VIBRATIONAL MEAN-SQUARE AMPLITUDE MATRICES
1489
Vibrational Mean-Square Amplitude Matrices. XVII.
Mean-Square
Perpendicular Amplitudes and Shrinkage Effects in Benzene Molecules
by W. V. F. Brooks, Department of Chemistry, Ohio University, Athens, Ohio
S. J. Cyvin, and P. C. Kvande Institute of Theoretied Chemistry, Technical University of Norway, Trondheim, Norway
(July 17, 1964)
Previous work on mean-square amplitude matrices (2) is briefly reviewed. This work has been continued by further studies on the benzene molecular model. In the present work, the algebraio expressions are given for both in-plane and out-of-plane mean-square perpendicular amplitudes in terms of the 2-matrix elements. Numerical values for benzene and benzene-da are reported. For the out-of-plane vibrations a new normal-coordinate analysis has been performed. Force constants, L-matrices, and 2-matrices are reported. The obtained mean-square perpendicular amplitudes are used for calculating the eight independent Bastiansen-Morino shrinkage effects in both benzene and benzene-df. These shrinkage effects are associated with the eight types of nonbonded.atom pairs in the molecules here considered. Also, the two types of linear shrinkage effects in the benzene molecules are reported. Spectroscopic calculations of this type are of great interest in molecular structure studies by gas electron diffraction.
Introduction
4n2C2Wk2 (h/8n2C~k) coth ( h c ~ , / 2 k T ) =
Theory of the 2-Matrix. The theory of the vibraThe tional 2-matrix has been treated matrix is defined in terms of its elements by the meansquare quantities zit
=
(Si2)
(1)
for the diagonal elements, and the mean products Et5
=
(SB5)
(2)
for the offdiagonal elements. Here Si and Sj designate some displacement coordinates, e.g., vibrational symmetry coordinates. The following secular equations involving the 2-matrix have been established.
I zG-l
-
0 I z F - (hk&)El = 0 =
(3) (4)
They are similar to the familiar secular equation in the problem of harmonic vibrations of polyatomic molecules.‘ The appropriate characteristic values of eq. 3 and 4 are given by
&
=
(5)
(6) where c is the velocity of light, h is Planck’s constant, k the Boltzmann constant, and T the absolute temperature. W k represents the vibrational wave numbers (also referred to as frequencies in cm.-l). The 2matrix elements may be computed by means of the Lmatrix (normal-coordinate transformation matrix; S = LQ) according to the useful relation
z = LAZ
(7) written in the matrix form. Here A is a diagonal matrix with the elements A, = ( Q k 2 ) given by eq. 6. The 2-matrix has also been proved to possess certain invariant propertied similar to the Wilson G-matrix‘ (1) 5. J. Cyvin, Acta Chem. S c a d . , 13, 2135 (1959). (2) S.J. Cyvin, Spectrochim. Acta. 15, 828 (1959). (3) Y. Morino and S. J. Cyvin, Acta Chem. S c a d . , 15,483 (1961).
(4) E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, “Molecular Vibrations,” McGraw-Hill Book Co., Inc., New York, N. Y., 1955, p. 65.
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W. V. F. BROOKS, S. J. CYVIN,AND P. C. KVANDE
and the “inverse” force-constant matrix6 (or compliance matrix’). Applications of the %Matrix. The mean amplitudes of vibration (u) have become very important in modern electron-diffraction s t u d i e ~ . ~It ? ~ has been shown that the mean-square amplitudes of vibration, viz., u2,may be expressed as linear combinations of the 2-matrix elements when dealing with small harmonic vibrations. Such expressions have been evaluated for a number of molecular models. A survey of this material is included in Table I. Morino and Hirotalo have introduced the generalized mean-square amplitudes for an arbitrary atom pair, i.e., (a) the mean-square parallel, (b) perpendicular amplitudes, and (c) the mean cross products. These quantities are defined in terms of the Cartesian displacements by (a) ((zf - zd2), (b) ((Zf - %,I2) and ((Y, YI)”, and (c) ( ( z t - Z , ) ( Z i - Z,>), ((Zf - Z,)(Yf yr)>, and ( ( u t - yj)(zi - q)). Here the z-axis is taken individually for each atom pair (zj) along the connecting line of the atoms in their equilibrium position. The mean-square parallel amplitudes may be identified (to the first approximation) with the mean-square amplitudes of vibration ( u 2 ; see above). All of the quantities (a)-(c) may be expressed as linear combinations of the 2-matrix elements (see Table I). The mean-square perpendicular amplitudes have particular interest because of their application in the computations of Bastiansen-Morino shrinkage eff e ~ t s . l l - ’ ~The shrinkage effect has been observed by electron diffraction and may be defined loosely as the difference between the observed interatomic distance and the calculated distance from molecular geometry. In the simplest case of three colinear atoms 1-2-3, a strict definition is given by
-
-813
= ~1~~-
(r2
+
T23‘)
(8)
where r t j g is the mean interatomic distance obtainable from electron diffraction. In this case the shrinkage effect may be computed spectroscopically according to -813
= K13 -
(Kl2
+ K23)
(9)
where K,, is composed of the mean-square perpendicular amplitudes as given by
Ki,
=
+ (AYt,2))/2r,,e
r f 3 e designates the equilibrium distance.
(10)
This procedure is always applicable to shrinkage effects for linear conformations of atoms. In special cases of molecules with high symmetry, a corresponding procedure has proved to be valid also for nonlinear shrinkage effects.’* The Journal of’ Physical Chemiatry
For a survey of molecular models previously treated with special attention to the &matrix, see Table I and the supplementary references.15-36
Treatment of Benzene Molecules In paper XV36of this series, details were given of the theoretical studies on in-plane and out-of-plane vibrations of the benzene molecular model. That paper includes the algebraic expressions for the mean-square amplitudes of vibration in terms of the 2-matrix elements (cf. Table I). In the subsequent articles7 the in-plane vibrations are studied, and numerical computations for benzene and benzene-ds are reported. Calculated mean amplitudes of vibration are given and compared with those from earlier s p e c t r o s ~ o p i c ~ ~ - ~ ~ (5) S.J. Cyvin, SpeCtrOSCOpia Mol., 9, 56 (1960). (6) S.J. Cyvin and N. B. Slater, Nature, 188, 485 (1960). (7) J. c. Decius, J. Chem. Phys., 38, 241 (1963). (8) Y. Morino, K. Kuohitsu, and T. Shimanouchi, ibid., 20, 726 (1952). (9) S. J. Cyvin, Kgl. Norske Videnskab. Selakabs, Skrifter, No. 2, 1 (1959). (10) Y.Morino and E. Hirota, J. Chem. Phys., 23, 737 (1955). (11) 0.Bastiansen and M. Traetteberg, Acta Cryst., 13, 1108 (1960). (12) Y.Morino, ibid., 13, 1107 (1960). (13) S.J. Cyvin, Tidsakr. Kjemi, BergveSen, Met., 22, 44,73 (1962). (14) Y.Morino, S.J. Cyvin, K. Kuchitsu, and T. Iijima, J. Chem. Phys., 36, 1109 (1962). (15) S.J. Cyvin, Spectrochim. Acta, 17, 1219 (1961). (16) E. Meisingseth and S. J. Cyvin, Acta Chem. S c a d . , 16, 1321 (1962). (17) G. Nagarajan, Bull. Soc. Chim. Belgea, 71, 337 (1962). (18) S. J. Cyvin and E. Meisingseth, A d a Chem. S c a d . , 15, 1289 (1961). (19) S.J. Cyvin, ibid., 13, 1809 (1959). (20) E.Meisingseth, ibid., 16, 778 (1962). (21) G.Nagarajan, Bull. SOC.Chim. Belgea, 71, 329 (1962). (22) E.Meisingseth and S.J. Cyvin, J. Mol. Spectry., 8, 464 (1962). (23) S.Sundaram, ibid., 7, 53 (1961). (24) M. Iwasaki and K. Hedberg, J. Chem. Phys., 36, 594 (1962). (25) S.J. Cyvin, J. Mol. Spectry., 5 , 38 (1960). (26) 5. J. Cyvin, ibid., 6, 333 (1961). (27) G.Nagarajan, Bull. Soc. Chim. Belges, 71, 347 (1962). (28) S.J. Cyvin, J. Brunvoll, B. N. Cyvin, and E. Meisingseth, ibid., 73, 5 (1964). (29) G.Nagarajan, J . Mol. Spectry., 12, 198 (1964). (30) K. Sathianandsn, K. Ramaswamy, S. Sundaram, and F. F. Cleveland, ibid., 13, 214 (1964). (31) B. H. Bye and S.J. Cyvin, Acta Chem. S c a d . , 17, 1804 (1963). (32) M. Kimura and K. Kimura, J. Mol. Spectry., 11, 368 (1963). (33) G.Nagarajan, Bull. Soc. Chim. Belgea, 72, 537 (1963). (34) 5. J. Cyvin and E. Meisingseth, Acta Chem. S c a d . , 17, 1805 (1963). (35) 9. J. Cyvin, Spectrochim. Acta, 16, 1421 (1960). (36) W.V. F. Brooks and S.J. Cyvin, ibid., 18, 397 (1962). (37) W. V. F. Brooks and S.J. Cyvin, Acta Chem. S c a d . , 16, 820 (1962).
VIBRATIONALMEAN-SQUARE AMPLITUDE MATRICES
Table I : References to Expressions for Certain Quantities in Terms of the 2-Matrix Elements
Molecular model
Mean-square parallel amplitudes
Mean-square perpendicular amplitudes
Linear symm. XY2 Linear YXZ Trigonal XS Bent symm. XYI Linear symm. XzY2 Tetrahedral X, Plane square X4 Planar XY3 Pyramidal XY3 Tetrahedral XY4
15 15,16 15 15,17 18 15 15 15, 19,20,21 23,24 15,25,26,27
14,15 15,16 15 15 18 15 15 14, 15,21
Planar XY, SOF4-type model Octahedral XYs Cyclopropane model Benzene model
29 30 31, 32, 33 35 36
a
Shrinkage effects
14
1491
perpendicular amplitudes in terms of these elements, the following previously useda6abbreviations have been adopted.
+ R2 + DR)"' R** = (30' + R 2 + 3DR)"' R*
=
(0'
(11)
(12)
In addition, one puts
14,20, 22
15,"26," 27,"28
22," 28
34
34
Here D and R denote the equilibrium X-X (C-C) and X-Y (C-H or C-D) distances, respectively. mx and my are used to denote the masses of the X (C) and Y (H or D) atoms, respectively. The ten algebraic equations follow.
The expressions are given erroneously.
and e l e c t r o n - d i f f r a ~ t i o n ~ *works. ~ ~ ~ - ~ ~In the present work the mean-square perpendicular amplitudes have been studied, both for the in-plane vibrations and the out-of-plane vibrations. A summary of the applied symbols for various mean-square amplitude quantities is given in Table 11.
Table 11: Summary of the Symbols Used for the Parallel and Perpendicular Mean-Square Amplitudes Associated with the Ten Types of Interatomic Distances
~
In-Plane Mean-Square Perpendicular Amplitudes The 26 in-plane, symmetrized E-matrix elements for a benzene-type molecule (x6Y6) are defined in paper XVa6 and given numerically for C6H6 and CsDa in paper XVI. 37 To express the in-plane mean-square
(38) 0. Bastiansen and S. J. Cyvin, Nature, 180, 980 (1957). (39) S. J. Cyvin, Acta Chem. S c a d . , 11, 1499 (1957). (40) €3. J. Cyvin, ibid., 12, 1697 (1958). (41) K. Kimura and M. Kubo, J . Chem. Phys., 32, 1776 (1960). (42) I. L. Karle, ibid., 20, 65 (1952). (43) A . Almenningen, 0. Bastiansen, and L. Fernholt, KQZ. Nora& Videnakab. Sehkabe, Skrifter, No. 3 , 1 (1958).
Volume 69, Number 6 M a y 1966
1492
W. V. F. BROOKS, S. J. CYVIN,AND P. C. KVANDE
[-D2 1 4
1 -3"'D(2D 12
1 -R(D 4
1 + R)Ziz(Ezg) - qDR223(EZg) +
1 + 2R)Z34(Ezg) - 24-3'"(R/D)R(D +
The Journal of Physical Chemistry
+ 81-DR - -R2 1 + -(R4/DZ)]Z11(Ezg) 1 + 16 36
[!D2 2
7 103 5 + -DR + z R z + i(R3/D) -t 8
VIBRATIONAL MEAN-SQUARE AMPLITUDE MATRICES
1493
ported by several authors. In the present work a new set of force constants was derived, using the experimental frequencies (see Table IV) from Brodersen and Langseth. 44 The details of the present recalculation of force constants shall not be reported. It is only referred to Table V for t’he final results. They are in good agreement with the force constants from the pioneer work of Miller and C r a ~ f o r d . Note: ~~ the digits as given in Table V and other tables are not necessarily all significant. Table IV : Observed and Calculated Frequencies for Benzenes” Molecule, species, and no.
Obsd.*
i id
I CaDa
a
Out-of-Plane Normal Coordinate Analysis Symmetry Coordinates. The out-of-plane symmetry coordinates are specified in Table 111. The valence coordinates from which the symmetry coordinates are formed, are 6 (torsion of a XX bond) and y (XY outof-plane bending). A diagram and more precise definition of coordinates are given in ref. 36. Force Constants. Numerical values of out-of-plane force constants for benzene molecules have been re-
B2E
E1g
Atu
990 707
10 11
{ ;; 10
Units: cm.-l.
i
Dev.. % ’
0 0
Equal to obsd.
398 967 599 829
Eig
Calcd.
496 345 787
0 0 0 0
594.9 831.9 657.1 494.0 346.0 786.6
-0.7 0.3 -0.3 -0.4 0.3 -0.05
Vapor frequencies from ref. 44.
Normal Coordinate Transformation Matrix L. The L-matrix elements (S = LQ) were calculated by the standard method of characteristic vector^.^ The results are given in Table VI. %Matrix. The %matrix elements are obtained by eq. 7. Their numerical values a t the temperatures of (44) 5. Brodersen and A. Langseth, Kol. Dan& Videnskab. Selskab, Mat. Fys. Skrifter, 1, No. 1, 19 (1956). (45) F. A. Miller and B. L. Crawford, Jr., J. Chem. Phys., 14, 282 (1946).
Volume 69, Number 6
M a y 1966
1494
W. V. F. BROOKS, S. J. CYVIN,AND P. C. KVANDE
Table V : Out-of-Plane Force Constants of Benzene" Value
Symbol
0.199731 0.515172 0.245032 0.335156 0.248058 0,160992 0.421606 0.170303 a
Table VI1 : Out-of-Plane %Matrix Elements for Benzenes"
ZII(B~~) &z( B2g) MB2g) z(Ed Z(Az,) Zil(E2u) ZZZ( Ezu ) WE2u)
Units: mdyne/A.
O'K.
298'K.
Benzene 0.0889533 0.0428819 - 0.0469440 0,0250380 0.0269433 0.0468275 0.0352435 -0.0259594
0,0942044 0.0439858 - 0.0489271 0.0259010 0.0291215 0.0607336 0.0369411 -0.0301109
Benzene46 0.0886795 0.0386251 - 0.0478540 0.0194124 0.0196944 0,0457971 0.0309843 -0,0269616
Table VI L-Matrix elements in (a.m.u.)-l/z for the out-of-plane vibrations of benzene
SS
4 1.749927 -0,554937
E1g2
SI0
10 1.120125
Ah:
SI1
11 1.036980
Ezu 3
SI6
&.
S,
517
16 0.972875 -0.273434
5
- 0.966657 1.444427
relevant for the practical purpose of the present numerical computations. 17 -0,621077 1.356248
L-Matrix elements in (a.m.u.)-'/z for the out-of-plane vibrations of benzene-ds B2p:
A2u ) EIU
:
SS
4 0,972298 0,0394798
SI0
10 0.871003
SI1
11 0.761115
S,
&e SI7
16 0.792770 - 0.147666
5 - 1,746798 1.377219
Out-of-Plane Mean-Square Perpendicular Amplitudes To obtain the out-of-plane mean-square perpendicular amplitudes, the Cartesian displacements perpendicular to the molecular plane were expressed in terms of the symmetry coordinates. Next, the expressions of the type ZI - ZZ, zz - 26, etc., were derived. The expressions finally obtained for the out-of-plane meansquare perpendicular amplitudes are given as follows. For explanation of the symbol M, see eq. 13.
17
-0.838892 1.181668
absolute zero and 298°K. are given in Table VII. Some comments should be made on the calculations for benzene-ds. In the A-matrix (see eq. 7) the observed rather than calculated frequencies were used. In the computation of the L-matrix, however, the calculated frequencies had to be used. In consequence, the reported 2-values for benzeneds are not exactly consistent with the given force constants reported in Table V. The F-matrix consistent with 2 is given by F = L-lAL-l, where the A-matrix contains the observed frequencies. This discussion is however not The Journal of Phyeical C h a i a t r y
0,0941244 0.0400691 -0,0495518 0.0210968 0,0236522 0.0608265 0.0328541 - 0.0306065
1 ((dz*)2) = %(l
+ %)m y 2 z ( E d + -211(Ezu) 16 2
1
(25)
1 ((dp**)') = -2 72
(26) 1 1 (rz2) = ~(R/D)zZn(Bzg) 4- g2zz(Bzg) 1 -3"'&2(B2g) 9
+ 3M
+
VIBRATIONALMEAN-SQUARE AMPLITUDE MATRICES
1495
Numerical Mean-Square Perpendicular Amplitudes Numerical values have been evaluated for the inplane and out-of-plane mean-square perpendicular amplitudes, using eq. 14-23 and 24-33, respectively. The obtained results for benzene and benzene-ds a t the temperatures of absolute zero and 298°K. are given in Tables VI11 (in-plane) and IX (out-of-plane). It is interesting to compare the magnitudes of the in-plane and out-of-plane amplitudes for the same atom pair. In all cases but one, viz., C1C4for C6H6, the contributions from out-of-plane vibrations are greater, in accord with our expectations. The inplane contributions, however, are by no means negligible. In similar calculations for the naphthalene skeleton4s the in-plane contributions were tentatively Table VI11 : In-Plane Mean-Square Perpendicular Amplitudes for Benzenes" Distance
O'K.
298'K.
Benzene 0.00238230 0.00225402 0.00182551 0.0140091 0.0117116 0.0136442 0,0140858 0.0155484 0.0229718 0.0247866
0.00247397 0.00237269 0.00191836 0,0140925 0.0118583 0.0138761 0.0143076 0.0155700 0.0232637 0.0250174
Benzene-de 0.00232283 O.OO219412 0.00176412 0.0102215 0.00868817 0,0100984 0,0103409 0.0112286 0,0166858 0.0176803
~~~~
0.00241936 0,00231832 0.00186076 0.0105126 0.00899589 0.0105493 0.0108062 0.0113368 0.0173248 0.0183406
~
Volume 69,Number 6
M a y 1966
W. V. F. BROOKS, S. J. CYVIN,AND P. C. KVANDE
1496
neglected. The situation for naphthalene is probably not substantially different from that of benzene. Hence, the approximation made in the case of naphthalene' seems to be wrong.
Table IX : Out-of-Plane Mean-Square Perpendicular Amplitudes for Benzenes" O'K.
Distance
clc4
CiHi C2Hi CiHi CiHi HiHz HiHr H1H4
--6(t**)
ClC2 ClCl
c1c4 CiDi CD1 CiDi C4Dl DiDz DID; DiD, a
--6(d**)
O -K ---.
--6(r*) = K(r*)
K(d**) - 2 K ( d )
(35)
3
+-
(36)
ClCZ ClC3 ClC4 CiDi CiDi CaDi C4Dl DiDz DiDs DlD4
(0.00238463) (0.00119726) (0.00071209) (0.0117440) (0.00611281) (0.00376978) (0.00293081) (0.00816501) (0.00498065) (0.00377309)
Units:
--8(r***)
--6(t)
= =
... 0.0034232 0.0048472
... 0.0073411 0.0147853 0.0180207 0.0070920 0.0261707 0.0334222
... 0.0029330 0.0040572 ... 0.0057507 0.0112840 0.0135825 0.0059636 0.0194909 0.0244842
(0.00279008) (0.00143025) (0.00075818) (0.0128650) (0.00721224) (0.00444443) (0.00324867 ) (0.00966669) (0,00586108) (0.00402659)
... 0,0034023 0.0048220 , . .
0.0059437 0.0123503 0.0151965 0.0059884 0.0212543 0.0272836
A.
Atoms
OOK.
HCC HCH
0.0127362 0.0275045
2980~.
Benzene 0.0131735 0.0285750
Benzene-ds DCC DCD
0.0095253 0.0204270
0.0103745 0.0224616
A.
(37)
K(r***) - 2 K ( d ) - K ( r )
(38)
K(t) - K ( 4 - K(r)
(39)
The Journal of Physical Chemhtry
... 0.0072806 0.0140713 0.0168731 0.0071443 0.0251213 0.0316414
(0,00275524) (0.00134901) (0.00066329) (0.0169448) (0.00915918) (0.00576304) (0.00443458) :0.0126081) (0.00795085) (0,00597798)
Table XI : Linear Shrinkage Effects in Benzenes"
Units:
K ( r ) cos b
... 0.0029865 0.0041369
Benzene-,ds
(34)
(
298'K.
Benzene
0.00537611 0.00460317 0.00237597 0.0173787 0,0220788 0.0196897 0.0143905 0.0366293 0.0330479 0.0216193
- K ( d ) cos a - K ( r ) sin a
,
(0.00238383) (0.00114241) (0.00063076) (0.0163579) (0.00839980) (0.00529345) (0.00425251) (0.0115975) (0.00734039) (0.00584213)
K(d*) - 3'/'K(d)
=
(41)
Diatanoe
Bastiansen-Morino Shrinkage Effects In the benzene model there are eight independent shrinkage effects corresponding to the eight types of nonbonded distances. All of them are obtainable from the harmonic-vibrat,ion analysis. The following equations have been d e d ~ c e d . ~ ' =
K(t**) - 2 K ( d ) - 2K(r)
Table X : K Values (given in parentheses) and Nonlinear Shrinkage Effects in Benzenes"
0.00522417 0.00415562 0.00178809 0.0226439 0.0276050 0.0253345 0.0200870 0.0469912 0.0450694 0.0343081
Units: A.2.
-6(d*)
=
(40)
The K values are defined in accord with eq. 10. a and b denote the CHC (or CDC) equilibrium angles given by a = CIHICzand b = CIHICS. The calculated K values and shrinkage effects for benzene and benzeneds are given in Table X.
Benzene& 0.00433982 0.00359984 0.00221502 0.0152395 0.0176495 0.0155504 0.0123905 0.0292862 0.0261201 0,0197638
K(t*) - 3'"K(d) - 3'"K(r)
298'K.
Benzene 0.00427811 0.00327450 0.00169918 0.0214549 0.0244798 0.0223714 0.0188967 0.0419983 0.0401148 0.0331907
ClCZ ClCt
-&(t*) =
(46) S. J. Cyvin and E. Meisingseth, Kgl. Norske Videnskab. Sehkabs, Skrifter, No. 2 , 1 (1962). (47) 5. J. Cyvin, Acta Chem. Scand., 17, 296 (1963).
PRESSURE EFFECTS ON GLASS TRANSITION IN POLYMERS
In benzenes there are two types of linear conformations of atoms, represented by HICICl and H1CIHd. The corresponding linear shrinkage effects may be derived from the eight quantities of eq. 3 4 4 1 in the following way.
1497
(43)
Conclusion Spectroscopic calculations of shrinkage effects will no doubt become very important in the future as the accuracy of electrondiff raction experiments increases. There have already been performed some tentative refinements of electron-diff raction interatomic distances in benzene, using the presently calculated shrinkage effects.
Numerical values of linear shrinkage effects in benzene and benzene-& are reported in Table XI.
(48) W. V. F. Brooks, B. N. Cyvin, 5. J. Cyvin, P. C. Kvande, and E. Meisingseth, Acta Chem. Scond., 17, 345 (1963).
-6
l i n (r ***)
-61in(l**)
=
-6(r***
= --6(t**)
1
+ 6(d**)
+ 6(d**)
(42)
Pressure Effects on Glass Transition in Polymers
by Umberto Bianchi Istitdo di Chimka Industriale, Sez. V del Centro Nazionule d i Chimica delle Mactomolecole, Vniveraitb, Genova, Italy (Received August 3, 1964)
Heating at constant volume of a glass (polyvinyl acetate) has been followed by measuring internal pressure, Pi,over a range of temperature comprising the glass transition temperature T,O at 1 atm. The large increase in Pi a few degrees above T,O indicates that the glass transition takes place even if the heating process is performed at constant volume. Free volume considerations are shown to be still capable of offering an explanation of pressure effects on glass transition provided the change in specific volume of the glass at different T , (and P,) is taken into account.
Introduction Starting from the fundamental consideration that pressure is the thermodynamic variable that, like temperature, can be used to change the volume of a body, many workers have studied the influence of pressure on the glass transition temperature, T,, of polymers.'-' In all of these papers it has been explicitly (or tacitly) assumed that we can speak of a significant glass temperature provided the time scale of the experiments is long enough to assure reproducibility of the most important physical properties (specific volume, thermal expansion coefficient, etc.) below and above T,. Experience has shown that this reproducibility can be achieved by changing the temperature of the sample
under investigation very slowly, at a rate of say 1-2'/ day, In these conditions, it also becomes possible6to apply thermodynamics to glass transition, with results which have proved very useful for a better understanding of some aspects of the transition itself. It is well known that the "free volume (1) N. Hirai and H. Eyring, J . Polymer Ai., 37, 51 (1959). and H. V. Belcher, J . Res. Natl. Bur. Sa., 67A,
(2) J. E. McKinney 43 (1963).
(3) J. M. O'Reilly, J. Polymer Sci., 57, 429 (1962). (4) K. H. Hellwege, W. Knappe, and P. Lehmann, Kolloid-Z., 183, 110 (1962). (5) G. Gee,P. N. Hartley, J. B. Herbert, and H. A. Laceley, Polymer, 1, 366 (1960).
Volume 69, Number 6
May 1966
