtures) cloudpoint curves observed for polymer mixtures on a temperaturepressure plane. The open circles indicate experimental points and the solid curves are just a guide for the eye. The curves were measured at Ф = 9.7, 19.4, 29.2, and 39.1(PEO v% at P=1bar), respectively. The region above the curves is the onephase homogeneous region, while the region below represents the twophase region. As shown, the demixing pressures increase with increasing temperature.
This binary system has been studied at four compositions under pressure. With the reservations concerning polymer concentrations, isobaric critical lines can be constructed from the isopleths of Figure 12. The thus obtained critical lines from 1 bar to 600 bar, respectively, show in Figure 13 on a Tϕ plane(the could points at P=1bar were measured with laser apparatus at atmosphere). In Figure 13 the coexistence boundaries are described on a Tϕ plane at the indicated pressures (bar). The shape of the coexistence curve depends
only slightly on pressure, which can be recognized by comparing the curve at 1bar with the curve at 600 bar, and this system shows that it is a UCST behavior system.
Scaling Parameters.
Within the framework of the SanchezLacombe theory, we set up the following combining rules for the molecular weight and the scaling parameters for P(EObDMS) block copolymer:
where
The scaling parameters for PEO were taken from the literature(Konowalow, 1903), those of P(EObDMS) were calculated with the chemical structure of P(EObDMS) and P(EObDMS)* were calculated with the new combine rules for the diblock copolymer parameters. In this paper, the parameters of P(EObDMS)* were used for the calculation as the parameters of block copolymer component.
In order to evaluate the new combining rules and the scaling parameters that obtained in Table 2, we used them for the calculations in SLLFT based on the experimental data in Figure 14 which obtained at atmospheric pressure. The calculated spinodals with the different scaling parameters and the measured data are shown in Figure 14. From Figure 14, it can be seen that the critical temperature (
Polymer  Mw/kg mol1  Ti*/K  Pi*/bar 

PEO  35.0  656  4922  1.178 
P(EObDMS)  1.8  609  4413  1.160 
P(EObDMS)*  3.0  622  4555  1.165 
Spinodals and binodals under pressure
Like the spinodals and binodals of ternary and quaternary polymer blends were calculated with the method(Horst 1995; Horst & Wolf, 1992) (the knowledge of the first and second derivatives of
Interaction energy parameters under pressure
In the reports(An et al. 1997; An & Wolf, 1998), the only one adjustable interaction energy parameter,
FH interaction parameter
In the FloryHuggins theory modified(Koningsveld & Staverman, 1968), the interaction parameter (γ) was considered as a function of composition and temperature. The FloryHuggins interaction parameter was dependent on the concentration of the components, temperature and pressure in the MFLG model and semiphenomenological treatment(van Opstal & Koningsveld, 1992). According to the results of our calculation, the FloryHuggins interaction parameter should be as a function of P, T, and ϕ for the system of PEO/P(EObDMS).
The FH interaction parameter,
Figures 17 shows the dependence of FloryHuggins interaction parameter on pressure and temperature against concentration. We can see that the FloryHuggins interaction parameter is almost linear increasing the composition of PEO increases at fixed temperature and pressure. It also can be observed that the FH interaction parameter reduces with the increasing of temperature at the fixed composition and pressure, which agrees with the experiment result that this system exhibits UCST behavior. The effects of pressure and temperature on γ are identical with that derived by de Loos et al. for the systems polyethylene/ethylene(de Loos et al. 1983).
Figure 18 shows the plots of the FloryHuggins interaction parameters against PEO concentration at fixed temperature (T=400K) under different pressures. It is shown that γ reduces with pressure at the fixed PEO concentration in the mixture, which is consistent with the experimental measured result that PEO/P(EObDMS) system shows pressureinduced miscibility behavior.
From these results calculated, the relations of temperature, pressure, and concentration of PEO/P(EObDMS) mixtures are presented. The influence of pressure on calculated FloryHuggins interaction parameter of PEO/P(EObDMS) mixtures shows that FloryHuggins interaction parameter deduces on raising the pressure, i.e., miscibility is enhanced.
Enthalpy of mixing
In order to investigate the pressure effects on
Volume Change of Mixing.
The volume change of mixing,
Figure 21 shows the temperature and polymer mixture composition dependences of
In order to observe the pressure effects on the volume change of mixing for PEO/P(EObDMS) mixture, we calculated the excess volume of PEO/P(EObDMS) mixtures at 400K for various pressures. Figure 22 shows
After calculating the
5. Abnormal pressure dependence of the phase boundaries in TL/ PEO/ P(EObDMS) ternary mixtures
The phase separation conditions for polymercontaining systems can change markedly as the systems pressured and the pressure effect on the phase behavior of mixtures consisting of polymers is of great technical and fundamental interest, and there are general observations on how the critical temperatures of polymer containing systems vary with pressure. The thermodynamic behavior of polymer blends is well understood in terms of the mean field FloryHuggins theory and their deviations near the critical point when thermal composition fluctuations become dominant (An & Wolf, 1998; Hammouda & Bauer, 1995). For those systems that display UCST behavior (that is, phase separation upon cooling), it is nearly always found that the critical temperature, T_{c}, increases with pressure, i.e., the effect of increasing pressure is to decrease the range of miscibility. Similarly, in mixtures that display LCST behavior (phase separation upon heating), T_{c} also increases with pressure; thus, increasing pressure almost always reduces the miscibility range in UCST systems and increases it in LCST systems. Most studies of polymer blends so far show an increase of the phase boundary with pressure which is quite obvious as the free volume decreases with pressure and thereby diminish the entropy of mixing(Janssen et al. 1995; Rudolf & Cantow, 1995; Schwahn et al. 1995).
The phase separation of polymer containing system is primarily driven by the reduced entropy of mixing as compared to small molecule analogs. This basic fact is captured by incompressible FloryHuggins (FH) theory. Note that the FH theory, being incompressible, would suggest that pressure is an irrelevant variable. In contrast to FH theory, as showed above, the experimental results proved that pressure can play an important role in the phase separation of polymer containing systems(Hajduk et al., 1995, 1996; Jiang et al. 2002).
The behavior of ternary polymer mixtures containing a diblock copolymer with homopolymer and toluene as a function of mixture composition and temperature were investigated to obtain experimental phase diagram for solvent/copolymer/ homopolymer mixture. In order to avoid the complications associated with the microphase separation of block copolymers, the molar mass of block copolymer was kept low in our experiment (Madbouly & Wolf, 2002).
The presently available experimental information concerning pressure effects on polymer containing ternary system reveals an abnormal phenomena: under demixing condition, (dT/dP) changed from positive to negative with different concentration of toluene (TL) in the ternary system. The purpose of this study is to determine the coexistence surface of the ternary system consisting of polymers with different pressures. The phase boundaries at normal pressure, chosen temperatures and the chosen compositions, various pressures are measured. Further, the pressure effects on (dT/dP) of ternary are compared with that of binary system without solvent.
The measured cloud point curves of polymer containing binary mixtures and ternary systems at normal pressure are shown in figure 23. Figure 23a shows the measured transition temperatures of PEO/P(EObDMS). The open circles indicate experimental points that got from the measurements of the transition temperature during the system changed from onephase to twophase, i.e. the heating process and the solid curve describes the behavior of the data points; the open squares indicate experimental points that got from the measurements of the transition temperature during the system changed from onephase to twophase, i.e. the cooling process and the dot curve describes the behavior of the data points. The lines are polynomial fits and serve as a guide for the eye (for other figures are the same). Figure 23b shows the cloud points of TL/PEO/P(EObDMS) ternary system at 35, 45, 55, and 60 ^{o}C under atmosphere pressure. The solid squares on the curves at 35 and 45^{ o}C represent the ternary system composition that were chosen for the pressure investigation, and the solid squares on the curves at 55 and 60 ^{o}C are the measured critical points at those temperatures and they were also investigated under pressure. It was found that toluene has much effect on the phase transition temperature of this ternary system, on the contrary, the shape of the coexistence curve depends only slightly on the ratio of PEO:P(EObDMS) in the ternary system, which can be recognized by comparing the curve at 35^{ o}C with the curve 60^{ o}C, so the position of cloudpoint curves depend on the concentration of TL in the ternary system.
The determination of the exact position of the critical point on the cloudpoint curve is a problem not only with multicomponent systems. In the case of binary mixtures the coexistence curves are often very flat in the neighborhood of the critical concentration is difficult to determine. It is established practice then to use the rectilinear diameter and assume that upon linear extrapolation the locus of the mean of the tie lines passes through the critical point. This implies that, if the volume fraction is used as the concentration variable, the volume ratio of the vanishing phases at the critical point should equal unity (Konowalow, 1903).
This does not mean that the diameter passing through the critical point is linear. For singlecomponent systems Bruhat has shown that the gasliquid diameter may be curved close to the critical point (Bruhat, 1930).
This argument can be extended to cover systems consisting of more than two components. Therefore, a necessary and sufficient criterion for determining whether a composition is the critical concentration appears to be the equality of the phase volumes as measured at a temperature very close to the cloudpoint of that concentration. As shown in figure 24, we just obtained the critical point for the mixtures at 55 ^{o}C and 60 ^{o}C which represented in figure 23b, however, at the lower temperatures, it is very difficult for us to observe the macrophase separation in this system even the experiment lasted for more than one month.
Figure 25 shows isopleths (i.e., lines of constant composition of the solution) cloudpoint curves observed for ternary polymer containing solutions on a temperaturepressure plane, and the pressure effect on the binary polymer mixtures has been mentioned in section 4. The open circles indicate experimental points and the solid curves describe the behavior of the data points. The region above the curve is the onephase homogeneous region, while the region below represents the twophase region. As shown in reference (Jiang et al, 2002), the demixing pressures decrease with increasing temperature for the binary polymer mixtures, and (dT/dP) of the binary system is negative, and this means that the pressure decreases the phase separation temperature of PEO/P (EObDMS) blends. Figure 25 shows the experimental measured phase transition temperatures under pressure for the compositions that showed in figure 23b as solid squares. As shown in Figure 25, the abnormal pressure effects on the ternary system was found, i.e., at low TL concentration of this system, pressure induces the compatibility, but for high TL concentration, pressure decreases the compatibility of this system.
With the reservations concerning the critical polymer concentration, indicated in the figure 23a, figure 26 was constructed from the phase diagram of PEO/P(EObDMS) binary mixtures as shown in figure 23a by plotting the (T_{H}T_{C}) vs. PEO WT% (T_{h} corresponding to the temperature of heating process, i.e., the spinodal temperature; T_{c} corresponding to the temperature of cooling process, i.e., the binodal temperature). The transition value of the (T_{H}T_{C}) was considered as the critical composition of this binary system.
The pressure effects on the demixing temperatures of polymer containing binary system and ternary system are shown in figure 27. Figure 27a shows the slopes of isopleths (dT/dP) in reference (Jiang et al, 2002) vs. PEO composition for PEO/P(EObDMS) binary mixtures. From this figure, we not only can recognize the degree of the pressure effect on the phase separation temperature of PEO/P(EObDMS) mixtures, i.e. dT/dP
The influence of pressure can be deduced by calculating the pressure coefficien (dT_{cr}/dP) of the critical solution temperature, which given by (Wolf & Blaum, 1977):
where
The same as classic thermodynamics, polymer thermodynamics is function of pressure, temperature and composite. But in many cases, pressure effects on polymer thermodynamics was neglected, because polymer thermodynamics were often studied under atmosphere. The classic theory of polymer thermodynamics is FloryHuggins hard lattice theory. In this theory, the hard lattice is incompressible. A rigorously incompressible system should be unaffected by pressure. However, since experimental results show that the critical temperature for polymer demixing system is strongly affected by pressure, it is clear that polymer containing systems show significant departures from this ideal limit. We wish above introduction would give readers a rough draft on the thermodynamics of polymer containing systems.
6. Acknowledgements
This work is supported by the National Natural Science Foundation of China (20974077，51073157) and subsidized by the Special Funds for National Basic Research Program of China (2010CB631102). We appreciated helpful discussion provided by Prof. Lijia An (Changhun Institute of Applied Chemistry) and Prof. Bernhard A. Wolf (Mainz University).