948
J. Phys. Chem. 1989, 93, 948-955
Pressure-Volume-Temperature Relations of Liquid, Crystal, and Glass of o -Terphenyl. Excess Amorphous Entropies and Factors Determining Molecular Mobility Motosuke Naoki* and Susumu Koeda Department of Textiles and Polymers, Faculty of Technology, Gunma University, Kiryu, Gunma 376, Japan (Received: July 2, 1987; In Final Form: February 24, 1988)
Pressure-volume-temperature relations of the liquid, the supercooled liquid, the crystal, and the glass are reported. From isochoric comparisons of the entropy of the amorphous states with the entropy of the crystal, the excess amorphous entropies are discussed with the help of the calorimetric data by Chang and Bestul. The ex- entropy of the glass is almost independent of volume and temperature, amounts to 1.4R, which consists of the intra- and intermolecular contributions, and is smaller than the residual entropy of the glass at 0 K. The excess entropy of the liquid considerably depends on volume and seems to be asymptotic to 4.6-5.OR at the zero density extreme near the room temperature. In a series of systematic tests on the factors determining the glass transition temperature and the dielectric relaxation time in the supercooled liquid state, various configurational quantities are examined. The most suitable factor is the configurational internal energy or entropy defined as the excess quantity over the crystal at constant temperature and pressure. These factors are expected to apply generally to the nonassociated liquids including polymer liquids.
Introduction o-Terphenyl can easily be vitrified at 244.5 K, and its supercooled liquid state is very stable when the sample has carefully been purified. A peculiar merit of this sample is that one can observe its liquid state, its supercooled liquid state, its crystalline state, and its amorphous solid state (the glassy state) near the room temperature. The pressure-volume-temperature (PVT)relations for each state are presented. The entropies of its various states differ from each other, but the difference is not necessarily a direct reflection of the difference in the spatial molecular configurations because of the strong volume dependence of each entropy. In the present work, we have attempted to compare the entropies at the same volume and to discuss the excesses of the entropy in the amorphous states, with the help of the calorimetric data presented by Chang and Bestul.' A series of systematic tests on the factors determining the glass transition temperature and the molecular mobility in the supercooled liquid state has previously been presented for poly(viny1 chloride)2 and a mixture of o-terphenyl and triphenylchloromethane (o-TP/TPCM).~ We will apply a similar test for oterphenyl, the simplest system of this series. Experimental Section o-Terphenyl was purchased from Tokyo Chemical Ind. Co. After repeated recrystallizations from methyl alcohol solutions, the crystalline sample was dried under vacuum for several weeks. The density of the crystal was measured by the floating method in NaCl/water solution. The test specimens of about 3-mm cubic crystal were cut from several bigger columnar single crystals (ca. 8 X 8 X 40 mm). The density of the supercooled liquid was also determined by a picnometer of volume 25 mL. The final densities of the crystal and the liquid are the averages of about 20 measurements. The PVT measurements of the crystal, the liquid, and the supercooled liquid were performed by using the mercury capillary dilatometer for V-T isobars at 1 atm and the pressure dilatomete# for V-P isotherms. As the sample vitrified through the glass transition region, the sample stuck to the Pyrex glass chamber and the dilatometer broke. Even if the dilatometer did not break, the high internal stresses stored in the glassy sample resulted in a poor reproducibility of the volumetric data. To avoid such internal stresses, we employed a thin rubber pouch that contained (1) Chang, S.S.;Bestul, A. B. J . Chem. Phys. 1972, 56, 503. (2) Naoki, M.; Owada, A. Polymer 1984, 25, 7 5 . (3) Naoki, M.; Matsumoto, K.; Matsushita, M. J . Phys. Chem. 1986.90,
4423. (4) Naoki, M.; Mori, M.; Owada, A. Macromolecules 1981, 14, 1567.
0022-3654/89/2093-0948$01.50/0
the sample and separated the sample from the Pyrex glass chamber and m e r c ~ r y .The ~ sample was loaded in the rubber pouch at 90 OC and degassed in a slight vacuum. Another difficulty was that mercury crystallized at -38.86 OC, about 10 "C lower than the glass transition temperature TBof o-terphenyl, and the region where we could carry out the volumetric measurements for the glassy state was limited. Reliable PVT data could not be obtained in so narrow (ca. 5 "C) a temperature region. In place of mercury, therefore, we employed a mixture of mercury and thallium (Hg/Tl, 91.5/8.5 weight ratio) whose crystallization temperature was about -60 OC. The Hg/Tl mixture was stored in an atmosphere of dried argon. Dilatometry at 1 atm was carried out at about 1.5 K intervals from 250 to 41 5 K for the liquid and the supercooled liquid and from 236 to 370 K through the melting temperature T, (ca. 330 K) for the crystal. The V-T isobar for the glass transition region was obtained by continuous cooling at a rate of -0.3 K/min from 310 to 218 K at 0.75 K intervals. Isothermal experiments to give the V-P relations were carried out at about 10 K intervals from 276 to 372 K for the liquid and at about 7 K intervals from 236 to 323 K for the crystal. To avoid the freezing of Hg/T1, we carried out the isothermal experiments for the glass at about 2 K intervals from 218 to 242 K under 0.1-29.4 MPa, from 222 to 242 K under 39.2 MPa, from 224 to 242 K under 49.0 MPa, and from 226 to 242 K under 58.5-78.5 MPa. The pressure was changed in increments of 9.8 MPa. The pressure vessel was immersed in an ethylene glycol/water or ethanol bath with temperatures were controlled to an accuracy of f0.03 K. The temperature of the inside of the pressure vessel was maintained within f0.008 K at each measurement. Silicone oil (KF-94, Shin-Etsu Chemical Co.) was used to transmit the hydrostatic pressure. The pressure was measured by a Heise bourdon gauge with an automatic compensator (Dresser Ind.), and the temperature by an alumel-chromel thermocouple inserted in the pressure cell. Illustrations and details of the pressure dilatometer and the manner for calculations of the specific volume were shown in the previous paper^.^^^ Experimental Results PVT Relations. From the density measurements, the value of the specific volume of the crystal, P,at 313.10 K is 0.861 03 f 0.00021 cm3 g-l, and the value of the specific volume of the supercooled liquid, V', at 303.15 K is 0.930 84 0.00004 cm3 g-l. In this paper, the superscripts c, 1, and g attached to the thermodynamic variables denote the crystal, the liquid including the supercooled liquid, and the glass, respectively. Temperature and
*
(5) Naoki, M.; Matsushita, M. Bull. Chem. Sot. Jpn. 1983, 56, 3549.
0 1989 American Chemical Society
No. 2, 1989 949
Figure 1. Variations of the specific volumes of the liquid, the supercooled liquid, the glass, and the crystal for o-terphenyl as a function of temperature. The thick solid lines are the representatives of the isobars under the elevated pressures. The iso-free-volume line is the intersection between the liquid and glass surfaces. The Clapeyron line is the trace of the melting temperature estimated as shown in the text. TABLE I: Parameters in the Polynomials for the Specific Volume (E4 1)' c(i,O) c(i,1) c(i,2) c(i,3) Crystal -3.232 X IO4 5.911 X c(0j) 0.818 11 -4.38 X 1.540 X 10" c(l j ) 5.54 X -3.812 X 1.134 X lo-" c(2j) 2.611 X c(0j) 0.741 19 c(1j ) 6.486 X IO4 c(2j) -3.125 X c(3j) 7.824 X c(0j) c(1j ) c(2j) a
0.921 39 -5.120 X lo4 1.6109 X 10"
Liquid 5.770 X lo4 -5.706 X 10" 1.136 X lo4 -1.431 X lo-"
-4.054 X 10" 2.8689 X 10" -5.831 X lo-" 8.328 X
-5.97 X lo* 6.436 X lo-" -1.605 X -1.479 X
Glass 1.0 X lo* -4417 X lo4 -2.00 X IO4 3.157 X 10" -9.51 X lo* 1.395 X 10''
V,T, and P are in units of cm3 g-l, K, and MPa, respectively.
pressure variations of the volume were calculated in reference to the volumes determined by the density measurements, i.e., p(7') and P(7')refer to V"(303.15), and P(T)to P(313.10). When the crystal was fused in the dilatometer, its volume agreed with the liquid value within the errors of f0.00015 cm3 g-'. The results are expressed by a polynomial in T and P: V = C&(ij)T'PJ i=oj=o
(1)
Values of the parameters, c ( i j ) ,are tabulated in Table I. The overall experimental errors including the instrumental uncertainties are collected in Table 11, which also shows the ranges to which eq 1 applies. In Table I11 the specific volume and the isobaric thermal expansivity, a = (a In V/aT), are compared with some literature values at 328.7 K ( T , obtained by Andrews and Ubbelohde).6 Our value of P is lower than the corresponding literature values.6-' The present specific volume of the crystal has been determined for the larger single crystals, and their values show more scatter. ag for the glass formed by the isobaric cooling varies from 2.22 X lo4 to 2.93 X lo4 K-' as the temperature changes from 220 to 240 K, and is considerably higher than that by Greet and
Turnbull.* Figure 1 shows the temperature variations of the specific volumes for the various states. To avoid the complexity, the data at elevated pressures are omitted, and only the isobars a t 39.2 and (6) Andrews, J. N.; Ubbelohde, A. R. Proc. R. Soc. London, A 1955,228, 435. (7) Clews, C. J. B.; Lonsdale, K. Proc. R. SOC. London, A 1937,161,493. (8) Greet, R. J.; Turnbull, D. J . Chem. Phys. 1967, 46, 1243.
TABLE 11: Uncertainty in the Specific Volume, the Thermal Expansivity, and the Isothermal Compressibility V,cm3 g-l a , K-I @, MPa-' temp region, K Atmospheric Pressure 236-323 crystal 2.2 X lo4 1.7 X liquid 8.0 X 2.8 X 255-415 21 8-242 glass 8.0 X 10" 2.6 X crystal 4.8 X IO4 liquid 3.1 X lo4 glass 3.8 X lo4
Elevated Pressures 7.9 X lo-' 3.0 X 10" 236-323 1.0 X 10" 3.9 X 10" 276-372 1.2 X 10" 4.5 X 10" 218-242 (0-30 MPa) 222-242 (39 MPa) 224-242 (49 MPA) 226-242 (58-79 MPa)
TABLE III: Comparison with the Literature Values at 328.7 K under Atmospheric Pressure lit. present work VC/cm3g-l 0.8671; O.86tlb 0.86451 f 0.00022 V'/cm3 g-I 0.9491,' 0.9488c 0.948 33 f 0.00008 1O ~ ~ C / K - I 1.33,' 1.97c 2.625 f 0.002 104~l/~-l 7.25: 7.49c 7.344 f 0.003 104~g/~-l 2.0c.d 2.575 f 0.003" OReference 6. bReference 7. CReference8. "At 230 K. 78.5 MPa are shown (thick solid lines). The atmospheric Tgwas determined from eq 1 as 244.5 K. The intersection between the liquid and glass surfaces in the PVT space is indicated as the iso-freevolume line, which is only a geometrical intersection and
not always equal to the real Tge9 Only the atmospheric Tgis the real Tgfor the glass vitrified by the atmospheric cooling as in the present case. On the both sides of the iso-free-volume line, variations of the volume along the line are identical. This leads to the relation" [dT/dPlisc-frw.volume
= AP/Aa
where AB and ACYare the discontinuities of the isothermal compressibility, P = -(a In V / d P ) ,and CY at the iso-free-volume line, respectively. This equation is alternatively derived from the assumption that the invariable quantity a t Tg is the free volume, Vf, which is characterized by the differences in the properties between the liquid and the glass.12 In other words, if Tg is (9) For instances, see ref 3 and 10. (10) McKinney, J. E.; Goldstein, M. J . Res. Nurl. Stand., Sect. A 1974, 78, 331. (1 1) Prigogine, I.; Defay, R. Chemical Thermodynamics; (translated by D. H. Everett) Longmans: London, 1954.
950
The Journal of Physical Chemistry, Vol. 93, No. 2, 1989
TABLE IV: Characteristic Temwratures (kelvin) of o-Terphenrl P / MPa 0.1 39.23 78.45 Intersections between Liquid and Glass Surfaces' VI and VBb 244.47c 256.9 270.2 S 1and SBd 242.0 248.8 255.0 H'and HS 242.0 249.0 255.5 Intersections between Liquid and Crystal Surfaces VI and V E SI and S v
146 202 174 170
H'and H C Vogel temp To/
137 208 178 177
136 213 181 185
"The calorimetric data are for the annealed glass in ref 1 . bThe iso-free-volume line in Figure 1. CThedilatometric glass transition temperature T8. dThe isoentropy line in Figure 6. 'The Kauzmann temperature TK.fAveraged Toin ref 21. 9
I
'
I
'
O 1 MPo-
Liquid I
39 2 MPa 78 5MPo-
b 3
I-
Glass
)
Crystal
't 0
1
i i
Naoki and Koeda by the least-squares analysis are as follows y1= 17.426 - 28.305V+ 11.813p yg
= 258.325 - 593.269V+ 341.831P
(3a) (3b)
+
262.408 - 854.790V 931.879P - 338.817V3 (3~) where y's are in units of MPa K-' and V in cm3 g-I. The employment of the cubic expression for the crystal is to avoid a monotonous increase in yc at higher volumes. 0 is calculated in two ways: one is the isothermal slope of eq 1, and the other from the relation 0 = a / y is shown in Figure 5 . The discrepancy at 78.5 MPa, the highest pressure, is remarkable for the liquid. Although eq 1 reproduces the volumetric data within the experimental errors, the polynomial may not be the best expression for the pressure variations of the volume. In the present case, this weakness turns up for the compressibility of the liquid under the highest pressure. Some appropriate expression such as the Tait equation is preferable for the pressure variations, but the expressions for the temperature variations become more complex without any improvement of the overall reproducibility of the volume. Entropy under Elevated Pressures. Following the familiar Maxwell relation, the change in S with P is identical with the change in V with T . Then the pressure variation of S can be calculated from eq 1: yc =
(4) Chang and Bestul have given the calorimetric data for two different samples of the glass, Le., the one referred to as the quenched glass vitrified by the cooling at a rate of about 6 K m i d , and the other, the annealed glass produced by annealing the quenched glass at 225 and/or 230 K for several days.' The difference in their entropies is about 0.5% at 220 K. Since the corrections with respect to the pressure variation by eq 4 is very small as seen in Figure 6, the experimental errors in eq 1 have scarcely exerted influence on the errors in the entropy. The entropy isobars are shown in Figure 6. The isoentropy line, which is an analogy of the iso-free-volume line in the PVT space, is an intersection between the liquid and glass surfaces in the pressure-entropy-temperature (PST) space. The line is only a geometrical intersection and does not necessarily correspond to I coincides with that of S g the real Tg. Since the variation of S along the isoentropy line, we have the relation
I 300 350 200
400
250
T /
K
Figure 2. Variations of the isobaric thermal expansivity calculated from eq 1.
governed by V, the iso-free-volume line should coincide with the glass transition line.21° The characteristic temperatures concerning Tg are summarized in Table IV. Some isobars of a are shown in Figure 2. There exist abrupt discontinuities between the glass and the supercooled liquid and between the crystal and the liquid. Since the temperature variation of a' is similar to that of a,, the increments at Tg and T , are in almost the same magnitude. The isochoric variations of P with T calculated from eq 1 are shown in Figure 3. Each isochore is almost linear and its slope, Le., the thermal-pressure coefficient, y = (dP/BT)", can easily be determined by the linear least-squares method. The second , the crystal is just zero in the present derivative, ( 8 2 P / 8 p ) y for experimental range, and that for the glass cannot be determined due to the limited experimental range. The second derivative for the liquid is small negative, but its quantitative determination is difficult due to the limited pressure range. The second derivative determines the volume dependence of the isochoric heat capacity, Cy,and we can roughly estimate its value for the liquid as 0.0023 MPa K-2 from the variation of the isobaric heat capacity, C,, with
P. Values of y are plotted as a function of volume in Figure 4. yc and y1decrease with volume, but y g increases rapidly with temperature. Such steep volume dependences of y* have been reported for some polymer glasses,13but the glasses of poly(viny1 chloride) formed under several pressure did not show such a volume d e p e n d e n ~ e . Since ~ the present PVT measurements in the glassy state are very close to the glass transition region, a slight volume relaxation, which was overlooked during PVT experiments, might take place at higher temperatures. The results determined (12) Ferry, J. D. Viscoelastic Properties of Polymers, 2nd ed.; Wiley: New York, 1961. (13) Nose, T. Polym. J . 1971, 2, 427.
where ACp is the discontinuity of C, at the line." This equation is alternatively derived from the assumption that the configurational entropy, S,, is invariable at Tg, which is characterized by the differences in the properties between the liquid and g l a ~ s . ' ~ ~ ' * It is well-known that this applies to many polymer^^*^^ and, in this case, the isoentropy line can be approximated as the Tg line in the PST space. Goldstein suggested a possibility of the configurational enthalpy, H,,for the factor determining the glass tran~iti0n.I~We have also calculated the enthalpy surfaces in the pressure-enthalpytemperature (PHT) space. The temperatures of the isoentropy and isoenthalpy lines are shown in Table IV. Clapeyron-Clausius Equation. The value of T, at 1 atm was determined as 329.35 K by Chang and Bestul.' T, at higher pressures can be determined on the basis of the ClapeyronClausius equation dT,/dP = AV,/AS, (5) where AV, and AS, are the latent volume and entropy at T,, respectively. From the values of T, and AV,/AS, at 1 atm, we obtained the provisional values of T,(P) under elevated pressures. At the beginning, values of AV,/AS, determined from the data of Vand S at each T,(P) disagreed with the slope dT,/dP. The (14) OReilly, J. M. J . Polym. Sci. 1962, 57, 429. (15) Goldstein, M. J . Chem. Phys. 1963, 39, 3369.
The Journal of Physical Chemistry, Vol. 93, No. 2. 1989 951
PVT Relations of o-Terphenyl I
'
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1
I
I
I
I
1
I
I
I
I
I
1
I
I
I
220
230
240
250
260
2iU
280
290
300
310
320
330
340
350
360
370
T
/
K
Figure 3. Pressure-temperature isochores for the crystal (0),the liquid (A),and the glass (0). The specific volumes in units of cm3 g-I are given in the figure. 1.91 ,
I
,
,
I
,
I
,
.a-
0 1 Mrn
.6.;--
Glass
.o %a0 O 082 6
084
086
om v
090
092
094
096
098
0.
220
XO
280
260
Figure 4. Variations of the thermal-pressure coefficient with the specific volume. Lines are from eq 3.
at
-1
K-I Smc/Jg-' &/.Ig-l K-I v,l/cm3 g-l vmc/cm3g-l AVm/cm3g-'
" Reference Crystal
300
250 T /
360
380
TABLE V Characteristics of the Melting
2cg-1K-'
200
340
Figure 6. Entropy isobars based on the calorimetric data by Chang and Bestul.' The isoentropy line is the intersection line between the supercooled liquid and annealed glass surfaces, and the Clapeyron line is the trace of the melting temperature.
0.1
0
320
300
T / K
/ cm3g-'
350
400
K
Figure 5. Variations of the isothermal compressibility with temperature. The solid lines are representative isobars calculated from eq 1 and the broken lines are the ones calculated from the thermal-pressure coefficient and the thermal expansivity.
calculation was then repeated to give the consistency between the values of AVm/AS, and dTm/dP. The results are summarized in Table V and shown as the Clapeyron line in Figures 1 and 6 .
329.352" 1.62750' 1.40087' 0.22664" d.94880 0.86466 0.08414
PIMPa 39.2 343.4 1.67323 1.44912 0.2241 1 0.93735 0.85902 0.0783
78.5 356.8 1.71656 1.49567 0.22089 0.9271 1 0.85371 0.0734
1.
Excess Amorphous Entropies Amorphous states contain an excess of entropy as compared with the corresponding crystal. From a microscopic point of view, the excess entropy excluding the contribution from volume dilation are much more interesting, because we can compare the configurations of molecules in each state at the same average free space condition (at constant We define the excess entropies of the liquid and glass as Si(T,V) = Si(T,V) - Sc(T,V) (6) V).'6317
(16) Oriani, R. A. J . Chem. Phys. 1951, 19, 93. (17) Naoki, M.; Tomomatsu, T. Macromolecules 1980, 13, 322.
952
Naoki and Koeda
The Journal of Physical Chemistry, Vol. 93, No. 2, 1989 S
........... ...... ...
4 -
P,
x 3 -
a \
-
mu
2 -
1
I
I
1 -
V
v' Figure 7. Schematic figure of the definitions of the excess entropies. Solid lines, experimental isotherms; dotted lines, extrapolations. vg
V C
5
0 0 95
100
I10
105 V-1
/
115
120
25
gcm-3
Figure 9. Isotherms of the excess entropies as a function of reciprocal specific volume (density). Open symbols, at the liquid volume or the glass volume; closed symbols, at the crystal volume. The square is the residual entropy at 0 K.I
independent of volume and a volume-dependent term. Since the entropy characterized by the frozen-in structure may be supposed as a very weak function of T and V from the present result, we specify the contributions to the entropies of the crystal and glass as
+ S$(T,l/)
(9a)
+ Sv'(T,l/) + S d
(9b)
SC(T,I/)= SkYT)
Sg(T,l/) = Skg(T)
c 0
i s '
h,
200
I
250
300 T /
,
f 350
K
Figure 8, Isobars of the excess entropies at 1 atm as a function of temperature. Solid lines, at the liquid volume; broken lines, at the crystal volume.
where i = 1 or g. When one wants to obtain the excess entropy at V = V', the entropy of the crystal should be extrapolated from F to and vice versa. This situation is schematically shown in Figure 7. The isothermal extrapolations of the entropy were achieved by integrations of the thermal-pressure coefficient expressed by eq 3 (7) where i, j = 1, g, or c. The excess entropies thus calculated for the atmospheric V', P,and V are shown in Figure 8 as a function of T and, in Figure 9, as a function of reciprocal specific volume (density). The small discrepancy at Tgin Figure 8 is due to the difference in the values of Tgobtained from the dilatometry and from the calorimetry. The excess entropy of the glass, Seg, is almost independent of T and V in the present experimental range. This suggests that the structure frozen-in at the glass transition gives a major contribution to Seg. Usually the entropy can be divided into two parts: one is a function of T only, and the other is a function of T and V. This is based on the experimental evidence that y for every state is well approximated as a function of volume only, as in the present case (Figure 3) [a(aP/aT)v/dTlv =
[ a ( a ~ / ~ v ) , / a T l v= [a(aE/al/),/aTl" = 0 (8)
which leads to the expressions in which the internal energy and the entropy are approximately regarded as a simple sum of a term
where S i is the kinetic entropy that is a function of T only, S;(T,l/), the volume entropy that corresponds to the so-called equation-of-state term, S d , the frozen-in structure contribution pertaining to the distorted lcttice structure of the glass. The present analysis gives Sd
= seg
(10)
This suggests that the contributions from the kinetic and lattice dilation are not largely different between the crystal and glass. The value of S d for the glass is about 1.4R, where R is the gas constant. This may be specified by two factors, Le., one from the intramolecular structure and the other from the intermolecular distortional configuration. The o-terphenyl molecule cannot turn into a plane structure due to the steric hindrance and the end phenyl groups tilt in the same direction in the crystalline state.' Then a possibility of two transient states of right-handed and left-handed tilts may be expected for the o-terphenyl molecule in the liquid state. These two types of the disks may randomly be frozen-in in the glassy state with the contribution of R In 2 . The frozen-in entropy pertaining to a number of ways of piling up the disks in the configurational space may further be expected as a common character of all sorts of amorphous materials, of which contribution is usually accounted as R In 2. Consequently one may expect S d for the glass to be 2R In 2 as the sum of the intra- and intermolecular contributions, which just agrees with the experimental value of 1.4R. It is very interesting that Sdfor the glass is smaller by about 0.4R than the residual entropy at 0 K indicated by the square in Figure 9.' If our estimation of the distortional entropy is approximately appropriate, the large amount of the residual entropy should come from the Sy term due to the difference in volume between the glass and the crystal at 0 K. This implies that y is finite a t 0 K. If one supposes the difference in volume between the glass and crystal at 0 K to be 0.038 cm3 g-' (the linear extrapolation of the atmospheric isobars), the value of y is estimated as about 0.4 MPa K-' at 0 K. The unusual large residual entropy of o-terphenyl' may be due to the contributions from the intramolecular isomerism and the significant volume imparity between the glass and crystal at 0 K.
The Journal of Physical Chemistry, Vol. 93, No. 2, 1989 953
PVT Relations of o-Terphenyl
i
TABLE VI: dT,/dP and ( a T / a P ) ,in Terms of Various Configurational Quantities 2,
Z,(definition) V f ( i ) ,V&i)
W ) S, , ( 4 Ec(i), E,(ii) TS,(i), TS,(ii) H c ( i ) ,H,(ii) Vdiii) S,(iii) E,(iii)
dT,/dP, (6T/6P), A@/Aa TVAa/ ACp ( T V A a + P V A @ ) / ( A C p- PVAa) T V A a / ( A C p+ S,) (TVAa - Vf)/ACp 1/Y1 TVy'/Cd (TVa' PV@')/(Cp'- PVal)
+
index 1
1
2 3
4
5 1'
2' 3'
The excess entropy of the liquid, S,', considerably depends on T and V a s shown in Figures 8 and 9. An excess of the liquid entropy over the glass one at the same T and V obtained by Sf(T,v) = S1(T,v)-Sg(T,v) Si(T,v) - S d (11) may reflect the fluid nature of the liquid. In the laiticelike liquid theories, Sfmay consist of the communal entropy and the entropies pertaining to the other degrees of freedom such as hindered rotations and vibrations. Kirkwood defined the communal entropy as the entropy difference between an unconstrained system and a restricted system having the center of each particle stay within its own cell and showed it to be a function of T and V.lS Hoover and Ree revealed that the communal entropy for hard spheres starts from R at the zero density, is nearly a linear function of density in the low-density region, and approaches zero near the solid density.lg Actually the plots of the liquid entropies of argon and carbon tetrachloride against the density seems to approach R at zero density.17 In Figure 9, S,' seems to approach asymptotically around 4.6-5.0R at the zero density extreme. If S d is independent of V in the liquid as well as in the glass, the value of Sfat the zero density is estimated as around 3.2-3.6R,19 which is much larger than the value R for the single molecular liquids. The liquid theory proposed by Rice predicts the communal entropy as 3R,20which is apparently close to Sf,though Hoover and Ree stated a criticism on the theory of Rice.19 Factors Determining the Glass Transition and the Relaxation Time in the Supercooled Liquid Following a series of an inclusive test on the factors determining Tgand the relaxation time T pertaining to the Brownian motion in the supercooled l i q ~ i d , we ~ , ~have applied a similar test to o-terphenyl by use of the dielectric resuk2' A more profitable point of c-terphenyl than the previous samples2v3is that the single crystal is available for o-terphenyl, and we can examine various configurational quantities concerned.22 In Table VI, the configurational quantities Z, and their expressions for dTg/dP and (dT/dP), reproduced from ref 2 and 3 are summarized. If Z , in the first column in Table V governs Tgor T,dTg/dP or (t3T/BP), should be equal to the relation in the second column. The definition of each configurational quantity indicated in the parentheses attached to Z, in the first column is illustrated in Figure 10, where TK is the Kauzmann temperature when the entropy surface of the liquid intersects that of the c r y ~ t a l , ~and ' To is the Vogel temperature when the dielectric relaxation time divergesaZ1Values of TK and Toare listed in Table IV. Z,(i) is the increment in Z of the liquid from the crystal, Le., Z,(i) = Z' - Zc,at the same temperature and pressure. Z,(ii) is the increment in 2 of the liquid from the hypothetical crystal (ii) of which thermodynamic second derivatives such as a,0, and C, are the same as those of the glass, Le., Z,(ii) = Z' - F ( i i ) . Z,(iii) is the increment in Z of the liquid from the hypothetical crystal (iii) of which properties are independent of T and P, Le., Z,(iii) = Z' - Zc(iii). (18) Kirkwood, J. G. J. Chem. Phys. 1950, 18, 380. (19) Hoover, W. G.; Ree, F. H. J . Chem. Phys. 1968, 49, 3609. (20) Rice, 0. K. J . Chem. Phys. 1944, 12, 1. (21) Naoki, M.; Endou, H.; Matsumoto, K. J. Phys. Chem. 1987,91,4169. (22) For instances, see ref 8, 12, 14, 15, and 23-26. (23) Davies, R. 0.; Jones, G. 0. Proc. R. SOC.London, A 1953,217,26. (24) Gibbs, J. H.; Dimarzio, E. A. J . Chem. Phys. 1958, 28, 373, 807. (25) Adam, G.; Gibbs, J. H. J . Chem. Phys. 1965,43, 139. (26) Goldstein, M. J. Phys. Chem. 1973, 77, 667. (27) Kauzmann, W. Chem. Rev. 1948, 43, 219.
3 C
r
T
4 4
--
:
,
---)
:
crystal(1 @ ~ I)
+
ZAid
I
----Hypothel lcal
crystal(h) I
r0
TK
Til
T,
T
Figure 10. Schematic figure for the definitions of the configurational quantities and the hypothetical crystals. Solid lines, experimental; dotted line, extrapolation; broken lines, hypothetical.
Absolute values of S, and Vf are required in the expressions for TS, and H, indexed as 4 and 5 in Table VI. We assume that S,(ii) of TS,(ii) indexed as 4 and Vf(ii) of H,(ii) indexed as 5 disappear at Toand estimate their values by integrations of ACp/T and A( Va)from However, the intersection between and HF is not identical with that between V' and P (Table IV), though the extrapolation is too long to state definitely. In H,(ii), therefore, some vagueness is included with respect to the supplemental assumption of simultaneous extinction of H,(ii) and Vf(ii). The results are summarized in Figure 11 (top) for 0.1 MPa and in Figure 11 (bottom) for 78.5 MPa. Slopes of the isofree-volume line in Figure 1 (eq 2a) and the isoentropy line in Figure 6 (eq 2b) are shown by the closed squares and triangles, which are equal to the corresponding quantities shown by the dotted lines indexes as 1 and 2, re~pectively.~ The value of dTg/dP determined from the differential thermal analysis (DTA) measurements by Atake and Ange1lZ8is also plotted, which is very close to the values of the dielectric (aT/dP),. For the viscous flow of the simple molecules with depressed intramolecular rotations, the free volumes Vf (indexed as 1 or 1') are expected to be suitable determining factors from the thermodynamic the0ries.2~ However, the Doolittle's type free volume, Vf(iii)(the broken line 1'),30 is very far from the experiments, and as is Vf(i)(the solid line 1). The Williams-Landel-Ferry (WLF) type free volume, Vdii) (the dotted line l), is better than the other free volumes, but the agreement is not as good as Vdii) for the o-TPITPCM m i x t ~ r e .For ~ o-terphenyl, Vdii) is always larger than the configurational entropy S,(ii) (the dotted line 2) in the present experimental region, that is, the iso-free-volume line is not identical with the isoentropy line. This leads to the experimental evidence that the glass transition of o-terphenyl cannot be approximated by the second-order transition of the Ehrenfest type,31 and the Prigogine-Defay ratio" is not unity. This conclusion is conspicuously different from the results on the o-TPITPCM mixture. The configurational entropies S, (indexed as 2 and 2'), which play an important role in the characteristics of polymer liquids in which the conformational entropy of intramolecular rotations holds above 50% of the latent entropy at Tm.17S,(i) and S,(ii) (indexed as 2) for o-terphenyl, especially S,(i) (the solid line 2), Le., the excess entropy of the liquid from the crystal at the same temperature and pressure, agree well with the experiments. S,(ii) (the dotted line 2), characterized by the difference between the
liquid and the glass, has been found to be a good factor for chlorinated p ~ l y e t h y l e n e ,poly(viny1 ~~ chloride),z and the oTP/TPCM r n i x t ~ r ewhereas ,~ a significant deviation is found for (28) (29) (30) (31) (32)
Atake, T.; Angell, C. A. J . Phys. Chem. 1979.83, 3218. Nose, T. Polym. J . 1971, 2, 437. Doolittle, A. K.; Doolittle, D. B. J . Appl. Phys. 1957, 28, 901. Davies, R. 0.; Jones, G. 0. Ado. Phys. 1953, 2, 370. Naoki, M.; Nose, T. J. Polym. Sci., Polym. Phys. Ed. 1975,13, 1747.
954
Naoki and Koeda
The Journal of Physical Chemistry, Vol. 93, No. 2, 1989
o-terphenyl as for the o-TPITPCM mixture, they are not very good factors as pointed out by Greet and T ~ r n b u l l . One ~ ~ may conclude that the Adam-Gibbs viscosity equation is not effective for the simple liquids. All H,(indexed as 5) proposed by Goldsteinls deviate far from the experiments. The discrepancy exceeds the vagueness of the determination of the value of V,. The results are summarized in the degrees of the agreements as follows: SJi) E,(i) > E,(ii) TS,(i) S,(ii) > Vf(ii) > H,(i) > TS,(ii) E,(iii) > H,(ii) S,(iii) Vdi) > Vdiii)
0 7 0 1 MPo
06
-
- -
--
-
For a summary of the results on the series of the examination in terms of dT,/dP and ( ~ T / c ~ P ) , ,the ~ , configurational ~,~~ internal energy or the configurational entropy may be one of the most suitable factors determining Tgand T of molecular liquids including polymer liquids. One may expect S,' or Sfin the last section as the factor governing T in the liquid. It is obvious that Sfis a major factor for the relaxation process pertaining to the Brownian motions, but it may be another problem that Sfis sufficiently enough for the factor. Since S d is approximately independent of T and P, S,] and Sfgive the same expression for (aT/dP),:
(3T/ aP)r = -(aSf/ap) I
0
230
2 50
r 07
I
270
260 i
I
1
7 8 5 MPa
~ '
;
(aS,/ aT)P
(12)
The value of (aT/aP), in eq 12 is very close to the value for S,(ii) (the dotted line 2) at 0.1 MPa and slightly higher than the latter at the higher pressures. Consequently a drastic improvement is not expected by the employment of the isochoric excess entropies of the liquid. Following the entropy separation in the last section, S,(i), one of the best factors, is expressed as
,
240
T/
'
S,(i) = [Sc]T + S d zz
[SI], + S d
+ SdT,Vl)
+ Sf(T,V)
(13)
The first terms on the right-hand side are construed as the pure lattice dilation from VC to V' relating the potential energy and the average free space of each particle in every cell. The present result suggests that the lattice dilation term plays an important role in the relaxation process in the supercooled liquid state.
i
0 31
-c-
z: . : : 2 . . : : ........ : : : : : . . : : : . : : : : . : : : : . : : . : 7 : .
i _ _ _ - - - -- - - -
I
o .i
01 259
3
m 5 . . . : : : ..................~..--5."..~...~.rrr..=.= _ - - - - .."........ ............
'
'
260
= I'
-
I
270 T
280 /
290
300
K
Figure 11. Examinationsof the factors (top) at 0.1 MPa and (bottom)
at 78.5 MPa. Solid lines with index numbers at left, the configurational quantities of definition i; dotted lines with index numbers at center, of definition ii; broken lines with index number at right, of definition iii. Indexes of the lines are indicated in the last column of Table V. (0), (aT/aP), from dielectric measurement;" (0),dT,/dP from DTA measurement by Atake and AngelI;28(H), slope of the iso-free-volume line; (A),slope of the isoentropy line.
o-terphenyl. It should be emphasized here that the calorimetric data for o-terphenyl is more reliable than those for the other samples interested here. E,(i) (the solid line 3) also agrees well with the experiments. E, (indexed as 3 and 3') is identical with the corresponding S, at the zero pressure. The Adam-Gibbs parameter TS, (indexed as 4)25was found to be one of the best factors for'polymer l i q ~ i d s ,but, ~ - ~for ~
Conclusion Conclusions on the analysis of the excess entropy of the amorphous solid (glass) may be summarized as follows: (1) the amount of S d is about 1.4R, almost independent of V and T; (2) sd is smaller than the residual entropy at 0 K, which suggests that, at 0 K, P may be larger than V and y may be finite; (3) S,Jmay consist of the contributions from the intramolecular isomerism and the intermolecular space packing. Conclusions on the analysis of the excess entropy of the liquid may be summarized as follows: (1) Sfstrongly depends on V and T; (2) the amount of S f is about 3.2-3.6R at the zero density extreme near the room temperature; (3) if one sets R aside for the communal entropy following the Kirkwood's theory,'* the other contributions amount to 2.2-2.6R, and if one sets 3R following Rice's theory,20 the other contributions amount to 0.2-0.6R. The present resolution of the amorphous entropy may give only crude estimations with the long extrapolations, and in addition to that, some of the separations of the entropy might be inappropriate. The analysis is constructed on the basic assumption that the volume entropy (the equation-of-state term) is nearly common to all of the states at the same volume and temperature, i.e., that the functionality of the intermolecular potential energy is not much different among the supercooled liquid, glassy, and crystalline states. From the tests in terms of dT,/dP and (aT/aP),, the most suitable factor determining the glass transition and the relaxation time are E, and S, defined as the increments to the crystal, and secondary are those defined as the increments to the hypothetical (33) Greet, R. J.; Turnbull, D. J . Chem. Phys. 1967, 47, 2185.
J. Phys. Chem. 1989, 93, 955-961 crystal containing the glassy characteristics. V, defined as the increment to the hypothetical crystal is not as good a factor for o-terphenyl as for the o-TPITPCM mixture, and the glass transition of o-terphenyl cannot be regarded as a quasi-equilibrium second-order transition of the Ehrenfest type. The Adam-Gibbs parameter is not effective for o-terphenyl. For a summary of the present we may be allowed to state that the thermody-
955
namic factor governing the molecular mobility of molecular liquids including polymer liquids is the configurational internal energy or the configurational entropy. Acknowledgment. We are grateful to Hironobu Okumura and Kouji Ujita for helpful correspondence. Registry No. o-TP, 84-15-1.
Triple Ion Formation in Acetonitrile Masashi Hojo,* Tohru Takiguchi, Minoru Hagiwara, Hironori Nagai, and Yoshihiko Imai Department of Chemistry, Faculty of Science, Kochi University, Kochi 780, Japan (Received: December 22, 1987; In Final Form: July 1 1 , 1988)
The effect of halide ions on the cathodic waves of cations was examined by means of dc polarography in acetonitrile. The formation constants of Li+(Cl-), (n = 1-4), NH4+(C1-), ( n = 2 and 3), Et3NH+(X-), (X = C1 or Br, n = 1 and 2), etc., were obtained. The present and the previous polarographic studies indicated the symmetrical formation of triple ions from trialkylammonium halides, Le., Kz = K3, in the following reactions: M+ + X- s M+X- ( K J , 2M+ X- s (M+)zX- (Kz), and M++ 2X- is M+(X-), (K3),where M+ stands for R3NH+(R = Me, Et, and n-Bu) and X- is CI-, Br-, or I-. The conductivity data of R3NH+X-in acetonitrile were quantitatively explained by the triple ions in addition to the ion-pair formation. Assuming that the limiting equivalent conductivity of the triple ions, AT, is one-third of that of the simple ions, Ao, the calculated A values of Et3NHCl ((0.4-6.0) X M) with K, = 2.8 X lo4 and Kz = K3 = 3.0 X lo6 fitted the observed values within 0.9% error, while the error was more than 9% at higher concentrations when only ion pairing was considered. Me3NHCl and n-Bu3NHClgave similar results. In the cases of bromide salts, less than 0.9% error in A values was given by the assumption of AT = &/2.
+
Introduction In a previous study,' we have examined the polarographic anodic (mercury dissolution) waves of halide ions in the presence of Li+, benzoic acid, p-bromophenol, and R3NH+ ( R = Et, etc.) in acetonitrile. The formation of the (M+),,X- type species (M+ = Li+, benzoic acid, etc.; n = 1 or 2; X = C1, Br, or I) was confirmed by a newly developed method. In the new method, the formation constants of "complexes" were evaluated by the positive shift in the half-wave potential of the mercury dissolution wave from a base (L) in a large excess of (Lewis) acids." The method was first proposed by Reilley et ale7in 1956 with a limiting feature for metal-EDTA (ethylenediaminetetraacetic acid) complexes in aqueous solutions. Casassas and Estebad presented a "general" equation for the method. Unfortunately, however, they introduced rather unrealistic conditions to the equation, which restricted the proper applications of the m e t h ~ d . We ~ have extended the new method further through the utilization of the cathodic wave of [HgLI2+instead of the mercury dissolution wave of Le4" Approximately the same formation constants of the (M+),J- species as those by the anodic wave of X- were obtained by the positive shift in the Ellz of the second cathodic wave of HgXz on the addition of the (Lewis) acidseg The effect of the (Lewis) acids on the half-wave potentials of two cathodic waves of CuClz and CuBrz gave similar result^.^ In the first part of the present paper, the formation of M+(X-), (M+= Li+, NH4+, Et3"+, C5H5NH+,etc.; X = C1, Br, or I; n = 1-4) type species in acetonitrile will be reported. The usual polarographic method is employed to determine the formation constants of the complexes: the negative shift in E l / 2 of the cathodic wave of the M+ cation on the addition of a large excess (1) Hojo, M.; Nagai, H.; Hagiwara, M.; Imai, Y. Anal. Chem. 1987, 59, 1770; Chem. Lett. 1987,449. (2) Hojo, M.; Imai, Y. Bull. Chem. Soc. Jpn. 1983, 56, 1963. (3) Hojo, M.; Imai, Y. Anal. Chem. 1985, 57, 509. (4) Hojo, M.; Imai, Y. J . Electroanal. Chem. 1986, 209, 297. (5) Hojo, M.; Hagiwara, M.; Nagai, H.; Imai, Y. J . Electroanal. Chem. 1987, 234, 251. (6) Hojo, M.; Imai, Y. Anal. Sei. 1985, 1 , 185. (7) Reilley, C. N.; Scribner, W. G.; Temple, C. Anal. Chem. 1956,28,450. ( 8 ) Casassas, E.; Esteban, M. J. Electroanal. Chem. 1985, 194, 11. (9) Hojo, M.; Sasayama, H.; Izumida, H.; Imai, Y., to be published.
0022-3654/89/2093-0955$01.50/0
of X- was utilized. We have proposed the formation of alkali-metal complexes with carboxylate ions: cyclic polyamines (cyclam6 and tetramethylcyclams), and acyclic polyamines, such as ethylenediamine and triethylenetetramine? in acetonitrile by the conventional and the newly developed methods. The coordination numbers of these complexes (Li', Na+, or K+ complexes) were all found to be 4. Now, it will be probable that four halide ions, especially chloride ions, coordinate an alkali-metal cation. Brooker'O stated that the coordination number of lithium ion has been the subject of several studies with differing conclusions. Incidentally, he also pointed out that tetrahedral coordination is not very common for hydrated ions with only [Be(H20)4]2+and possibly [Li(HZO),]+ exhibiting this structure. On the basis of the confirmation of the Li+(Cl-)4, NH4+(CI-),, Et,NH+(X-),, and CSH5NH+X-species in acetonitrile, in the second part of the present paper we will deal with the triple ion formation from trialkylammonium halides, R3NH+X- (R = Me, Et, and n-Bu; X = C1, Br, and I), by electrical conductivity data. The concept of triple ion formation was first introduced by F u w and Kraus" as early as 1933 to explain the minimum in the relation between the equivalent conductivity (A) and the concentration (C)of tetraisoamylammonium nitrate in dioxane-water mixtures (dielectric constant e < 12). They assumed the triple ion formation from a neutral molecule and a simple ion (AB B- s AB2- and AB A+ F! A,B+) by the action of electrostatic forces. Sellers et al.Iz interpreted the conductance behavior of weak acids and bases in nonaqueous solvents in terms of complex equilibria. They suggested that H(HA),+ ( n 3 1, H A = an acid) type species should be considered for a possibility as part of a general equilibrium scheme and that the formation of HAH' would turn out to provide an alternate explanation for certain literature data. On the basis of IR and Raman spectra, Bacelon et al.I3 proposed the formation of triple ions (Li+NCS-Li+ and
+
+
(10) Brooker, M. H. In The Chemical Physics of Soluation. Part B: Spectroscopy of Solvation; Dogonadze, R. R., Kalman, E., Kornyshev, A. A., Ulstrup, J., Eds.; Elsevier: Amsterdam, 1986; Chapter 4, p 160. (11) Fuoss, R. M.; Kraus, C. A. J . Am. Chem. Soe. 1933, 55, 2387. (12) Sellers, N. G.; Eller, P. M.P.; Caruso, J. A. J . Phys. Chem. 1972, 76, 3618. (13) Bacelon, P.; Corset, J.; Loze, C. J. Solution Chem. 1980, 9, 129.
0 1989 American Chemical Society
