J. Phys. Chem. 1993,97, 12356-12362
12356
Pressure-Volume-Temperature Relations and Configurational Energy of Liquid, Crystal, and Glasses of D-Sorbitol Motosuke Naoki,' Koji Ujita, and Seiji Kashima Physical Chemistry Section, Bioscience Department, Faculty of Engineering, Gunma University, Kiryu, Gunma 376, Japan Received: August 30, 1993"
Pressure-volume-temperature relations of the liquid, the supercooled liquid, the crystal, and two glasses of D-sorbitol are presented, and characteristics of the glass transition and the melting are reported. The glasses were vitrified from the liquid by isobaric cooling a t a rate of -0.2 K/min under 0.1 and 78.5 MPa. Vitrification pressure alters effectively the properties of the glasses. The thermal expansivity and the internal pressure of the higher-density glass vitrified under 78.5M P a are higher than those of the lower-density glass vitrified under the atmospheric pressure. With the aid of the result of the dielectric activation process in the liquid, these properties are interpreted by the number of frozen-in high-energy and high-density regions which have presumably been produced from defects and weak portions of the intermolecular hydrogen-bonding networks in the liquid a t the vitrification points. A simple configurational energy consisting of a Lennard-Jones contribution and a hydrogen-bond contribution is proposed and has been applied to the glasses and the crystal.
I. Introduction D-Sorbitol is a sample which vitrifies easily near room temperature and its supercooled liquid is very stable. From an analysis of the dielectric activation process in the liquid of D-sorbitol, it has been deduced that the number of intermolecular hydrogen bonds decreases with increasing pressure and temperature.' The peculiar properties of the glass transition temperature Tgand the molecular mobility in the liquid are well explained by the change in the number of intermolecular hydrogen bonds with pressure and temperature.* If the distribution of hydrogen bonds in the liquid at Tg exists frozen in the glass, configurational properties of the glass may be influenced by vitrification pressure. It has been found in several pressure-densified glasses of poly(vinyl chloride) (PVC) that the internal pressures of those glasses lie on a single curve when the internal pressures are plotted as a function of volume. That is, the law of the corresponding states holds for the PVC glasses and the functional form of the configurational energy is not altered by vitrification p r e ~ s u r eIf. ~ the number of hydrogen bonds in the D-sorbitol glasses depends on vitrification pressure, the functional form of the configurational energy will differ with each glass and the law of the corresponding states will not hold. In the present work, it is one of the objectives to study effects of the distribution of intermolecular hydrogen bonds in the liquid on configurational properties in the associated glasses. 11. Experimental Section
A. Sample. D-Sorbitol was purchased from Nakarai Chemical Ltd. The original sample melted to a slightly milk-white liquid and showed small endothermic shoulders near 70 and 110 OC in the DSC (differential scanning calorimetry) chart. After drying under vacuum at 140 OC for 2 h, the sample became transparent and those shoulders in the DSC chart vanished. The sample dried under vacuum at 130 OC for 1 day was used for the experiments on the liquid and the glassy state. The liquid sample obtained above was cooled down to 70 O C to crystallize in the atmosphere of Ar gas. The present preparation of the sample may not perfectly exclude water in the sample, and the sample may be contaminated by some moisture. The density of the liquid was measured by a pendant-drop type pycnometer ofvolume about 3.5mL. After thesupercooled liquid a
Abstract published in Advance ACS Absrracts, October 15, 1993.
sample was poured in the pycnometer and degassed under vacuum, the temperature was raised slowly to read the temperature at which the sample began to overflow the pycnometer. The volume of the pycnometer was measured by the degassed water at several temperatures around the overflow temperature by the same way. The density of the liquid was determined at 362.68 K. The density of the crystal at 305.56 K was determined by the floating method with n-hexane and CC1, in a pycnometer of volume about 50 mL. In the present experiments, three glasses were formed by isobaric cooling. The sample vitrified under the atmospheric pressure at a cooling rate of -0.2 K/min is referred to as 1 Glass, and under 78.5 MPa at the same cooling rate it is referred to as 800 Glass. The sample vitrified under the atmospheric pressure at a cooling rate of -2 K/min is referred to as Q Glass. The experiments on the volume-temperature (V-T) isobars under atmospheric pressure was carried out for all of the glasses, but the experiments on the volumepressure (V-P)isotherms were only for 1 Glass and 800 Glass. B. V-T Isobars under Atmospheric Pressure. Mercury capillary dilatometers were used to obtain V-T isobars of the crystal, the liquid, and the supercooled liquid. When the liquid was vitrified through the glass transition region, the sample stuck to the Pyrex glass chamber and usually the dilatometer broke. Even if the dilatometer did not break, high internal stresses stored in the glassy sample resulted in poor reproducibility of the volumetric data. To avoid such internal stresses, we employed a thin natural-rubber pouch to prevent the adhesion between the sample and the glass chamber. The liquid sample was loaded in the rubber pouch at 130 O C and degassed in a slight vacuum, and the mouth of the pouch was tied by the same rubber piece. Although the sample penetrated into the rubber pouch, the excess mixing volume between D-sorbitol and natural rubber was ignored in calculations of the specific volume of the glasses. In order to avoid a loss in weight of the sample due to the vapor pressure when the sample was evacuated, the dilatometer was cooled beforehand to -20 OC for the dilatometric measurements of the liquid and to -35 "C for the glasses and the crystal. Then the dilatometer was evacuated and mercury was inserted in it. At -20 OC, D-sorbitol is in the glassy state and sticks on the glass chamber, but does not break the chamber at this temperature. Dilatometries under the atmospheric pressure were carried out at about 1 K intervals from 274 to 505 K for the liquid and
0022-3654/93/209712356$04.00/0 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97,No. 47, 1993 12357
Liquid, Crystal, and Glasses of D-Sorbitol n
in
" ' I "
I
I
Y
(1 Giassi
IscrFrmlm Line
o,a200sI
"
250
"
"
300
"
350
400
T /
"
450
"
500
'
I
550
K
Figure 1. Specific volumes of D-sorbitol: (0)0.1 MPa, ( 0 )78.5 MPa of the liquid and the supercooled liquid; (0)0.1 MPa, (m) 78.5 MPa of the crystal; (A)0.1 MPa, (A) 78.5 MPa of 1 Glass; (*) 0.1 MPa, (-) 78.5 MPa of 800 Glass. the supercooled liquid, from 221 to 315 K for the glasses, and from 260 to 360 K for the crystal. C. V-P Isotherms. Isothermal experiments to give V-P relations were performed by a pressure dilatometer which is a straight mercury capillary dilatometer of length about 50 cm contained in a pressure vessel. Changes in the mercury level in capillary of the dilatometer were read by measuring the emf of a variable transformer (1501-9, Shinko Electric Co.) induced by a piece of Permalloy set on the top of mercury column. The pressure vessel was immersed in an ethylene glycol/water, an ethanol, or a silicon oil bath. The temperature of the inside of the pressure vessel was maintained within fO.O1 K at each measurement. The temperature was measured by an alumelchrome1 thermocouple inserted near the sample cell in the pressure vessel. Silicone oil (KF-94, Shin-Etsu Chemical Co.) was used to transmit the hydrostatic pressure. The pressure was measured by a Heisebouldon gauge with an automatic compensator (Dresser Ind.). Since D-sorbitol vitrifies below --19 OC and mercury crystallizes a t --39 OC, the temperature region for the experiments on the glasses is narrow when the mercury capillary dilatometer is used. In order to expand the temperature range, we employed a mixtureof mercury and thallium (Hg/Tl, 91.5/8.5 weight ratio) in place of mercury for the experiments on the glassy state. The Hg/Tl mixture with a melting temperature T,,, of about -60 O C was stored in an atmosphere of dried argon. Illustrations and details of the pressure dilatometer and the methods for calculations of the specific volume were shown in the previous paper^.^^^ Isothermal experiments were carried out at 14 temperatures from 293.82 to 41 7.49 K for the liquid and the supercooled liquid, a t 15 temperatures from 256.47 to 339.23 K for the crystal, at 16 temperatures from 231.27 to 252.49 K for 1 Glass, and a t 15 temperatures from 230.71 to 252.22 K for 800 Glass. At every temperature, the pressure was changed in increments of 9.8 MPa from 0.1 to 78.5 MPa. 111. Results and Discussion
A. Specific Volumes. The specific volume u of the liquid at 362.68 K is 0.712 96 f 0.000 18 cm3/g, which is larger than 0.7073 cm3/g obtained by Barkatt and Angell.s The u of the crystal a t 305.56 K is 0.678 00 0.000 16 cm3/g a t 305.56 K. Values of u are plotted as a function of temperature in Figure 1. Every isobar for the liquid (and the supercooled liquid), the crystal, 1 Glass, 800 Glass, and Q Glass was fitted to a polynomial,
*
where Tis in unit of K. Values of c(i), the standard deviation (STD) in fitting eq 1, and the overall experimental uncertainty in u are listed in Table I. When the crystal was fused in the
Figure 2. Schematic diagram illustrating liquid and glass surfacesin the P-V-Tspace. 1 Glass and 800 Glass are the glasses formed at a cooling rate of -0.2 K/min under the pressures 0.1 and 78.5 MPa, respectively. T,(P) line is the locus of the vitrification points determined by isobaric cooling experiments.
TABLE I: Parameters in the Polynomial (Eq 1) for the Specific Volumes (cm3 g')at 1 atm liquid crystal 1 Glass 800 Glass Q Glass 40) 0.61421 c(1) X lo4 2.736 4 2 ) X lo7 -2.024 c(3) X 1Olo 5.480 region, K 274-482 STD 0.000120 uncertainty 0.00030 region, K 489-505 STD 0.000413 uncertainty 0.00059
0.66013 0.1394 1.458
0.69903 -2.312 6.407
0.69931 -2.926 8.472
260-301 223-266 229-258 0.000080 0.000141 0.000050 0.00024 0.00032 0.00023
0.71034 -2.841 6.787 237-265 0.000085 0.00027
dilatometer, its volume agreed with the liquid value within the errors of *O.OOO 12cm3/g, which is below the experimental errors in the density measurements. Specific volumes under 78.5 MPa are also plotted in Figure 1. The P-V-T relation of the liquid and the crystal is expressed by a polynomial in T and P v = Fxc(ij)T'P (2) where P is in a unit of MPa. Values of c ( i j ) , the STD, and the uncertainties in u are listed in Table 11. Although eq 2 reproduces the experimental volumes of the liquid and the crystal within the experimental errors, the representation by the polynomial is not necessarily appropriate to describe variations of volume with pressure. Besides, the temperature range of the glassy state was very limited and reliable variations of volume with temperature under elevated pressures could not be obtained. In this regard, we employed the Tait equation: u/vo = 1 - C l n (1
+P/B)
(3) where uo is the zero-pressure specific volume, and B and C are temperature-dependent constants. Following Simha et al.,637C is fixed as 0.0894 in every state. In the present system, B can be expressed by
B=B,+B,T (4) Values of BO,B 1 ,C,the STD,and the uncertainties in u are listed in Table 111. In the present experiments, two glasses of different pressure histories are examined. In order to clarify the complex relationships between them, a schematic diagram illustrating liquid and glass surfaces in the P-V-T space is shown in Figure 2. The glass transition temperature (the vitrification point) of 1 Glass, Tg,,~l~~~~ is the intersection of the isobars of 0.1 MPa between the liquid and 1 Glass. The iso-free-volume line of 1 Glass is the intersection linebetween the liquidsurfaceand the 1Glass surface,
12358 The Journal of Physical Chemistry, Vol. 97, No. 47, 1993
Naoki et al.
TABLE II: Parameters in the Polynomial (Eq 2) for the Specific Volumes (cm3 g i ) under the Elevated Pressures c(i,O) c(i, 1 ) c(i,2) c(i,3) Liquid C(0J) C(1J) c(2J) d3J)
regions STD uncertainty
0.61423 2.736 X lo" -2.026 X l t 7 5.484 X 294-417 K 0.00041 0.00071
-2.305 X lo" 2.966 X lo-' 1.578 X le9 -4.499 x 10-'2 0.1-78.5 MPa
0.66014 1.392 X 1.459 X l C 7 256-328 K 0.000059 0.00030
-1.047 X lo" 2.222 x io-7 -5.597 x 10-10 0.1-78.5 MPa
2.097 X 10-6 -5.088 X l C 9 4 . 3 5 8 X lo-'* 1.345 X
c(i,4)
-2.166 X 1W8 4.168 X lo-" 5.444 x 10-15 -1.643 X
9.876 X lO-'-" -1.278 X -2.500 X 1O-I' 6.250 X
-8.605 X 1C8 2.265 X -9.000 X
5.236 X -1.413 X 1O-I' 8.780 X
Crvstal C(0J) 4 1J ) C(2J)
regions STD uncertainty
TABLE I V Characteristics of Glass Transition and Melting Glass Transition 265.29 t 1.54 TB.1c d O . 1 MPa), K 0.68278 t 0.00031 u(TS,lclass, 0.1 MPa), cm3g-I 254.21 & 1.49 rw-frcbvolumc.800 Glaas (0.1 MPa), K 0.67966 t 0.00029 O( Tiso-free-volumc,800 clam,0.1 MPa), cm3g-' Melting T,(O.l MPa), K 388.29 t 1.91 diquid(Tm, 0.1 MPa), cm3g-l 0.72201 t 0.00101 ucvatal( T,, 0.1 MPa), cm3g-l 0.68752 t 0.00049 0.03449 f 0.00045 Au,(T,, 0.1 MPa), cm3g-I
0.685
0.680
4.541 X 10-6 -1.148 X 1W8 8.034 X IO-"
-
1 1 K than Tiso-free-volume of800 Glass at 0.1 MPa, Tiso-fra-volume,800ciiaJs (0.1 MPa). The slope of the Tg(P) line in Figure 2, Le., dT,/dP, was determined as 0.043 K/MPa by Atake and Angell, which is much smaller than those of molecular liquids.2 By the use of this dT,/dP,Tg,8w Glass is calculated as 268.7 K. From Tg,800 Glass 0.670 220 240 280 280 300 and ~mfree.volume,800class (0.1 MPa), the slopeof theiso-free-volume T / K line is calculated as 0.185 K/MPa, which is almost the same as Figure 3. Specific volumes in the glassy region at 1 atm: (A) 1 Glass dT,/dP of molecular liquids. (cooling and heating experiments);(A)800 Glass (heating); (m) Q Glass The specific volume of the crystal at Tg,l G I is~0.67409 ~ ~ cm3/g (heating); (0)crystal. and the free volume fraction, ( V I - ucrystal)/ulGlass, is about 0.013, which is almost half of 0.025 deduced from the WLF TABLE 111: Tait Parameters (Eq 3) equation for many polymers.9 liquid crystal 1 Glass 800 Glass The Tgof Q Glass, Tg,QGlassr is seen to be the same as Tg,I0lass BO,MPa 777.9 2203 1297 3105 in Figure 3. However, due to the fact that the experimental B I ,MPa K-1 -0.787 -4.209 -0.5296 -7.509 region is very close to the glass transition region and the magnitude STD, cm3g-l 0.000269 0.0001 12 0.000132 0.000137 of relaxation (annealing effects) of the quenched glass is larger uncertainty,cm3g-l 0.00057 0.00035 0.00045 0.00047 than those of the glasses vitrified more slowly, a reliable Tg,QG I ~ cannot be obtained. The u of Q Glass at 240 K is larger by about along which one of the Ehrenfest equations applies: 0.12% than that of 1 Glass. The volume of the crystal increases with temperature above 320 K. When the temperature was maintained constant below where a and 8, are the thermal expansivity (a = (a In u/aT)p) 350 K, thevolume fellon theisobar ofeq 1 within theexperimental and the isothermal compressibility (j3 = -(a In u/aP)r), respecerrors. The melting temperature Tm was determined from the tively. If the free-volume theory applies to the present system, intersection between the liquid isobar (eq 1) and the extrapolation the iso-free-volume line should be identical with the Tg(P) line, of the ascending transition curve. The characteristics of the which is the locus of vitrification points determined by isobaric melting are listed in Table IV. Another ascending transition cooling experiments at a constant cooling rate. curve in the lower temperature region was also observed and its In Figure 3, the lowest-temperature region in Figure 1 is extrapolation intersected at 373.8 f 2.1 K with the liquid isobar. enlarged to show the glassy state and the glass transition region This apparent secondary transition may be due to moisture in detail. The u of 800 Glass is smaller by about 0.0028 cm3/g remaining in the sample. than that of 1 Glass at 230 K; that is, 800 glass is more densified B. Tbermal-hessureCuefficients and Internal Pressures. The by about 0.42% than 1 Glass, which is the same order of T-P isochores were obtained from the polynomial (eq 2) and the densification as those observed in poly(viny1 acetate)8and PVC.3 Tait equation (eq 3) for the liquid and the crystal, and from the The surfaces of 1 Glass and 800 Glass are parallel with each Tait equation alone for 1 Glass and 800 Glass. Results are shown other as illustrated in Figure 2 and the free volume theory does in Figures 4a-d. Each isochore of every state is linear within the not apply to D-sorbitol. experimental errors over the present experimental range. Therefore, the slope of each T-P isochore, Le., the thermal-pressure The numerical results for the glass transitions are listed in Table IV. The vitrification point of 800 Glass, Tg,800 G I ~and ~ ~ , coefficient y = (aP/aT),, is a function of u only and independent of T: the iso-free-volume temperature, Tiso.rrce-vo~umc, under the higher pressures were not determined due to the inaccuracy of extrapolations of the temperature dependence of the Tait equation for theglasses. Tg,l~l~~~ is 265.29 1.54 K, which is higher by about where S is the entropy. Figure 5 shows variations of y with v. 0.675
t
*
~
~
The Journal of Physical Chemistry, Vol. 97, NO. 47, 1993
Liquid, Crystal, and Glasses of D-Sorbitol
12359
(a1 l i q u i d
T /
i30
K
235
240
245
250
255
250
5
T / K (d) BOO Glass
(b) C r y s t a l
i30
235
240
245
T I K
T / K
Figure 4. P-T iswhores. Lines are linear fittings. (a) Liquid: (0)v = 0.69 cm3//g;( 0 )0.695; (0) 0.70; (m) 0.705; (A) 0.71; (A) 0.715; (*) 0.72. (b) Crystal: (0) v = 0.672 cm3/g; ( 0 )0.673; (0)0.674; (m) 0.675; (A)0.676; (A)0.677. (c) 1 Glass: (0)v = 0.677 cm3/g; ( 0 )0.6775; (0)0.678; (m) 0.6785; (A)0.679; (A)0.6795; (*) 0.680. (d) 800 Glass: (0)v = 0.6745 cm3/g; ( 0 )0.675; (0)0.6755; *(D) 0.676; (A)0.6765; (A)0.677; (*) 0.6775.
TABLE V Parameters of the Thermal-Pressure Coefficients (I2417) go. 819 STD, uncertainty, MPa K-' g MPa K-l ~ m - MPa ~ K-' MPa K-' -7.372 0.027 0.079 liquid 7.914 -24.683 0.008 0.114 crystal 18.152 0.025 0.28 -12.053 20.025 1 Glass 121.391 0.013 0.53 800 Glass -79.776
-. .
e. I1
the crystal, but the y of 800 Glass is higher than those of 1 Glass and the crystal. From the thermodynamic transformation, 0. U
0. I 9 0.10 v / cm3ql
0. 11
0. 1 2
Figure 5. Thermal-pressure coefficients: (0)liquid; (0)crystal; (A) 1 Glass; (A) 800 Glass. Solid lines are eq 7. Broken lines indicate the experimental uncertainty limits.
The y is expressed by a linear equation:
Y = go + g,o
(7)
Values of go and gl, the STD, and the overall uncertainties are listed in Table V. We have analyzed the data for the glasses, in which retardation effects were not observed our experimental time scale. Notwithstanding this, some possibility of small persisting retardations is unavoidable and the uncertainty in the glass increases. The y of 1 Glass is almost the same as that of
[a(aslav)~laTlu= (aPi/aT)u = 0
(8)
where Pi is the internal pressure defined by Pi = (aE/aV)T and E is the internal energy. Equation 8 indicates that Pi is also a function of volume only. Figure 6 shows variations of Pi with u. All Pi increase with volume. The Pi of 1 Glass is lower than that of the crystal, but the Pi of 800 Glass is higher than those of 1 Glass and the crystal. This point will be discussed in a later section. The uncertainties in Pi are 27.6, 34.2, 69.5, and 132 MPa for the liquid, the crystal, 1 Glass, and 800Glass, respectively. C. Thermal Expansivities and Isothermal Compressibilities. Figure 7 shows the original dilatometric data on the thermal dilations, (aV/aT)p, at 0.1 MPa. The dilation of 1 Glass is scattered. This is mainly due to the scatter in the interdiameter of the capillary of the dilatometer, rather than the change in
Naoki et al.
12360 The Journal of Physical Chemistry, Vol. 97, No. 47, 1993 I , 200
4 Liquid
01 0.68
"
"
"
"
0.81
0.80
0.89
0.10
v
0.11
0.12
0.00 1
I
"
'
0.73
0.14
/ cm39-'
Figure 6. Internal pressures: (0)liquid; (0) crystal; (A) 1 Glass; (A) 800 Glass. The thick solid line is the theory for the crystal and the thick broken line is the theory for 800 Glass. Thin broken lines indicate the experimental uncertainty limits. 0.00010
0.OOOSO
250
350
300
T
/
I
400
K
Figure8. Isothermal compressibilities calculatedfrom the Tait equation, eq 3. Thick lines are 0.1 MPa, and thin lines are 78.5 MPa.
hydrogen bonds in the liquid to all possible ones. When all of six OH groups of a molecule combine with the other molecules by hydrogen bonds, t i s unity. Although this is a crude estimation, it has given significant information on understanding the peculiar dielectric properties in the liquid of D-sorbitol.' Along the glass transition line, the T,(P) line in Figure 2,the change in the configurational entropy is considered to be small. Therefore, the contribution of S* in eqs 9 and 9' may be ignored along T,(P). Then the difference in t: between T ~ Glass , I and Tg,8w Glass in the liquid can be estimated from eqs 9,9', and 10 as
*
0.00000 I
250
300 T
/
350 K
400
150
I
Figure 7. Thermal dilations at 1 atm: (0)liquid; (0)crystal; (A) 1 Glass; (A)800 Glass. Solid lines are eq 1.
volume of the sample. The average diameter of the capillary was measured by mercury, but small deviations could not be corrected in this way. The dilation of 1 Glass is almost the same as that of the crystal, but the dilation of 800 Glass is higher than those of 1 Glass and the crystal. It may be interesting that the higher-density glass (800 Glass) dilates more than the lower-density glass (1 Glass). This was not for case of the densified glasses of PVC.3 In nonassociated liquids and glasses, the higher density usually corresponds to lower entropy, and then the lower pressure dependence of entropy (=the lower thermal dilation). Therefore, this peculiar difference in the thermal dilation between 1 Glass and 800 Glass may be due to a contribution of hydrogen bonds. The difference in the distribution of hydrogen bonds between 1 Glass and 800 Glass may presumably by formed at the vitrification points, Le., Tg,lGlass = TJ265.29 K,0.1 MPa) and Tg,8wclass= Tg(268.7K,78.5 MPa). Since the rearrangement of hydrogen bonds may not extensively occur in the glassy state, the distributions of hydrogen bonds frozen-in at Tg,1classand T,,sw class may have direct effects upon the glass properties. From an analysis of the dielectric activation process in the supercooled liquid of D-sorbitol, it has been deduced that the number of intermolecular hydrogen bonds changes with pressure and temperature-I The changes in the number of intermolecular hydrogen bonds estimated a t 294 K are ( a f / 1 3 7 9= ~ -S*/A* - 0.0023 f 0.0003K-'(0.1MPa) (9) = -S*/A* - 0.0026 f 0.0005K-'(78.5 M P a )
(9')
( a € / a P ) , = -0.0015f 0.0003 MPa-' (10) where S* is the intramolecular conformational contribution, A* is the total activation affinity, and 5 is the extent of intermolecular hydrogen bonds,1° which is a number ratio of intermolecular
f l Glass - E800 Glass = o*126 0.025 (1 1) That is, the number of hydrogen bonds at Tg,1 classis larger by about 10% than that at T,,sw ~ l Most ~ of ~ this ~ difference . in eq 1 1 arises from eq 10. In the PVC glasses, the change in the relaxation magnitude of the local mode of motions is corresponding to the change in the configurational (P-V-T) pr0perties.l' The local mode of motions has been considered to occur in small high-energy and high-entropy regions distributed in liquids and glasses.I2 Most configurational properties also may originate in the high-energy regions." If the distributions of hydrogen bonds in the liquid are frozen-in at the vitrification points, the difference in the number of hydrogen bonds at the vitrification points may approximately remain in the glasses. Defects and weak portions of intermolecular hydrogen-bonding networks in the liquid may produce localized high-energy and high-entropy regions in the glasses. That is, the number of high-energy regions in 800 Glass may be larger by about 12% than that in 1 Glass. The higher entropy may correspond to the higher pressure dependence of entropy (=thermal dilation). This may be the reason why the thermal dilation of 800 Glass is larger than that of 1 Glass. The isothermal compressibilities B calculated from the Tait equation are shown in Figure 8. The thermal expansivity and the isothermal compressibility are related to the thermal-pressure coefficient as
./a = Y (12) For the liquid and the crystal, values of CY can be calculated from the polynomial (eq 2) and also from eq 12 with y and @ of the Tait equation (eq 3). The discrepancy in the values obtained in the two ways gives the total uncertainty in the present determination. Similarly, the uncertainty in B is the discrepancy between the values of @ from the Tait equation and those from eq 12 with y and (Y of the polynomial. The uncertainties (the averages of the root mean square of discrepancies) under the elevated pressures for the liquid and the crystal are shown in Table VI. For the glasses, unfortunately, no polynomial equation under the elevated pressures was obtained due to the limited temperature range. Values of have been determined from the Tait equation
The Journal of Physical Chemistry, Vol. 97, No. 47, 1993 12361
Liquid, Crystal, and Glasses of D-Sorbitol
TABLE VI: Uncertainties in a and B a x 104, K-l 0.1 MPa elevated P liquid crystal
0.065
1 Glass
0.190 0.095
800 Glass
0.062
0.134 0.136
dependent term: B X lo4, MPa-l 0.049 0.068
0.212
0.135
0.348
0.127
alone and the uncertainty cannot be obtained from eq 12. In Table VI, the uncertainty estimated from the scatter in u and y are listed in Table IV. The crystal and the two glasses exhibit almost the same compressibility and the same pressure dependence of the compressibility. Although the uncertainties in @ of the glasses are large, the @ of 800 Glass appears to be slightly smaller than that of 1 Glass, as observed in the PVC g l a ~ s e s . ~This may be interpreted as follows. In molecular glasses, the localized highenergy and high-entropy regions correspond to the low-density regions and external pressure may effectively compress the lowdensity regions to avoid the concentration of internal stresses. The number of the low-density regions in the lower-density (normal) glass is larger than that in the higher-density (densified) glass, and the@in the normal glass is larger than the @ in densified glass. In the D-sorbitol glasses, on the contrary, the high-energy and high-entropy regions correspond to the high-density regions. In this case, external pressure may effectively work on positions of comparatively smaller defects and/or weaker hydrogen bonds in the low-density, low-energy, and low-entropy regions. Therefore, the low-density 1 Glass consisting of larger low density regions may be more easily compressed than the higher-density800Glass. The densified 800 Glass was formed from the liquid under 78.5 MPa and was forced to be expanded to 0.1 MPa by the release of external pressure. If some distortion of hydrogen bonds occurred in this vitrification process, it may be possible that the hydrogen bonds in the densified glass are uniformly weaker than those in the normal glass. This also may explain the experiment that a in 800 Glass is larger than that in 1 Glass. It should be noted that the change in the number of intermolecular hydrogen bonds in the liquid (eqs 9, 9‘, and 10) has been estimated based on assuming that the energy of hydrogen bonds is constant. If we assume that the number of hydrogen bonds in the liquid is independent of P,we may have a similar expression in terms of the energy of hydrogen bonds in place of [ in eq 10, which also may explain the molecular mobility in the liquid. This problem might be solved by a study of the pressure dependence of the Raman effect below 200 cm-’ in the supercooled liquid of D-sorbitol. The hydrogen bond has the covalent and ionic structure and its potential is very anharmonic. This may give a comparatively large thermal dilation to hydrogen-bonded solids. The energy of hydrogen bond is about 9RT, which is much higher than the energy of a pair molecular contact (ca. -2RT by the dispersion and dipole forces) near room temperature. When many molecular contacts remain in the glass as in the case of D-sorbitol, these weaker molecular contacts may mainly suffer from the thermal energy (ca. -RT). The large a in 800 Glass may be largely due to the molecular bonds and weaker hydrogen bonds. On the contrary, external pressure may distort or bend the frozen-in intermolecular hydrogen-bonding networks which form possibly imperfect and highly disordered frame. When some hydrogen bond becomes weaker, its energy may compete with the energy of two or three molecular contacts and it may easily be affected by external pressure.
IV. Configurational Energy Equation 8 shows that the internal energy E can be regarded as a simple sum of a term independent of volume and a volume-
E = E,(T) + E,(T,V) (13) where Ek(T) and Ec(T,V) are denoted as the kinetic energy and the configurational energy, respectively. Therefore, the internal pressure is the slope of the configurational energy:
pi = (aEc/dV), (14) As shown previously, the internal pressure of the densified glass is larger than that of the normal glass. In order to clarify this, a simple double potential model is proposed. If there is no hydrogen bond and the molecules interact with their environment by a dispersion force and a dipole force, the potential energy may be described by the Lennard-Jones potential.” The simplest form of its average is ( 1 / 2 ) N z t-~2y2),14 ~~ where N is the Avogadro’s number, z is the coordination number per molecule, BLJ is the Lennard-Jones’ energy constant, and y is the reduced density defined by y = u*/u (15) where u* is the van der Waals volume of the molecule. In a mole of the liquid, [ N H B /of~ intermolecular hydrogen bonds exist, where NHB is the number of the groups which can form hydrogen bonds, Le., the number of the hydrogen-bond functional groups. Then the average coordination number per molecular may decrease from z to z(Nz - g[NHB)/Nz. The g is the parameter which corrects the excluded volume effect of each hydrogen bond. Then the dispersion (LJ) energy, ELJ,is written as
EL, = (1/2”1
- gxtkLJCv4- 2Y2)
(16)
where x = NHB/Nz (17) is the fraction of all hydrogen-bond contacts in all pair contacts. When a hydrogen-bondfunctional group of the second molecule approaches a certain position relative to that of the first molecule, a hydrogen bond may be formed. The probability that we find the second molecule in the confined cell of the volume u* around the first molecule is proportional to u * / u ( = y ) . The probability that the hydrogen-bond functional group of the second molecule points faces in a suitable direction to constitute a hydrogen bond with the first molecule is x . Since the number of hydrogen-bond pairs is (1/2)[NHB, the attractive term of the hydrogen-bond energy is expressed by -( 1/2)X[NHBBHBy,where B H B is the energy constant of the hydrogen-bond potential. Adding the repulsive energy (y4) in the same form as that in the Lennard-Jones potential, we may write the hydrogen-bond (HB) energy, EHB, as
= (1/2)XtNHBBHBb4 -Y)
(18)
Then the total configurational energy is expressed by
E, = ELJ+ EHB
= ~ * [ (- ig x t ) b 4 - 2 y 2 ) + e x 2 t ( y 4 - y ) ]
(19)
E* = (1/2)NzeLJ
(20)
where
e = CHB/ZCLJ (21) Equation 19 may apply to any state of the system containing the intermolecular hydrogen bonds. In the liquid state, however, should be determined as to minimize the free energy at every temperature and pressure. A full expression of the partition function, which gives the free energy, requires much more insight into the complex configuration of each molecule. When the number of hydrogen bonds is assumed to be constant, as in the case of solid states such as the crystal and the glasses, the potential
Naoki et al.
12362 The Journal of Physical Chemistry, Vol. 97, No. 47, 1993 V. Conclusion
I. 0
I. 5
2. 0
2. 5
Y-'
Figure 9. Configurational energies and their components as a function of the reduced volume. Thick lines are the crystal (t = 1) and thin lines, 800 Glass ( E = 0.774). Solid lines, the configurational energies (Ec); broken lines, Lennard-Jones contribution (ELJ);dotted lines, hydrogenbond contribution ( E H B ) .
energy can be handled separately. If 5 is constant, we have the internal pressure as
Pi= -(E*/o*)[4(1
-gxf)(yS - y 3 )
+ ex2f(4y5- y 2 ) ]
(22)
We here assume f = 1 for the crystal, Le., all of the hydrogenbond functional groups form the intermolecular hydrogen bonds. When the theoretical log(-[4(1 - gx)(ys - y 3 ) + ex2(4yS-y2)l) against y is superposed to the experimental log Pi of the crystal, the magnitude of the shift of the superposition is log(E*/u*). In the present system, for simplicity, we set gx = 1/4 and exz = 1/ 2 , Le., g 1, x 1/4, and e 8. The van der Waals volume calculated by Bondi's methodl5 is 95.82 cm3/mol and then u* = 0.5259 cm3/g. The superposed theoretical curve is shown in Figure 6. From this superposition, we have E* = 734.4 J/g. The theoretical Pi increases with volume in accordance with the experiment, but the theory does not describe well the volume variation of Pi of the crystal. In Figure 6,the theoretical Pi with f = 0.774 is also drawn for 800 Glass. The theory agrees with the experiment within the experimental uncertainty but does not describe the steep slope of 800 Glass. In this case, f of 1 Glass is equal to 0.900 from eq 11 and its 7 is slightly larger than that of the crystal. In Figure 9, the configurational energy and their components calculated with the same values of the parameters are shown. When [decreases, EHB becomes shallower, ELJbecomes deeper, and the minimum of total Ec shifts to a smaller volume. Then Pi,the slope of the E, curve near y-1 = 1.3, increases. This may be the reason why the internal pressure and the thermal-pressure coefficient of the higher-density glass (800 Glass) are higher than those of the lower-density glass (1 Glass) and the crystal. In the present analysis, several significant assumptions have been used: effects of intramolecular hydrogen bonds are not taken into account; all OH groups form intermolecular hydrogen bonds in the crystal; effects of the ionic character of the hydrogen bond are not formally included in the potential; and the excluded volume effect of the hydrogen bond in the crystal, g, is ignored. Because of the large experimental uncertainties in the glasses due to the limited experimental range, however, such a fine adjustment of the parameters may have not any physical significance. It should be emphasized that the peculiar properties of the densified glass are mainly due to that the number of frozen-in intermolecular hydrogen bonds in the densified glass is smaller than that in the normal glass.
- -
-
The P-V-Trelations for the liquid, the supercooled liquid, the crystal, and the normal glass (1 Glass) and the pressure-densified glass (800 Glass) of D-sorbitol are presented. The thermal expansivity and the isothermal compressibility in every state of D-sorbitol are smaller than those of the corresponding states of organic nonassociated molecules, whereas the thermal-pressure coefficient and the internal pressure of the pressure-densified glass and the liquid are comparatively high. The phenomena that the thermal expansivity, the thermal-pressure coefficient, and the internal pressure of the pressure-densified glass are higher than those of the normal glass has not been observed in the nonassociated glasses of poly(viny1 chloride). The analysis of the dielectric activation process in the supercooled liquid of D-sorbitol suggests that the number of intermolecular hydrogen bonds decreases with increasing temperature and pressure. If the distribution of intermolecular hydrogen bonds in the liquid is frozen in, at the vitrification point, defects and weak hydrogen bonds in transient hydrogen-bonding networks in the liquid may presumably form localized high-energy, high-entropy, and high-density regions in the glass. The configurational properties and the local modes may largely arise from these high-energy regions distributed in the low-energy glassy frame. The difference in the configurational properties between the normal and the densified glass may be produced by the difference in the number of the high-energy regions in each glass. Vitrification pressure effectively alters the properties of the D-sorbitol glasses. This may be due to the fact that D-sorbitol molecule has six OH groups and forms easily transient intermolecular hydrogen-bonding networks in the liquid. The networks may be highly disordered due to the flexible D-sorbitol molecule. When the networks' frame is distorted or bent by external pressure, weak portions of the networks may be broken or weakened. In the case of simple alcohols with one or two OH groups, pair, cyclic, and short-chain structures rather than the networkstructure may be dominant, and external pressure may not so effectively rearrange the hydrogen-bonding structures in their liquids as in the D-sorbitol liquid. In order to clarify the experimental results, a simple configurational energy consisting of a Lennard-Jones term and a hydrogen-bond term is used. The latter potential has a minimum at a longer distance (a larger volume) than the former potential. The resulting configurational energy describes well the large internal pressure of the pressure-densified glass. References and Notes (1) Naoki, M.; Katahira, S . J. Phys. Chem. 1991,95,431. (2) Atake, T.; Angell, C. A. J. Phys. Chem. 1979, 25, 3218. (3) Naoki, M.; Mori, H.; Owada, A. Macromolecules 1981, 14, 1567. (4) Naoki, M.; Matsushita, M. Bull. Chem. SOC.Jpn. 1983, 56, 3549. (5) Barkatt, A.; Angell, C. A. J. Chem. Phys. 1979, 70, 901. (6) Nanda, V. S.;Simha, R. J. Chem. Phys. 1964, 41, 3870. (7) Quach, A.; Simha, R. J. Appl. Phys. 1971,42,4592. (8) Mckinney, J. E.; Goldstein, M. J. Res. Natl. Bur. Stand., Sect. A 1974, 78, 331. (9) Ferry, J. D. Viscoelastic Properties of Polymers, 2nd ed.; Wiley: New York, 1961. (10) Prigogine, I.; Defay, R. Thermodynamique Chimique; Editions Desoer: Liege, 1950. ( I 1) Naoki, M. J. Chem. Phys. 1989, 91, 5030. (12) Johari, G. P. Phase Transitions 1985, 5, 277. (13) Lennard-Jones, J. E.; Devonshire, A. E. Proc. R. SOC.London Ser. A 1937, 163, 53. (14) Lennard-Jones, J. E.; Ingham, A. E. Proc. R. Soc. London Ser. A 1925, 107,636. (1 5 ) Bondi, A. Physical properties of Molecular Crystals, Liquids, and Glasses, Wiley: New York, 1968.
