The Structure of the Free Volume in Poly(styrene-co-acrylonitrile ...

500 Full Paper
Summary: The structure of the free volume and its temperature
dependence between 25 and 190 8C of copolymers
of styrene with acrylonitrile, SAN (0 to 50 mol-% AN), is
studied by pressure volume temperature (PVT) experiments
and positron annihilation lifetime spectroscopy (PALS). In
this first part of the work, PVT data are reported which were
analysed with the Simha-Somcynsky equation of state (S-S
eos) to estimate the volume fraction of holes, h, which constitute
the excess free volume. The temperature and pressure
dependence of the specific volume V, the specific occupied
and free volume, V occ ¼ (1 h)Vand V f ¼ hV, and the corresponding
isobaric expansivities and isothermal compressibilities
for both the rubbery and glassy state, are estimated. We
obtained the unexpected results that (i) the occupied volume
changes its coefficient of thermal expansion at T g from
a occ,g 0.5a g 1 10 4 K 1 below T g to almost zero above
T g and (ii) the isothermal compressibility of the occupied
volume at zero pressure is rather high, kocc 2
10 4 MPa 1 , and decreases only slightly at T g. The variation
of total, occupied, and free volume parameters with the
composition of the SAN copolymers is discussed.
The Structure of the Free Volume in
Poly(styrene-co-acrylonitrile) from Positron Lifetime
and Pressure Volume Temperature (PVT) Experiments
I. Free Volume from the Simha-Somcynsky Analysis of
PVT Experiments
Günter Dlubek,* 1 Jürgen Pionteck, 2 Duncan Kilburn 3
1
ITA Institut für Innovative Technologien GmbH, Köthen, Aussenstelle Halle, Wiesenring 4, D-06120 Lieskau (bei Halle/S),
Germany
Fax: þ49-40-3603241463; E-mail: gdlubek@aol.com
2
Institut für Polymerforschung Dresden e.V., Hohe Strasse 6, D-01069 Dresden, Germany
3
University of Bristol, H.H. Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1 TL, UK
Received: September 16, 2003; Accepted: November 11, 2003; DOI: 10.1002/macp.200300103
Keywords: free volume; glass transition; microstructure; pressure-volume-temperature (PVT) experiments; styrene copolymers
Introduction
The concept of free volume in polymers has proved useful in
discussing physical properties such as viscosity, viscoelasticity,
the glass transition, volume-recovery experiments,
and mechanical properties. [1–8] In general, the free (not
Specific volume V of SAN50 as a function of temperature T
and as selection of isobars (in MPa).
occupied) volume, V f, is defined as difference between the
total volume V and the occupied volume Vocc. [8] Vf and its
volume fraction f are given by
Vf ¼ V Vocc ð1aÞ
f ¼ 1 Vocc=V ð1bÞ
Macromol. Chem. Phys. 2004, 205, 500–511 DOI: 10.1002/macp.200300103 ß 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The Structure of the Free Volume in Poly(styrene-co-acrylonitrile) from Positron Lifetime ... 501
However, no unambiguous definition of the occupied
volume is available. Due to this, very different values for Vf
and f and sometimes misleading conclusion can be found in
the literature.
The simplest and most clear way is to identify V occ with
the van der Waals volume V W, that is the space occupied by a
molecule, which is impenetrable to other molecules with
normal thermal energies. [8,9] VW can be calculated using the
group contribution method developed by Bondi. [10,11] In
this case Vf : VfW and fW ¼ 1 VW/V represents the total
(van der Waals) free volume or empty space. This type of
empty space is well known in polymer crystals where it is
frequently termed interstitial free volume, [8] V c ¼ V W þ V fi.
It has values typically somewhat larger than 1 C hdp ¼
0.26 where C hdp ¼ 0.74 is the packing coefficient of the
most dense packing of hard spheres (hdp or fcc lattice) [11]
and expands with the temperature due to the anharmonicity
of molecular vibrations.
Amorphous polymers contain an additional free volume
which lowers the density by about 10% compared with
the crystalline state of the same material. [11] This excess
free volume appears in form of many irregularly shaped
cavities or holes of atomic and molecular dimension (local
free volumes) which arise because of disordered molecular
packing in the amorphous phase (static and pre-existing
holes), and molecular relaxation among the molecular
chains and terminal ends (dynamic and transient holes). The
diffusion of small molecules through glassy polymers and
the dynamics of rubbery polymers, as observed in mechanical
or dielectrical relaxation and in viscosity experiments,
[1–6] are related to this type of (excess or hole) free
volume. For calculating the hole free volume, Vf : Vfh ¼
V Vocc, fh ¼ 1 Vocc/V, various approximations of the
occupied volume, Vocc, which now includes the interstitial
free volume, Vocc ¼ VW þ Vfi, fi ¼ 1 Vfi/V, are used. The
total (van der Waals) free volume is given by VfW ¼ Vfi þ
V fh, the corresponding fractional free volume by
f W ¼ f i þ f h. Following the traditional terminology we will
use in the following paragraphs the symbols V f (:V fh) and
f (:fh) to describe the (excess) hole free volume and its
volume fraction and VfWand fW for the total, van der Waals,
free volume. Most common approximations of the occupied
volume are the zero point volume estimated from the
extrapolation of the densities of crystalline or liquid
(rubbery) materials down to 0 K, [1,2,10–12] V occ(0) ¼
V r(0) ¼ V c(0) 1.3 V W, [10,11] and the crystalline volume
at room temperature, V occ(25 8C) ¼ V c(25 8C)
1.45 VW. [11,13,14]
Due to the anharmonicity of molecular vibrations the
occupied volume will show a thermal expansion with a
coefficient aocc, Vocc(T) ¼ Vocc(0)(1 þ aoccT). Since aocc is
usually unknown, it is standard to identify it with the
coefficient of thermal expansion of the glassy state, a occ ¼
a g. [12] From this follows a frequently used relation for the
calculation of the temperature dependence of the fractional
excess (hole) free volume above Tg:
f ðTÞ ¼fg þ afr *ðT TgÞ ð2aÞ
afr * Da ¼ ar ag ð2bÞ
where fg is the fractional excess free volume at Tg and
a fr* ¼ (1/V) (dV f/dT) ¼ (1/V) [d(V V occ)/dT] is the fractional
coefficient of thermal expansion of the excess free
volume in the rubbery state of the polymer. a fr* is usually
approximated by the difference of the coefficients of
thermal volume expansion in the rubbery and glassy
states, afr* ¼ Da ¼ (ar ag). [12] William, Landel, and Ferry
(WLF) [3] estimated from viscosity data values of Da ¼
4.8 10 4 K 1 and fg ¼ 0.025 (WLF equation with B ¼ 1)
which were believed for a long time to be universal.
Later [15] it was found, however, that the hole fraction
increases from f g 0.02 for polymers with T g ¼ 200 K to
fg 0.08 for polymers with Tg ¼ 400 K, a results which was
confirmed recently. [16] From this result it was concluded
that the glass transition is not an iso-free volume transition
but rather determined by the structural relaxation time
(tg 100 s).
A method which proved to be very successful in estimation
of the fraction of free volume holes is the analysis of
pressure volume temperature (PVT) experiments using the
Simha-Somcynsky hole theory equation of state (S-S
eos). [15,17–19] This theory describes the structure of a liquid
by a cell or lattice model which allows an occupied latticesite
fraction y ¼ y(V, T) of less then one. The volume of an
occupied lattice-site contains the van der Waals volume of
the mer and a free volume inherent to the lattice cell. The
latter can be imagined to be similar to the interstitial free
volume of an elementary cell in crystals. In atomistic
modelling it has its correspondence in a large number of
holes being too small to contribute essentially to the
transport of small molecules, for example. It is assumed that
N molecules, consisting of n chemical repeating units, nmers,
with molecular weight Mrep, are divided into s
equivalent segments, s-mers, with molecular weight M 0,
sM 0 ¼ nM rep. The number N of s-mer molecules occupy
the fraction y ¼ sN/(sN þ N h SS ) of the total available sites,
(sN þ Nh SS ), where Nh SS is the number of unoccupied lattice
sites or holes of the S-S theory. The excess free-volume
fraction is given by the fraction of unoccupied lattice-sites
(holes) which is denoted in this theory by h, h(V, T) ¼ 1 y.
The h of the S-S eos corresponds to our definition of f, f : h;
in this paper we will use both symbols synonymously. The
macroscopic volume expands mainly due to the formation
of new empty lattice sites.
The S-S theory expresses the configurational or Helmholtz
free energy F in terms of the volume V, temperature T
and occupied lattice-site fraction y ¼ y(V, T), F ¼ F(V, T, y).
The value of y is obtained through the pressure equation
P ¼ (@F/@V)T and the minimisation condition (@F/
@y) V,T ¼ 0. The S-S eos is the most successful theory in
Macromol. Chem. Phys. 2004, 205, 500–511 www.mcp-journal.de ß 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

502
describing the variation of the specific volume as function
of the temperature T and the hydrostatic pressure P. Several
universal relationships where found which allow an
approximate estimation of h and fW. [9,17–19]
Despite a great deal of interest in investigations of free
volume in polymers, only limited information about its real
structures, the hole dimensions and the size and shape
distributions, are available, mainly due to a lack of suitable
probes for open volumes of molecular dimensions. During
the past decade, positron annihilation lifetime spectroscopy
(PALS) has developed to be the most important method for
studying sub-nanometer size holes in polymers. [20–23] In
this technique, positronium in its ortho-state, o-Ps, is used
as a probe for local free volumes. o-Ps is formed in or
trapped by sub-nanometer size holes that form the (excess)
free volume of amorphous polymers. Due to collisions with
the walls of the holes, the lifetime of o-Ps decreases with
decreasing size of holes. Assuming spherical holes, the hole
size can be calculated from the o-Ps lifetime using a semiempirical
model.
PALS itself is able to measure the mean volume of the
holes and, with larger limitations, their size distribution, [24]
but not directly the hole density and the hole fraction.
However, a correlation of PALS results with the hole free
volume fraction estimated from the macroscopic volume by
the help of the S-S eos theory allows to estimate the hole
density. [25–30] In one of our previous works [31] some of us
have shown that in thermal expansion experiments the
direct comparison of the hole volume from PALS with the
macroscopic volume even allows to estimate the number
density of holes and their entire volume fraction. In this
way, all parameters of hole free volume can be determined.
This method has been successfully applied by several
groups. [14,16,29,30] In the analysis of the experiments it is,
however, usually assumed that the occupied volume does
not expand. [14,16,29] As the thermal expansion of crystals
shows [32] this simplification may be too strong.
In order to obtain detailed information on the microstructure
of the free volume in glassy and rubbery states of
polymers, we have studied in the present work the temperature
dependence of the specific volume, V, by pressurevolume-temperature
(PVT) experiments [33,34] and of the
size of local free volumes (holes) by PALS. For this study
we have chosen a series of copolymers of styrene (S,
[–CH 2CH(C 6H 5)–]) with acrylonitrile (AN, [–CH 2CH-
(CN)–]), SAN, with a content of AN comonomer between
0 and 50 mol-%. Polystyrene (PS) has been frequently
studied in the literature by PALS, [14,16,26,28] while for SAN
only two papers presenting a series of room temperature
measurements [28,35] are known to us. SAN is an interesting
system for our study, since the specific volume of the
copolymers change with variation of the AN content while
the glass transition temperature, T g, is almost constant.
Our work is subdivided into two parts. In this part, we
present PVT experiments and determine from these, using
the S-S eos, [15,17–19] the hole fraction h. The temperature
dependence of the specific volume V, the specific occupied
and free volume, Vocc ¼ yVand Vf ¼ hV, and the corresponding
isobaric expansivities and isothermal compressibilities
for both the rubbery and glassy states, are estimated. The
variation of total, occupied and free volume parameters
with the composition of the SAN copolymer will be
discussed. From our data we will confirm that fg 0.07 >>
0.025 for SAN and show that the traditional approximation
afr* ¼ Da ¼ ar ag is not correct.
The second part of the work shows measurements of the
local free volume using PALS. For the analysis of the
positron lifetime spectra we used the new routine LT in its
latest version 9.0 [36,37] which allows both discrete and log
normal distributed annihilation rates. From these distributions
the mean size and the size distribution of free volume
holes can be calculated. We will also discuss the analysed
hole size distribution in relation to theoretical models
describing the thermal volume fluctuation. From the comparison
of specific total and free volumes, Vand Vf, with the
mean hole volume the hole number density is estimated.
Moreover, we discuss variations in the structure of the
free volume (fraction, hole size and number) as a function of
the content of AN co-monomer and temperature. Finally,
we will show relations between the bulk modulus and the
free volume.
Experimental Part
G. Dlubek, J. Pionteck, D. Kilburn
Poly(styrene-co-acrylonitrile) (SAN) test samples were provided
by the BASF AG Ludwigshafen. Their characteristics are
giveninTable1.ThemolecularweightsweredeterminedbyGPC
withRI-detector using PS standards and by GPC coupled withan
LALLS-detector. The mean values of both methods of two
measurements each were very similar (deviation in M w within
3%). The acrylonitrile co-monomer content was calculated from
the nitrogen concentration determined by elemental analysis by
means of an EA1108 (Carlo Erba). The standard deviation from
three measurements was within 0.1% (except SAN50: 0.25%).
The DSC measurements were performed with a DSC-7 (Perkin-
Elmer), PYRIS-software, Version 4.01. The glass transition
temperaturesT g(half stepmethod) andthestepheightsDc p given
in Table 1 were determined from the 2 nd heating run of a circle 1 st
heat, 1 st cool, 2 nd heat with a scan rate of þ10/ 80 K min 1 in
the temperature range of 10 to 200 8C.
The PVT experiments were carried out by means of a fully
automated GNOMIX high-pressure dilatometer. [33,34] The
data were collected in the range from room temperature to
250 8C in steps of 10 K. At every constant temperature the
material was pressurised from 10 to 200 MPa. The specific
volumes for ambient pressure were obtained by extrapolating
the values for 10 to 30 MPa in steps of 1 MPa according to the
Tait equation using the standard GNOMIX PVT software. The
accuracy is within 0.002 cm 3 g 1 . The data obtained in a
cooling run after the heating showed a disappearing small
hysteresis to the heating data and are therefore not discussed.
The densities of the samples at room temperature were
Macromol. Chem. Phys. 2004, 205, 500–511 www.mcp-journal.de ß 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The Structure of the Free Volume in Poly(styrene-co-acrylonitrile) from Positron Lifetime ... 503
Table 1. Sample characterisation and volume parameters estimated from PVT data (see text).
Quantity Uncertainty PS SAN22 SAN38 SAN50
AN content (mol-%) 0.2% 0 22.1 38.4 50.6
AN content (wt.-%)
Mn (kg mol
0.1% 0 12.6 24.1 34.3
1 )
Mw (kg mol
175 108 74 67
1 )
Tg (DSC, 8C)
Dcp (DSC, J g
2
394
104
203
104
153
110
131
108
1 K 1 )
Tg (spec. vol., 8C)
V25 (cm
2
0.32
98
0.35
99
0.39
101
0.44
98
3 g 1 )
VW (cm
0.001 0.9506 0.9404 0.9335 0.9190
3 g 1 )
Vg (cm
0.6033 0.5990 0.5958 0.5934
3 g 1 )
Eg (10
0.002 0.9660 0.9540 0.9451 0.9306
4 cm 3 g 1 K 1 )
Er (10
0.05 2.17 1.85 1.61 1.65
4 cm 3 g 1 K 1 )
ag (10
0.05 6.14 5.90 5.55 5.27
4 K 1 )
ar (10
0.05 2.25 1.94 1.71 1.77
4 K 1 )
kg (10
0.05 6.36 6.18 5.87 5.66
4 MPa 1 )
kr (10
0.1 3.62 3.61 3.08 2.96
4 MPa 1 )
T* (K)
V* (cm
0.1
40
5.76
11650
5.90
11797
5.41
12160
5.24
12380
3 g 1 ) 0.002 0.9373 0.9281 0.9235 0.9134
P* (MPa) 5 848.5 860 880 890
Mrep (g mol 1 )
M0 (g mol
104.1 97.98 91.35 86.25
1 )
o (A˚
0.3 40.48 40.94 41.45 42.19
3 ) 0.5 60.2 60.3 60.7 61.1
Vocc,g (cm 3 g 1 )
Eocc,g (10
0.003 0.895 0.886 0.882 0.872
4 cm 3 g 1 K 1 )
Eocc,r (10
0.05 0.89 0.92 0.93 0.90
4 cm 3 g 1 K 1 )
*
aocc,g (10
0.05 0.20 0.18 0.18 0.18
4 K 1 )
*
(10
0.05 0.91 0.96 0.98 0.97
4 K 1 ) 0.05 0.21 0.19 0.19 0.20
aocc,r
aocc,g (10 4 K 1 ) 0.05 0.98 1.04 1.05 1.04
aocc,r (10 4 K 1 ) 0.05 0.22 0.21 0.20 0.21
*
kocc,g (10 4 MPa 1 ) 0.05 2.22 2.20 2.13 2.15
*
(10 4 MPa 1 ) 0.05 2.12 2.12 2.00 1.92
kocc,r
kocc,g (10 4 MPa 1 ) 0.05 2.18 2.17 2.10 2.12
kocc,r (10 4 MPa 1 ) 0.05 2.17 2.09 1.97 1.90
Vfg (cm 3 g 1 ) 0.003 0.070 0.067 0.063 0.058
fg ¼ hg 0.003 0.073 0.070 0.067 0.063
T0 0 (8C) 5 25 18 15 15
Hh (kJ mol 1 ) 0.1 7.24 7.92 8.02 8.16
Efg (10 4 cm 3 g 1 K 1 ) 0.06 1.19 0.94 0.69 0.74
Efr (10 4 cm 3 g 1 K 1 ) 0.06 5.89 5.74 5.39 5.07
* 4 1
afg (10 K ) 0.06 1.34 0.98 0.73 0.80
* 4 1
afr (10 K ) 0.06 6.10 6.02 5.70 5.44
afg (10 3 K 1 ) 0.08 1.84 1.39 1.09 1.28
afr (10 3 K 1 ) 0.08 8.41 8.56 8.67 8.70
* 4 1
kfg (10 MPa ) 0.3 1.8 1.6 1.3 1.0
kfr
* (10 4 MPa 1 ) 0.1 3.41 3.11 3.00 2.89
kfg (10 3 MPa 1 ) 0.4 2.5 2.2 1.9 1.7
kfr (10 3 MPa 1 ) 0.1 4.71 4.43 4.50 4.64
determined by means of a Ultrapycnometer 1000 (Quantachrome)
with an accuracy of 0.03%.
Results and Discussion
Specific Volume
In the following section we will discuss firstly the ambient
pressure volume parameters as a function of the SAN
composition. In one of the last paragraphs, the pressure
dependence of the specific volume will be viewed. Figure 1
displays the temperature dependence of the specific volume
Vof the SAN copolymer series for P ¼ 0.1 MPa. V increases
almost linearly within the two temperature regions below
and above Tg with a distinct increase in its slope at Tg. The
experimental data were linearly fitted in the temperature
ranges from 25 8C toTg 10 K and from Tg þ 10 K to
200 8C. Table 1 shows the volumetric Tg’s together with the
DSC values. Further parameters shown are the specific
volume at room temperature, V 25, the van der Waals
volume, V W (van Krevelen [11] ), the specific volume at T g,
Vg, and the isobaric (P ¼ 0.1 MPa 0) expansivities of the
Macromol. Chem. Phys. 2004, 205, 500–511 www.mcp-journal.de ß 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

504
Figure 1. The specific volume Vof SAN copolymers as function
of temperature T at ambient pressure. Empty symbols: experimental
data, dots: S-S eos fits using Equation (8), straight lines:
linear fits in the ranges from 25 8CtoT g 10 K and from T g þ 10 K
to 200 8C.
specific volume, E g ¼ [dV/dT] P (T < T g) and E r ¼ [dV/dT] P
(T > T g), in the glassy and rubbery state of the polymers.
The volumetric Tgs are systematically lower by 6 to 10 K
than those from DSC. This slight discrepancy may be
attributed to the different physical quantities probed by
these techniques, and to the way to calculate Tg as ‘‘midpoint’’
temperature of the glass transition. The Tgs show a
slight increase with increasing content of AN comonomer
which appears more clearly in DSC experiments and
follows the relation
T DSC
g ¼ 103:5ð 2Þþ0:11ð 0:06Þ XAN ð3Þ
where Tg is given in 8C and is XAN the content of AN
comonomer in mol-%.
As displayed in Figure 2, while T g increases slightly, the
values of V25 and Vg decrease parallel to the van der Waals
volume VW and follow the relations
V25 ¼ 0:952ð 0:003Þ 5:9ð 1Þ 10 4 XAN ð4Þ
Vg ¼ 0:968ð 0:003Þ 6:7ð 1Þ 10 4 XAN ð5Þ
In this paper all specific volumes are given in units of
cm 3 g 1 . The ratio V25/VW shows a weak variation from
1.58 (PS) to 1.54 (SAN50) while Vg/VW changes from 1.60
to 1.56 ( 0.02). A linear extrapolation of the equilibrium
specific volume from above Tg down to 0 K delivers Vr(0)/
VW ¼ 1.23 ( 0.05). When using the S-S eos analytic
expression for extrapolation (see Equation (8)), one obtains
V r(0)/V W ¼ 1.39 ( 0.05). Both values can be compared
with Vr(0)/VW ¼ 1.3 (Bondi [10,11] ) frequently assumed as
general value of the occupied volume of amorphous
polymers at 0 K.
The specific expansivities Eg and Er decrease roughly
linearly with the mole content of AN comonomer,
G. Dlubek, J. Pionteck, D. Kilburn
Figure 2. The Tg from DSC and the various specific volumes of
SAN copolymers as a function of the content of AN comonomer.
Shown are the specific total, occupied, and free volume at Tg, Vg,
Vocc,g, Vfg, the total volume at 25 8C, V25, the S-S eos scaling
volume, V*, and the van der Waals volume, VW.
E g ¼ (2.2 1.6) 10 4 cm 3 g 1 K 1 and E r ¼ (6.1
5.3) 10 4 cm 3 g 1 K 1 (Table 1). As shown in Figure 3,
the isobaric coefficients of thermal expansion, a ¼ (1/V)
[dV/dT]P, in the glassy and rubbery state, ag ¼ Eg/Vg and
ar ¼ Er/Vg, show the same behaviour with increments of
0.014 10 4 K 1 and 0.020 10 4 K 1 per mol-%
AN, while their differences, Da ¼ (ar ag) ¼ (4.1 0.2)
10 4 K 1 , are almost constant. Moreover, we remark that
the value of DaT g is constant within their uncertainties,
DaT g ¼ 0.15 0.01, while a rT g may show a slight decrease
from 0.24 to 0.21. Here Tg is given in K. The last two
relations where discovered in the classical work of Simha
and Boyer [12] and confirmed recently. [16]
Figure 3. Coefficients of thermal expansion of the total volume
for the glassy and rubbery state, ag and ar, and their difference
Da ¼ ar ag (empty symbols), and the compressibilities kg, kr,
and Dk ¼ kr kg (filled symbols), as a function of the content of
AN comonomer in SAN.
Macromol. Chem. Phys. 2004, 205, 500–511 www.mcp-journal.de ß 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The Structure of the Free Volume in Poly(styrene-co-acrylonitrile) from Positron Lifetime ... 505
Free Volume from S-S eos
The Simha-Somcynsky equation-of state [15,17–19] follows
from the pressure equation P ¼ (@F/@V)T, where F ¼ F(V,
T, y) is the configurational (Helmholtz) free energy F of the
liquid,
~P~V
~T ¼ 1 yð21=2y~VÞ þ y
~T
2:002ðy~VÞ 4
1=3 1
2:409ðy~VÞ 2
ð6Þ
~P, ~V, and ~T are reduced variables, ~P ¼ P/P*, ~V ¼ V/V*,
~T ¼ T/T*, where P*, V*, and T* are characteristic scaling
parameters. The occupied volume fraction y ¼ 1 h is
coupled with ~T and ~V in a second equation derived from the
minimisation condition (@F/@y)V,T ¼ 0 and assuming for
polymers s !1and a flexibility ratio of s/3c ¼ 1. 3c is the
degree of freedom of a s-mer molecule. [24–26] The scaling
parameters are linked by the equation
ðP*V*=T*ÞM0 ¼ðc=sÞR ¼ R=3 ð7Þ
where M0 is the molecular mass of a s-mer occupying a
lattice site, sM0 ¼ nMrep.
It was shown that both equations may be replaced in the
temperature and pressure ranges ~T ¼ 0.016 to 0.071 and
~P ¼ 0 to 0.35 by the universal interpolation expression
(Utracki and Simha, [19] see here also for the values of the
constants a0 to a5)
ln ~V ¼ a0 þ a1 ~T 3=2 þ ~P½a2 þða3 þ a4 ~P þ a5 ~P 2 Þ~T 2 Š ð8Þ
There is no universal relationship for the h-function in the
glassy state at present. The S-S eos Equation (6) is derived
under the general assumption of equilibrium, however the
specific assumption that the free energy is a minimum, has
not been made. Therefore, it is usual to calculate the h
values from the specific volume below Tg (respectively
Tg(P), the pressure dependent glass transition) via Equation
(6) assuming constant scaling parameters P*, V* and T*.
These h values are considered to be sufficiently good
approximations for conditions not too far from equilibrium.
[15,17–19,38–40]
A first set of scaling parameters we have determined by
using the standard GNOMIX PVT software. Since we are
most interested in good fits of the ambient pressure isobar,
we have estimated the final parameters in the following
way. First T* and V* were determined from non-linear least
squares fits of Equation (8) to the volume data for P ¼ 0 MPa
(ambient pressure) plotted in the temperature range from
T g þ 10 8C to 250 8C. As shown in Figure 1, Equation (8)
describes well the experimental data above Tg. Typical
coefficients of determination of Cd ¼ 0.9999, correlation
coefficients squared of r 2 ¼ 0.9998 and standard deviations
of s ¼ 0.0003 were obtained. The maximum deviation of
the fits from the experimental data amounts to jDVj¼
0.0009 cm 3 g 1 corresponding to 0.09%. Both, the
experimental data and the fits to Equation (8) increase with
temperature slightly faster than the linear fits from the range
Tg to 200 8C.
In a second fit including the data from the PVT field in the
range (T g(P) þ 10) 8C to 200 8C and 0 to 200 MPa and fixing
T* and V* to the former values the scaling pressure P* was
found. The scaling parameters describe well the experimental
curves in the whole fitting range, the maximum
deviation between fits and experiments is now somewhat
larger than for P ¼ 0 and amounts to jDVj¼0.002 cm 3 g 1 .
Our scaling parameters for PS deviate somewhat from
previous ones [41] but are similar to those found recently by
Schmidt et al. [29]
The S-S eos originally derived for homopolymers has
been applied also for polymer blends and copolymers. [42,43]
In case of random mixing the derived parameters are mean
values. In particular, M0 is then the mean molecular weight
of a s-mer determined by averaging the molecular weights
M0i of the component s-mers over the number sixi of
component mers where xi is the mole fraction and si the
number of s-mers of a molecule of the component i.
We found that all of the scaling parameters vary linearly
with the content of AN comonomers (Table 1 and V* in
Figure 2). From linear fits we obtained
V* ¼ 0:938ð 0:002Þ 4:5ð 0:6Þ 10 4 XAN ð9Þ
T* ¼ 11586ð 90Þþ14:8ð 3Þ XAN ð10Þ
P* ¼ 846ð 3Þþ0:85ð 0:1Þ XAN ð11Þ
where V*, T*, and P* are given in cm 3 g 1 , K, and MPa,
respectively. With these scaling parameters the specific
volume Vas function of temperature Tand pressure P can be
predicted from Equation (8) for the SAN copolymer system
with an AN content between 0 and 50 mol-%. The observed
linear composition dependence of the scaling parameters
suggest an absence of strong specific interactions between S
and AN comonomers. The scaling volume V* has a constant
ratio to the van der Waals volume of V*/VW ¼ (1.54 0.01).
This ratio is close to that found by other groups for a larger
variety of polymers, 1.45 [15] and 1.57 to 1.60. [16]
From the scaling parameters the molecular mass of a smer
occupying a lattice cell, M0, and the cell volume o at a
given temperature can be estimated using Equation (7). Our
scaling parameters deliver values for M0 increasing linearly
between M 0 ¼ 40.6 g mol 1 for PS and 42.2 g mol 1 for
SAN50. The ratio M 0/M rep ¼ n/s varies linearly with X AN,
M0=Mrep ¼ 0:384ð 0:008Þþ0:291ð 0:02Þ 10 2 XAN
ð12Þ
In the literature the values of M0/Mrep for different
polymers vary between 0.25 and 1.25; [41] for PS 0.49 [41] and
0.38 [28] , and for SAN25 wt.-% 0.40 [28] have been estimated.
From our values for M0/Mrep a cell volume at Tg, og ¼
Macromol. Chem. Phys. 2004, 205, 500–511 www.mcp-journal.de ß 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

506
(M0/Mrep) Vrep(Tg) ¼ M0Vocc,g/NA (NA – Avogadro’s number),
showing a slight increase with raising AN content from
og ¼ 60.2 to 61.1 ( 0.5) A˚ 3 follows.
Using the scaling parameters P*, V*, and T* we have
calculated the fractions of occupied sites, y, and hole sites,
h ¼ 1 y, from a numerical solution of Equation (6) for
both temperature ranges, above and below T g. Figure 4 and 5
show the results of our analysis displayed as the specific
occupied volume Vocc ¼ yV, and specific free volume Vf ¼
hV. As in the S-S eos, we assume that the partial volumes
Vocc and Vf behave additively and are exposed to the same
temperature T and hydrostatic pressure P as applied
externally to the sample, V(T, P) ¼ V occ(T, P) þ V f(T, P).
Under these assumptions, their expansivities and compressibilities
behave also additively. One observes that the
occupied volume shows a linear expansion below and above
Tg with an abrupt decrease in its expansivity at Tg. The free
volume expands even almost linearly in both of the temperature
regions, below and above Tg and shows a strong
increases in the expansivity at Tg.
The total increase of the specific free volume V f comes
from the increase in N h SS , i.e. the creation of new empty
cells, and to a small extent from the thermal expansion of
the cells assumed to have the uniform size o ¼ Vocc/Ns. In
order to study the real variation in the number of created
holes, Nh SS , we have to consider the hole number related to
the number of occupied lattice sites, Ns. Its variation per 1 K,
d(Nh SS /Ns)/dT ¼ d(Vf/Vocc)/dT ¼ d(h/y)/dT, amounts for PS
to 6.71 10 4 K 1 above and 1.42 10 4 K 1 below T g.
Both values decrease with increasing content of AN comonomer
and amount to 5.94 10 4 K 1 and 0.79
10 4 K 1 , respectively, for SAN50.
When we consider the holes as quasi-point defects, their
concentration may be calculated from [44,45]
h ¼ expðSh=kBÞ expð Hh=kBTÞ ð13Þ
Figure 4. The specific occupied volume, Vocc ¼ yV, of SAN
copolymers as function of temperature T at ambient pressure.
(Symbols as in Figure 1, straight lines: linear fits).
G. Dlubek, J. Pionteck, D. Kilburn
Figure 5. The specific hole free volume, Vf ¼ hV, of SAN
copolymers as function of temperature T at ambient pressure
(Symbols as in Figure 1, straight lines: linear fits).
where h is the number of holes per lattice site, Hh is the
hole formation enthalpy, Sh is the hole formation entropy, kB
is the Boltzmann constant, and T is the absolute temperature.
Equation (13) is derived from minimising the free
enthalpy with respect to the hole number. Arrhenius plots
of ln h vs. 1/T show straight lines above T g (Figure 6) with a
typical variance of 0.004 and r 2 value of 0.9997. From
the fits a molar activation enthalpy for the hole formation in
PS of Hh ¼ 7.24 kJ mol 1 can be estimated which increases
systematically to Hh ¼ 8.16 kJ mol 1 for SAN50,
while Sh/R fluctuates around 0.9. The fit parameters correspond
to Hh ¼ (2.3 2.6) RTg (R – gas constant) which is
close to H h 3 RT g estimated by Perez [44] from calorimetric
data.
We may imagine the hole creation like the formation of a
Schottky defect, i.e. a s-mer migrates to the surface leaving
Figure 6. Arrhenius plot of the fraction of empty lattice cells
above Tg of SAN copolymers (Symbols as in Figure 1, straight
lines: linear fits).
Macromol. Chem. Phys. 2004, 205, 500–511 www.mcp-journal.de ß 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The Structure of the Free Volume in Poly(styrene-co-acrylonitrile) from Positron Lifetime ... 507
an unoccupied lattice site in internal regions of the sample.
For the migration, the bonds of a mer to its neighbours must
be broken. In the average, half of them are reconstructed at
the surface. From this consideration follows that the hole
formation enthalpy Hh should approximately correspond to
half of the cohesive energy. One may calculate the cohesive
energy per mole lattice site from Ec ¼ d 2 Vmol(M0/Mrep) ¼
d 2 (Mrep/r)(M0/Mrep) ¼ d 2 (M0/r) where r is the density,
Vmol ¼ Mrep/r is the molar volume of the polymer, and Mrep
and M0 are, as before, the molar mass of a n-mer and a s-mer,
respectively. d is the solution parameter defined as the square
root of the cohesive energy density, d ¼ (Umol/Vmol) 1/2
with Umol the mean value of the intermolecular interaction
energy per mole. In van Krevelen’s collection [11] d
18 J 1/2 cm 3/2 for PS and d 28 J 1/2 cm 3/2 for polyacrylonitrile
(PAN, r ¼ 1.184 g cm 3 , Mrep ¼ 53.1 g mol 1 )
can be found. These values lead to Ec ¼ 12.4 kJ mol 1 for
PS and a mean value of Ec ¼ 16.2 kJ mol 1 for SAN50.
The half of both values is not very far from the Hh values
found for PS and SAN50.
Thermal Expansion and Free Volume
From the curves shown in Figure 4 and 5 one may derive the
expansivities of the specific occupied volume, E occ,g ¼
dV occ/dT (T < T g) and E occ,r ¼ dV occ/dT (T > T g), and of the
specific free volume, Efg ¼ dVf/dT (T < Tg) and Efr ¼ dVf/
dT (T > Tg) as well as the characteristic volumes at Tg,
Vocc,g ¼ Vocc(Tg) and Vfg ¼ Vf(Tg) using linear fits. The
expansivities determine the corresponding fractional coefficients
of the thermal expansion of the occupied and free
volumes defined by
aocc;g * ¼ Eocc;g=Vg ð14aÞ
afg * ¼ Efg=Vg ð14bÞ
when T < Tg and
aocc;r * ¼ Eocc;r=Vg ð14cÞ
afr * ¼ Efr=Vg ð14dÞ
when T > Tg.
The fractional coefficients are related to a by
a ¼ aocc * þ af*. They are related to the coefficients of thermal
expansion of the occupied and free volume, aocc ¼ (1/
Vocc)(dVocc/dT) and af ¼ (1/Vf)(dVf/dT), via aocc * ¼
(1 f)aocc and af* ¼ faf (f : h). The volumes and coefficients
are shown in Table 1. We notice the relations
afg* aocc,g * 0.5 ag (T < Tg) and afr* ar, aocc,r * 0(T > Tg). From linear fits the relations
Vocc;g ¼ 0:8958ð 0:002Þ 4:5ð 0:6Þ 10 4 XAN ð15Þ
Vfg ¼ 0:0712ð 0:001Þ 2:4ð 0:3Þ 10 4 XAN ð16Þ
follow. The specific free volume at Tg, Vfg ¼ hgVg, decreases
from 0.070 cm 3 g 1 for PS to 0.058 cm 3 g 1 for SAN50
(Figure 2) which comes from a decrease in the hole fraction,
fg ¼ hg, from 0.073 to 0.063, and a corresponding change in
the total volume Vg. This shows in agreement with previous
results [15–19] that the fractional free (hole) volume can be
distinctly larger than the WLF value [3] of fg ¼ 0.025. Table 1
shows also a temperature denoted as T 0 0
. This is the tem-
perature where the equilibrium free volume disappears
when extrapolated linearly from the temperature region
above Tg. We notice that T0 0 ¼ Tg (113 123)K. T0 0 should
be closely related with the temperature where the structural
relaxation time becomes infinite, this is the Vogel temperature
T0. [3] In Part II of our work we will discuss this
point more in detail taking into account the results from
PALS.
afr and afg are in the order of 10 3 K 1 while, due to the
low free volume fraction, afg* and afr* have values in the order
of 10 4 K 1 . The parameters Efg, Efr, and afg*, afr* decrease
with increasing content of AN comonomer. The differences,
(afr* afg*)¼ (4.8 0.2) 10 4 K 1 (Figure 3), and
the values (afr* afg*) Tg ¼ (0.18 0.01) and
afr*T g ¼ (0.22 0.01) are constant within their statistical
uncertainties. We notice that the values of (afr* afg* ) are
somewhat larger than the corresponding values estimated
from the specific volume data. This difference is due to the
abrupt decrease in the thermal expansivity of the specific
occupied volume, Vocc ¼ yV, atTg (Figure 4). The physical
reasons for this behaviour are still unknown but there are
additional indications for this from our PALS results (see
Part II of this work). These show that this behaviour is not
the possible effect of the application of the S-S eos for the
non-equilibrium glassy phase. We speculate that the change
in the expansivity of the occupied volume is probably
related to the change in the polymer dynamics at Tg. [46]
With increasing temperature the b process (trapped
motions) sets in somewhat below Tg and shows a weak
but observable volumetric effect. [47] At Tg the a process (notrapped
motions, segmental dynamics) is activated which
has a large volumetric effect. Future research may resolve
this question.
The specific expansivity of the occupied volume for all of
our samples changes at Tg from Eocc,g ¼ dVocc/dT ¼ 0.91
10 4 cm 3 g 1 K 1 (T < Tg) toEocc,r ¼ dVocc/dT ¼ 0.18
10 4 cm 3 g 1 K 1 (T > Tg) which corresponds to a
jump in the fractional coefficient of thermal expansion
of the occupied volume from aocc,g * ¼ Eocc,g/Vg ¼ 0.95
10 4 K 1 to aocc,r
* ¼ Eocc,g/Vg ¼ 0.21 10 4 K 1 . These
coefficients are smaller than those of polymer crystals. For
polyethylene crystals, for example, ac ¼ 1.9 10 4 K 1
has been estimated. [32] Because of aocc,r 0.1 ag, the
fractional coefficient of thermal expansion of excess free
volume should be approximated by a fr* a r rather than
by a fr* Da ¼ (a g a r) as usually done (see Equation (2)).
The same result we have obtained from investigations of
Macromol. Chem. Phys. 2004, 205, 500–511 www.mcp-journal.de ß 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

508
a series of styrene-maleic anhydrite (SMA) copolymers
(to be published) and differently plasticized polyvinylchloride
(PVC). [30] This shows the general nature of these
conclusions.
As a further aspect we observed that the ratio V occ,g/V*,
which is denoted in the literature by K(~Tg), [15,17–19] has a
constant value of V occ,g/V* ¼ 0.9548 0.0005. The hole
fraction h in thermal equilibrium is frequently calculated
from the approximation h ¼ 1 K(~T)/~V, where K(~T) being
a very slowly varying function. V* may be estimated from
the ratio V*/VW, which is 1.54 in case of our SAN system.
From our data, we found also a constant ratio Vocc,g/
V W ¼ 1.47 0.01. This value is larger than the traditionally
accepted ratio of V r(0)/V W ¼ 1.3 which is estimated for
T ¼ 0 K (Bondi [10,11] ), but may be compared with
the universal relationship Vc/VW ¼ 1.45 (T ¼ 25 8C) where
Vc is the specific crystalline volume (van Krevelen [11] ).
The result Vocc,g/VW Vc/VW justifies our previous judgement
[13,48,49] to identify the occupied volume with the
crystalline one when Vocc is not known.
Hydrostatic Compression and Free Volume
In this chapter we discuss the results of the S-S eos analysis
of the PVT measurements for nonzero pressures P. Figure 7
shows as an example for the copolymer system the temperature
dependence of the specific volume of SAN50 for
selected pressures. As shown in the Figure, Equation (8),
together with the estimated scaling parameters, fit very well
the experimental data above T g(P).
Figure 8 displays the variation of the specific occupied
volume, Vocc, with the temperature. Again, the abrupt
change in the expansivity Eocc from below Tg to above Tg(P)
Figure 7. Specific volume V of SAN50 as a function of
temperature T and as selection of isobars (in MPa). Tg(0) and
T g(P) represent the zero-pressure glass transition and the glass
transition temperature as a function of pressure. Empty symbols:
experimental data, dots: S-S eos fits using Equation (8) in the range
of 125–250 8C and 0–200 MPa. (straight lines: linear fits).
G. Dlubek, J. Pionteck, D. Kilburn
Figure 8. As in Figure 7, but specific occupied volume, Vocc ¼
yV, of SAN50 (Symbols as in Figure 7, straight lines: linear fits).
can be observed. In Figure 9 the variation of the specific free
volume Vf ¼ hV with the temperature is shown. Vf exhibits a
linear expansion below T g(0) and above T g(P). Between
T g(0) and T g(P) it is compressed to values lower than the
glass that was originally loaded into the PVT device.
In Figure 10, the pressure dependence of the specific
total, occupied, and free volume is shown for selected temperatures.
As can be observed, the free volume, and therefore
also the total volume, decrease highly nonlinear with P
at temperatures above Tg. V(P) and Vf(P) could be fitted by a
polynomial of forth degree. At room temperature V and V f
show a small, linear variation with P. V occ exhibits in all
temperature ranges an almost linear decrease with P.
The specific expansivities of all of the volumes, V, Vocc
and Vf (Figure 7–9), decrease with increasing pressure as
follows for SAN50 as example (in units of 10 4 cm 3 g 1
K 1 ): Eg from 1.65 at P ¼ 0.1 MPa to 0.86 at P ¼ 200 MPa,
Er from 5.27 to 3.17, Eocc,g from 0.90 to 0.74 and Eocc,r from
Figure 9. As in Figure 7, but specific free volume, V f ¼ hV, of
SAN50 (Symbols as in Figure 7, straight lines: linear fits).
Macromol. Chem. Phys. 2004, 205, 500–511 www.mcp-journal.de ß 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The Structure of the Free Volume in Poly(styrene-co-acrylonitrile) from Positron Lifetime ... 509
Figure 10. Specific total, V, occupied, Vocc ¼ yV, and free,
V f ¼ hV, volume of SAN50 as a function of pressure P for selected
temperatures. Circles: 253 8C, squares: 209 8C, up-triangles:
149 8C, down-triangles: 26 8C. The lines are fits of the data to a
polynomial function of first (Vocc and all data for 26 8C) and fourth
degree (V f, V), respectively.
0.18 to 0.07, Efg from 0.74 to 0.10 and Efr from 5.07
to 3.27. An Arrhenius plot of ln h vs. 1/T (Equation (13))
gives straight lines in the temperature range above Tg(P)
with activation enthalpies for hole formation, Hh, decreasing
slightly with increasing pressure.
The isothermal compressibilities of the specific volumes
for the glassy and rubbery states of the copolymers at Tg and
for P ! 0 are shown in Figure 3 as function of the SAN
composition. It was found that both kg ¼ (1/Vg)[dV/dP]T
(T < Tg, T ! Tg) and kr ¼ (1/Vg) [dV/dP]T (T > Tg,
T ! Tg) could be estimated best from linear fits to the bulk
elasticity modulus, B ¼ 1/k, belowandaboveTg. kg and kr
exhibit a decrease with an increment of 0.0143
10 4 MPa 1 and 0.0114 10 4 MPa 1 per mol-% AN.
Their differences, Dk ¼ (kr kg) are constant at Dk ¼
(2.26 0.1) 10 4 MPa 1 . From this follows also the
constancy of dTg/dP(0) ¼ Dk/Da ¼ 0.55 K Mpa 1 (Ehrenfest
relation).
Figure 11 shows the behaviour of the temperature dependent
isothermal compressibility k(T) ¼ [1/V(T)][dV(T)/
dP]T (P ! 0) for PS and SAN50 together with the fractional
compressibilities of the occupied and free volume,
kocc * (T) ¼ [1/V(T)][dVocc(T)/dP] T and kf*(T) ¼ [1/V(T)]-
[dVf(T)/dP] T where k ¼ kocc * þ kf*. Moreover, the compressibility
of the free volume itself, kf(T), is shown. The
relations kocc * ¼ (1 f)kocc and kf* ¼ fkf (f ¼ h) with
kocc(T) ¼ [1/Vocc(T)][dVocc(T)/dP]T and kf(T) ¼ [1/
Vf(T)][dVf(T)/dP]T hold.
The compressibility k of PS and the SAN copolymers
shows a slight increase from 2.5 10 4 MPa 1 at room
temperature to 3.5 10 4 MPa 1 at Tg, an abrupt jump to
6 10 4 MPa 1 above Tg and a further increase to
10 10 4 MPa 1 at 250 8C. The fractional compressibility
Figure 11. Temperature dependence of the isothermal compressibilities
for P ! 0 of the total volume, k, and fractional
compressibilities of the occupied, kocc * , and free volume, kf*, for
PS (circles) and SAN50 (squares). Moreover, the compressibilities
of the free volume, kf, are shown (see text).
of the occupied volume, kocc * (T), shows an only slight
decrease with values around kocc * 2 10 4 MPa 1 . kf*(T)
behaves like k(T) but is reduced by kocc * (T). The value of
kocc * (T) is close to that of kg(T), and somewhat larger than
the compressibility of crystals of PE. From the data
published by Jain and Simha [50] zero pressure compressibilities
of PE crystals of 1.5 10 4 MPa 1 (20 8C) and
1.6 10 4 MPa 1 (66 8C) can be estimated. The values of
kocc * and kocc seem to be unexpected high. Bohlen and
Kirchheim [14] have estimated the specific number of local
free volumes from isothermal compression experiments by
a comparison of the specific volume, V, and mean local free
(hole) volume derived from PALS, assuming kocc ¼ 0. The
estimated values are higher by a factor of 1.5 to 2.5 than
those from thermal expansion experiments. An agreement
between the results from both types of experiments is,
however, obtained when taking into account the above
estimated values for kocc.
Surprisingly, the compressibility of the free volume,
kf(T), is at room temperature not much larger than that of the
occupied volume, kocc(T). kf(T) increases strongly with
increasing temperature and becomes an order of magnitude
larger than k(T) near below and above Tg. Due to this larger
values, the fractional compressibility kf*(T) dominates the
total compressibility although the volume fraction is
between 5 and 15% only.
As mentioned, the SAN copolymers show coefficients of
thermal expansion and compressibilities of the free volume,
af and kf, which are, for not too low temperatures, by about
one order of magnitude larger than the corresponding
values for the total volume. afr shows an increase with
increasing content of AN comonomer while kfr shows a
weak decrease above Tg. All the coefficients of the total
Macromol. Chem. Phys. 2004, 205, 500–511 www.mcp-journal.de ß 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

510
volume, a and k, and the fractional coefficients of the free
volume, af* and kf*, decrease with increasing content of AN
comonomer. The behaviour of af and kf can be attributed to
the increase in the cohesive energy density with increasing
content of AN comonomer. [11] That is also the reason that
the hole formation enthalpy, H h, increases and with that the
hole concentration C h(T), respectively the specific free
volume at any temperature T, Vf(T), decrease. The decrease
in the coefficients of thermal expansion and compressibilities
of the specific (total) volume, a and k, is mainly due
to the decreasing fraction of free volume with increasing
content of AN comonomer, and to a smaller extent to the
variation in a f and k f. The decrease in the specific free
volume V(T) comes from both, the decrease of the occupied
volume due to the decrease of the van der Waals volume,
and the decrease of the free volume (Table 1).
Conclusions
Using the S-S eos for the analysis of PVT data of SAN
copolymers with 0–50 mol-% AN we have calculated the
hole or excess free volume fraction, h, and from this the
specific occupied and the free volume, V occ ¼ (1 h)V and
Vf ¼ hV where the specific total volume is given by
V ¼ Vocc þ Vf. We studied the behaviour of these sub-volumes
as function of temperature and pressure.
The unexpected result was found that the fractional
coefficient of thermal expansion of Vocc changes at Tg from
aocc,g * 0.2
* 0.5 ag 1 10 4 K 1 (T < Tg)toaocc,r 10 4 K 1 (T > Tg). From this it follows that the traditional
approximation, whereby the fractional coefficient of
thermal expansion of the free volume in the rubbery state
is set equal to the difference of the coefficients of total
volume expansion above and below Tg, afr* ¼ Da ¼
(ar ag) 4 10 4 K 1 , which comes from aocc,r * ¼ ag,
is incorrect. Since aocc,r * 0, afr* ar 6 10 4 K 1 is the
distinctly better approximation. From the temperature
variation of V f the fractional coefficient of thermal expansion
of the free volume was estimated to be a fg* ¼ (1.3
0.8) 10 4 K 1 and afr* ¼ (6.1 5.4) 10 4 K 1 while the
corresponding values of the free volume itself are
afg ¼ (1.84 1.28) 10 3 K 1 and afr ¼ (8.41 8.70)
10 3 K 1 .
The fractional compressibility of the occupied and free
volume, kocc * , and kf* show the following behaviour. kocc *
exhibits an only small change at Tg with values of
kocc,g * kocc,r * 2 10 4 MPa 1 , a value which seem to
be unexpected high but corresponds well to the compressibility
of PE crystals, for example. kf* varies parallel to k,
kf* ¼ k kocc * . The compressibility of the free volume itself,
kf, has values of kf 2 10 3 MPa 1 below Tg and
kf 5 10 3 MPa 1 above.
The variation of total and free volume parameters with
the composition of the SAN copolymers is discussed. It was
found that the hole number per mer follows an Arrhenius
G. Dlubek, J. Pionteck, D. Kilburn
law with an activation enthalpy approximately half of
the cohesive energy. The free volume at Tg, Vfg, and the
expansivities and compressibilities decrease linearly with
increasing content of AN comonomer. This was attributed
to the increasing cohesive energy. In the second part of
the work the mean size, mean number density and size
distribution of sub-nanometer size free volume holes are
determined from the PALS experiments and their comparison
with PVT data. We show that the unexpected findings
aocc,r * 0 and kocc,g * kocc,r * 2 10 4 MPa 1 have a large
effect on the estimation of the hole densities.
Our results show clearly that the occupied volume shows
a thermal expansion and is also compressible since it
contains the interstital free volume, Vocc ¼ VW þ Vfi. Its
expansivity and compressibility differ from that of the
excess free volume Vf. Vf depends on the route in the T-P
plane on which the constant V is reached. From this follows
that the frequently made observation of different relaxation
properties for the same total volume or density [40,51] does
not necessarily contradict the free volume theory as it has
been sometimes concluded. [51] The total volume is not the
controlling parameter of the mobility, but, as shown
recently by some of us, [52] the excess free volume is.
Acknowledgement: We thank M. Weber, BASF AG, for
providing the SAN samples. For the characterisation of the
samples by GPC, DSC and EA we thank D. Voigt, L. Häußler, and
R. Schulze (all from Dresden).
[1] [1a] T. G. Fox, P. J. Flory, J. Appl. Phys. 1950, 21, 581; [1b]
T. G. Fox, P. J. Flory, J. Polym. Sci. 1954, 14, 315.
[2] A. K. Dolittle, J. Appl. Phys. 1951, 21, 1471.
[3] M. L. Williams, R. F. Landel, J. D. Ferry, J. Am. Chem. Soc.
1955, 77, 3701.
[4] [4a] M. H. Cohen, D. Turnbull, J. Chem. Phys. 1959, 31,
1164; [4b] D. Turnbull, M. H. Cohen, J. Chem. Phys. 1970,
52, 3038.
[5] H. Fujita, Fortschr. Hochpolym. -Forsch. 1961, 3,1.
[6] [6a] J. S. Vrentas, J. L. Duda, J. Polym. Sci., Part B: Polym.
Phys. 1977, 15, 403; [6b] J. S. Vrentas, C. M. Vrentas,
J. Polym. Sci., Part B: Polym. Phys. 1992, 30, 1005.
[7] R. E. Robertson, R. Simha, J. G. Curro, Macromolecules
1985, 18, 2239.
[8] J. Frenkel, ‘‘Kinetic Theory of Liquids’’, Oxford University
Press, London 1946.
[9] R. Simha, G. Carri, J. Polym. Sci., Part B: Polym. Phys. 1996,
32, 2645.
[10] [10a] A. Bondi, J. Phys. Chem. 1964, 68, 441; [10b] A.
Bondi, ‘‘Physical Properties of Molecular Crystals, Liquids,
and Gases’’, Wiley, New York 1968, p. 450.
[11] D. W. Van Krevelen, ‘‘Properties of Polymers’’, Elsevier Sci.
Publ. Co., Amsterdam 1990.
[12] R. Simha, R. F. Boyer, J. Chem. Phys. 1962, 37, 1003.
[13] G. Dlubek, J. Stejny, Th. Lüpke, D. Bamford, K. Petters, Ch.
Hübner, M. A. Alam, M. J. Hill, J. Polym. Sci., Part B: Polym.
Phys. 2002, 40, 65.
Macromol. Chem. Phys. 2004, 205, 500–511 www.mcp-journal.de ß 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The Structure of the Free Volume in Poly(styrene-co-acrylonitrile) from Positron Lifetime ... 511
[14] J. Bohlen, R. Kirchheim, Macromolecules 2001, 34, 4210.
[15] R. Simha, P. S. Wilson, Macromolecules 1973, 6, 908.
[16] R. Srithawatpong, Z. L. Peng, B. G. Olson, A. M. Jamieson,
R. Simha, J. D. McGervey, T. R. Maier, A. F. Halasa, H.
Ishida, J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 2754.
[17] R. Simha, T. Somcynsky, Macromolecules 1969, 2, 342.
[18] R. Simha, G. Carri, J. Polym. Sci., Part B: Polym. Phys. 1994,
32, 2645.
[19] L.A.Utracki,R.Simha,Macromol.TheorySimul.2001,10,17.
[20] O. E. Mogensen, ‘‘Positron Annihilation in Chemistry’’,
Springer-Verlag, Berlin, Heidelberg, New York 1995.
[21] Y. C. Jean, P. E. Mallon, D. M. Schrader, Eds., ‘‘Principles
and Application of Positron and Positronium Chemistry’’,
World Scientific, Singapore 2003.
[22] R. A. Pethrick, Progr. Polym. Sci. 1997, 22,1.
[23] Y. Ito, T. Suzuki, Y. Kobayashi, Eds., ‘‘Proc. of the 6 th Int.
Workshop on Positron and Positronium Chemistry (PPC 6)’’,
7–11 June 1999, Tsukuba, Japan, Rad. Phys. and Chem.
2000, 58, no. 5-6, p. 401 ff.
[24] J. Liu, Q. Deng, Y. C. Jean, Macromolecules 1993, 26, 7149.
[25] Y. Kobayashi, W. Zehng, E. F. Meyer, J. D. McGervey, A.
Jamison, A. R. Simha, Macromolecules 1989, 22, 2302.
[26] Z. Yu, U. Yashi, J. D. McGervey, A. M. Jamieson, R. Simha,
J. Polym. Sci., Part B: Polym. Phys. 1994, 32, 2637.
[27] M. Schmidt, F. H. J. Maurer, Macromolecules 2000, 33, 3879.
[28] M. Schmidt, M. Olsson, F. H. J. Maurer, J. Chem. Phys. 2000,
112, 11095.
[29] M. Schmidt, F. H. J. Maurer, Polymer 2000, 41, 841.
[30] G. Dlubek, V. Bondarenko, J. Pionteck, M. Supey, A.
Wutzler, T. Krause-Rehberg, Polymer 2003, 44, 1921.
[31] G. Dlubek, J. Stejny, M. A. Alam, Macromolecules 1998, 31,
4574.
[32] R. P. Quirk, M. A. A. Alsamarrie, ‘‘Physical Constants of
Polyethylene’’, in: Polymer Handbook, J. Brandrup, E. H.
Immergut, Eds., 3 rd Edition, John Wiley & Sons, New York
1985, p. V/15.
[33] P. Zoller, C. J. Walsh, ‘‘Standard Pressure-Volume-
Temperature Data for Polymers’’, Technomic Publ Co.
Inc., Lancaster, Basel 1995.
[34] J. Pionteck, S. Richter, S. Zschoche, K. Sahre, K. F. Arndt,
Acta Polymer 1998, 49, 192.
[35] C. Wästlund, F. H. J. Maurer, Polymer 1998, 39, 2897.
[36] J. Kansy, Nucl. Instrum. Methods Phys. Res., Sec. A 1996,
374, 235.
[37] J. Kansy, LT for Windows, Version 9.0, June 2002, PL-40-007
Katowice: Inst. of Phys. Chem. of Metals, Silesian
University, Bankowa 12, Poland, private communication.
[38] R. K. Jain, R. Simha, P. Zoller, J. Polym. Sci., Part B: Polym.
Phys. 1982, 20, 1399.
[39] A. Quach, R. Simha, J. Phys. Chem. 1972, 76, 416.
[40] R. E. Roberson, in: ‘‘Computational Modelling of Polymers’’,
J. Bicerano, Ed., Marcel Dekker, Midland, MI 1992,
p. 297.
[41] B. Hartmann, R. Simha, A. E. Berger, J. Appl. Polym. Sci.
1991, 43, 983.
[42] R. K. Jain, R. Simha, J. Polym. Sci., Part B: Polym. Phys.
1982, 20, 1399.
[43] P. Zoller, R. K. Jain, R. Simha, J. Polym. Sci., Part B: Polym.
Phys. 1986, 24, 687.
[44] J. Perez, ‘‘Physics and Mechanics of Amorphous Polymers’’,
A. A. Balkema, Ed., Brookfield, Rotterdam 1998, p. 18.
[45] N. Hirai, H. Eiring, J. Polym. Sci. 1959, 37, 51.
[46] Y. Tanabe, in: Macromolecular science and engineering,
Springer series in materials science, Vol. 25, Y.
Tanabe, Ed., Springer, Berlin, Heidelberg, New York 1999,
p. 267.
[47] O. Olabisi, R. Simha, Macromolecules 1975, 8, 211.
[48] G. Dlubek, K. Saarinen, H. M. Fretwell, J. Polym. Sci., Part
B: Polym. Phys. 1998, 36, 1513.
[49] G. Dlubek, D. Bamford, A. Rodriguez-Gonzalez, S.
Bornemann, J. Stejny, B. Schade, M. A. Alam, M. Arnold,
J. Polym. Sci., Part B: Polym. Phys. 2002, 40, 434.
[50] R. K. Jain, T. Simha, J. Polym. Sci., Polym. Lett. Ed. 1979, 17,
33.
[51] J. T. Bendler, J. J. Fontanella, M. F. Shlesinger, M. C.
Wintersgill, Electrochim. Acta 2003, 48, 2267.
[52] D. Bamford, A. Reiche, G. Dlubek, F. Alloin, J.-Y. Sanchez,
M. A. Alam, J. Chem. Phys. 2003, 118, 9420.
Macromol. Chem. Phys. 2004, 205, 500–511 www.mcp-journal.de ß 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim