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Soft Matter under Exogenic Impacts
edited by
Sylwester J. Rzoska
Dept. of Biophysics and Molecular Physics
Institute of Physics
Silesian University
Katowice
POLAND
and
Victor A. Mazur
Dept. of Thermodynamics
Institute of Energy and Ecology
State Academy of Refrigeration
Odessa
UKRAINE
TABLE OF CONTENTS
Part I: General Issues
Asteroid impact in the black sea; a black scenario
R. D. Schuiling, R. B. Cathcart, V. Badescu............................................1
The conductivity of hydrogen in extreme conditions
V. T. Shvets, S. V. Savenko, J. K. Malynovski...........................................9
Dynamic Crossover and liquid- liquid critical point
in the TIP5P model of water
P. Kumar, S. V. Buldyrev, H. E. Stanley..................................................22
Amorphization of ice by collapse under pressure,
vibrational properties and ultraviscous water at 1 GPa
G. P. Johari, O. Andersson......................................................................33
Coupled ordering in soft matter: competition of mesoscales
and dynamics of coupled fluctuations
M. A. Anisimov..........................................................................................73
All standard theories and models of glass transition
appear to be inadequate: missing some essential physics
K. L. Ngai.................................................................................................89
Part II: Glass forming liquids
Positron annihilation lifetime spectroscopy
and atomistic modeling - effective tools for
the disordered condensed systems characterization
J. Bartoš, D. Račko, O.Šauša, J. Krištiak................................................110
Segmental and chain dynamics in Polymers,
C. M. Roland and R. Casalini.................................................................130
Isobaric and Isochoric properties of glass-formers,
R. Casalini and C. M Roland..................................................................137
Molecular structure and relaxation processes
in diisooctyl phthalate and diisooctyl maleate.
S. Pawlus, M. Paluch, M. Mierzwa, S. Hensel – Bielowka,
E. Kaminska, K. Kaminski, S. J. Rzoska and S. Maslanka.....................144
Orientationally disordered glassy phases
J. Ll. Tamarit, S. Pawlus, A. Drozd-Rzoska and S. J. Rzoska..................155
Part III: Liquid Crystals
Glassy dynamics of rod – like liquid crystals:
the influence of molecular structure
A. Drozd-Rzoska, S.J. Rzoska, M. Janik...................................................182
Ordering effects on dynamics in glassforming
mixture of liquid crystals
M. Mierzwa, M. Paluch, S. J. Rzoska, J. Zioło, U. Maschke....................193
Nonlinear dielectric spectroscopy near SmA - SmC*
transition in ferrolectic liquid crystal DOBAMBC
S. J. Rzoska and A. Drozd-Rzoska ...........................................................208
Confined liquid crystaline 5CB – combined
temperature and high pressures dielectric relaxation studies
S. Pawlus, S. J. Rzoska, W. Osińska, S. Cordoyiannis, S. Kralj...............220
Annihilation of defects in liquid crystals
M. Svetec, M. Ambrožič, S. Kralj..............................................................230
Waves at the nematic-isotropic interface:
nematic—non-nematic and polymer-nematic mixtures
V. Popa-Nita, T. J. Sluckin........................................................................243
Part IV: Critical Liquids
Global phase behaviour of supercritical water –
environmentally significant organic chemicals mixtures
S. V. Artemenko, V. A. Mazur.................................................................259
Properties of water near its critical point
V. Kulinskii, N. Malomuzh.......................................................................277
Fluctuational equation of state and slopes of
critical curves near the critical point of solvent
V. Rogankov, O. Byutner.........................................................................295
Combined models of thermophysical properties
along the coexistence curve
E. E. Ustjuzhanin, B. F. Reutov, V. F. Utenkov, V. A. Rykov...................314
Intermolecular potential for simple liquids and gases
in the high pressure region
V. Yu.Bardic, L. A.Bulavin, V. M.Sysoev,
N. P.Malomuzh, K. S.Shakun..................................................................328
Homogeneous nucleation and growth from highly
supersaturated vapor by molecular dynamics simulation,
N. Lümmen, B. Fischer, T. Kraska...........................................................341
How to generate and measure negative pressure in liquids?
A. R. Imre.................................................................................................367
Indirect methods to study liquid-liquid miscibility
in binary liquids under negative pressure
A. R. Imre, A. Drozd-Rzoska, T. Kraska, S. J. Rzoska............................ 378
Part V: Bio-iquids and related problems
Critical properties of soft matter at
restricted geometry as emerging problem:
fundamentals and biological applications
A. V. Chalyi, L. A. Bulavin and
K. A. Chalyy , L. M. Chernenko...............................................................387
Water-biomolecule systems under extreme conditions:
From confinement to pressure effects
M.-C. Bellissent-Funel.............................................................................401
Recent progresses in understanding of water interacting
with biomolecules, and inside living cells and tissues
R. C. Ford, J. Li........................................................................................420
Self-assembly of polypeptides.
The effect of thermodynamic confinement
G. Floudas and P. Papadopoulos.............................................................434
Coulomb liquids under electric field –
application of a new computer simulation method
E. S. Yakub................................................................................................443
Solvation effect in near-critical polar liquids
A. Onuki....................................................................................................452
PARTICIPANTS:
Directors:
Sylwester J. Rzoska
Institute of Physics, Silesian University, Katowice, POLAND
Victor M. Mazur
Academy of Refrigeration, Odessa 65026, UKRAINE
Key speakers:
Gyan P. Johari
Dept. of Materials Science and Engineering, McMaster University, CANADA.
Christiane M. Alba-Simionesco
Lab. Chimie Physique, C.N.R.S, Université Paris-Sud Orsay, FRANCE.
Marie-Claire-Bellisent
Laboratoire Leon Brillouin, Gif-Dur-Yvette, FRANCE
Thomas Kraska
Institute for Physical Chemistry, University at Cologne, Köln, GERMANY.
George Floudas,
Department of Physics, Univ. of Ioannina, Ioannina, and FORTH-BRI
GREECE,
Attila R. Imre
KFKI Atomic Energy Institute, Budapest, HUNGARY
Simone Capaccioli
Istituto Nazionale per la Fisica della Materia and Dip. di. Fisica Univ. di Pisa,
ITALY
Rolaf D. Schuiling
Utrecht University, Utrecht, The NETHERLANDS
Aleksandra Drozd-Rzoska
Institute of Physics, Silesian University, Katowice, POLAND.
Marian Paluch,
Institute of Physics, Silesian University, Katowice, POLAND.
Stefan Jurga,
A. Mickiewicz University, Department of Macromolecular Physics, Poznan,
POLAND
Jerzy Zioło
Institute of Physics, Silesian University, Katowice, POLAND.
Samo Kralj
Univerza v Mariboru, Oddelek za fiziko, Maribor, SLOVENIA.
Josep Lluís Tamarit
ETSEIB, Universitat Politècnica de Catalunya, , Barcelona, Catalunya, SPAIN.
Jichen. C. Li
Dept. Physics, Biomolecular Sci, Manchester University, Manchester, UK
Eugene H. Stanley
Center for Polymer Studies, Dept Phys., Boston University, Boston, USA
Mikhail A. Anisimov
Dept. Chem. Engn. & Inst. for Phys. Sci.&Tech. Univ. of Maryland College
Park, USA
C. Mike Roland
Naval Research Laboratory, Washington, USA
Kia L. Ngai
Naval Research Laboratory, Washington, USA
Eugene Yakub
Computer Science Dept., State Economic University, Odessa, UKRAINE.
Leonid Bulavin
Taras Shevchenko Kiev National University, UKRAINE.
Nikolay Malomuzh
Odessa National University, Odessa, UKRAINE
Aleksandr V. Chalyi
Physics Dept., National Medical University, Kiev, UKRAINE
Kirill Schmulovich
Institute of Experimental Mineralogy, Chernogolovka, RUSSIA.
Akira Onuki
Department of Physics, Graduate School of Science, Kyoto University, Kyoto,
JAPAN.
Other Participants:
George Cordoyiannis
National Centre for Scientific Research “Demokritos”, Aghia Paraskevi,
GREECE
Michał Mierzwa
Institute of Physics, Silesian University Katowice, POLAND
Wiesław Sułkowski
Institute of Chemistry, Silesian University, Katowice, POLAND
Sebastian Pawlus
Institute of Physics, Silesian University, Katowice, POLAND
Małgorzata Janik
Institute of Physics, Silesian University, Katowice, POLAND
Sławomir Maślanka
Institute of Chemistry, Silesian University, Katowice, POLAND
Milan Svetec
Regional Dev. Agency and Fac. of Educ., University of Maribor, Maribor,
SLOVENIA.
Josef Bartos
Polymer Inst., Slovak Academy of Sciences, Bratislava, SLOVAKIA.
Ricardo Casalini
Naval Res. Lab., Washington DC and George Mason University, Fairfax, USA
Vlad Popa-Nita
Faculty of Physics, University of Bucharest, ROMANIA
Dmitry Yu. Ivanov
St. Petersburg State Uni. of Refrig. and Food Engn, Saint Petersburg, RUSSIA.
H. Schvets
Odessa Natl. University, Odessa, UKRAINE.
Vitaly Rogankov
Odessa Natl. University, Odessa, UKRAINE.
Vladimir Kulinski
Odessa Natl. University, Odessa, UKRAINE.
Sergey Artemenko
Odessa Natl. University, Odessa, UKRAINE.
ARW NATO “Soft matter under exogenic impacts” Odessa, Ukraine, 2005
conference photo at the terrace of Hotel “Morskoy”. Arrows indicate directors
of the Workshop.
POSITRON ANNIHILATION LIFETIME SPECTROSCOPY AND
ATOMISTIC MODELING – EFFECTIVE TOOLS FOR THE
DISORDERED CONDENSED SYSTEM CHARACTERIZATION
FREE VOLUME FROM PALS AND MODELING
J.BARTOŠ∗, D.RAČKO
Polymer Institute of SAS, Dúbravská cesta 9, 842 36 Bratislava
Slovakia
O.ŠAUŠA, J.KRIŠTIAK
Institute of Physics of SAS,Dúbravská cesta 9,842 28 Bratislava,
Slovakia
Abstract. The complex structure–property relationships in the disordered
systems under normal and exogenic conditions can be understood after
characterizing the spatial arrangement of constituents. Here, an integral
approach including the relevant experimental technique, phenomenological and
theoretical analyses as well as atomistic modeling is presented. Application of
such a combined approach is demonstrated for the cases of glycerol and
propylene glycol.
Keywords: free volume; positron annihilation; free volume models; atomistic modeling
1. Introduction
Many important condensed materials are characterized by the irregular spatial
arrangements of the particles‘ constituents. Understanding the complex
structural – property relationships in such systems under normal and exogenic
conditions needs as complete as possible a characterization of the irregular sites
of the matrix. At present, the relevant experimental technique is positron
annihilation lifetime spectroscopy (PALS). In order to utilize fully the
characterization potential of this method there is required a complete
understanding of the universal as well as the specific features of the PALS
______
∗
To whom correspondence should be addressed. E-mail:
111
FREE VOLUME FROM PALS AND MODELING
response of a given material. To solve this complex task requires a combined
approach including phenomenological analyses of the PALS response using the
thermodynamic and dynamic data, its appropriate theoretical treatments, i.e.,
free volume analyses (FVA) and theoretical and atomistic modeling approaches
consisting in appropriate free volume models of liquid state or molecular
dynamics (MD) simulation followed by the cavity analysis (CAVA),
respectively.The individual aspects of this combined approach will be presented
together with application on two model systems.
2.Positron annihilation lifetime spectroscopy
Positron annihilation lifetime spectroscopy (PALS) is based on the unique
annihilation behavior of ortho-positronium (o-Ps)1-5 being very sensitive
indicator of the local regions of lowered electron density in condensed matrix
such as vacancies in the ordered (crystal) systems or the so-called free volume
holes in the disordered (amorphous or semicrystalline) ones.
When a positron, e+, from the positron source 22Na enters the condensed
system it loses its kinetic energy very quickly under forming a positron
radiation track which contains the ionized entities such as electrons, e-, and
cations of constituents, M+ and the excited species. The large majority of these
ionized species are immediately neutralized by their mutual recombinations
within the positron track but some of electrons can escape it and be
preferentially trapped in the local positively charged regions of the matrix. The
electrons from track and the trapped electrons can capture the original or next
entering positron at forming a neutral bound positron-electron particle, the socalled positronium (Ps). Depending on the mutual spin orientation of positron
and electron in Ps, two distinct states differing in the respective lifetimes, τ ,
can arise. Singlet with antiparalel spins: para-positronium (p-Ps) has a typical
lifetime, τ1 ≅ 0.12 ns being almost independent on temperature. On the other
hand, a triplet state with parallel spins forms the ortho-positronium (o-Ps)
which is more stable with a typical lifetime,τ3 , ranging from ~ 0.6 – 5 ns in
normal organic substances. This specific form of Ps annihilates by an
interaction of the positron with some electron from the surrounding medium
with antiparallel spin in the so-called pick-off decay mechanism being
temperature dependent. Finally, in addition to the above-mentioned bounded
forms of positron, the unbounded “free“ positron can survive in the matrix with
a typical lifetime, τ2 ≅ 0.3 - 0.4ns which annihilate by an interaction with some
electron in the matrix.
112
2.1 TYPICAL PALS RESPONSE, ITS PHENOMENOLOGICAL ANALYSIS AND
CORRELATIONS
Figure 1 shows the typical o-Ps responses of two small molecular glassformers: glycerol (GL) HO-CH2-CH(OH)-CH2-OH 6 and 1,2-propylene glycol
(PG) HO-CH2-CH(OH)-CH37. In both the chemical compounds, the o-Ps
lifetime, τ3, as a function of temperature exhibits four regions of different
thermal behavior depicted as regions I, II, III and IV. Linear analyses of these
regions define the characteristic PALS temperatures in the liquid state: TgPALS,
TLb1 and TLb2 8. In the case of GL: TgPALS = 189K, TLb1 = 241K and TLb2 ≅ 290K,
while for PG: TgPALS = 172K, TLb1 = 220K and TLb2 ≅ 265K.
3.0
2.8
GL vs. PG
τ3 , ns
2.6
IV
2.4
III
2.2
2.0
Propylene glycol
1.8
Glycerol
1.6
II
1.4
I
1.2
L
T b2
L
1.0
PALS
T b1
Tg
0.8
0
50
100
150
200
250
300
T,K
Figure 1. The o-Ps lifetime, τ3, as a function of temperature for glycerol (GL) and propylene
glycol (PG)
The first pronounced change in slope on τ3 – T plot takes place near the
corresponding quasi-static glass to liquid transition temperatures, TgTHERM as
obtained by classical macroscopic thermodynamic techniques such as
dilatometry9,10 as well as to the dynamic glass temperatures, TgDYN, being
commonly defined as the temperature at which the mean relaxation time of the
primary α relaxation reaches some arbitrary value, e.g., τα = 100s11,12. These
findings mean that the expansion behavior of the localized free volume regions
detectable via the annihilation of the o-Ps probe is closely related to the
expansion behavior of the macroscopic volume as well as to the cooperative
113
FREE VOLUME FROM PALS AND MODELING
dynamics which is believed to be responsible for the most characteristic
dynamic process, i.e., the primary α relaxation.
In the glassy state region I up to TgPALS, GL and PG are characterized by the
relatively low o-Ps lifetimes, whereby the values of τ3 for GL are systematically
smaller then those for PG over the whole temperature range investigated. By
comparing GL and PG with small molecular systems of van der Waals type as
studied by us such as meta-tricresyl phosphate (m-TCP)5 or diethyl phthalate
(DEP)13 it indicates that both the systems are relatively effectively packed ones.
This appears to be consistent with the hydrogen-bonding nature of both the
systems, whereby the lower values for GL may be attributed to the higher
density of hydroxyl groups capable to form more intermolecular hydrogen
bonds compared to PG. Anyway, the presence of H-bonding leads to the more
compact and stiffer microstructure of GL matrix with respect to the PG one as
well as to some typical van der Waals substances such as m-TCP and DEP 5,13.
log τα (s) , -
2
0
PG
Schoenhals 1993
Park 1999
-2
Stickel 1995
Leon 1999
-4
τα(T
L
Lunkenheimer 2001
)
b1
-6
Bergman 2002
L
τα(T
-8
-10
b2
)=τ3
Tg=168K
-12
160
180
L
L
T b1=220K T b2=265K
SCH
TB =268K
200
220
240
260
280
300
320
340
T,K
Figure 2. Compilation of the primary α relaxation times for propylene glycol (PG) together
with the characteristic PALS temperatures (Tg , TLb1 and TLb2 ) and the DS temperature TBSCH
showing empirical correlation between TLb2 and TBSCH
In the liquid state above TgPALS, the two further bend effects are evident in
Figure 1: i) a slight change in slope in strongly cold liquid at TLb1 = 1.275 TgPALS
for GL and TLb1 = 1.28TgPALS for PG which defines mild crossover between
regions II and III and finally ii) a large change in slope in weakly cold liquid at
higher temperatures to a quasi-plateau level at TLb2 = 1.53TgPALS for GL and
TLb2 = 1.54 TgPALS for PG defining drastic crossover between regions III and IV.
114
First, we analyse the former of the mentionned effects. Figure 2 represents
a compilation of the mean relaxation time of the primary α process, τα , as
a function of temperature for PG as obtained from several dielectric relaxation
studies in the literature. As can be seen, at TLb1 the mean dielectric relaxation
time of the primary α process reaches just the value of 10-6 s, now being
depicted as Tα(-6). Similar situation can be found also for further glass-forming
systems. Figure 3 demonstrates such an empirical correlation between the TLb1’s
values and the characteristic DS temperature Tα(-6) for a series of small
molecular as well as polymer glass-formers. This finding suggests that a change
in the expansion behavior of the localized free volume regions at around 1.2 1.3 TgPALS seems to reflect some universal feature of glass-forming liquids
which occurs when the mean α relaxation time reaches just a microsecond
level.
300
b1
T
L
,K
290
T
280
L
- Tα(-6) correlation
b1
G
270
3
high-MW compounds:
260
1 - c-t-1,4-PBD
2 - PPG 4000
3 - PIB
250
240
C
D
230
F
low MW compounds:
E
A - DEP
B - PG
C - DPG
D - TPG
E - GL
F - m-TCP
G - OTP
2
B
220
A
210
1
200
200
210
220
230
240
250
260
270
280
290
300
Tα(-6) = Tα(log τα = − 6 ) , K
Figure 3. The empirical correlation between the first characteristic PALS temperature, TLb1, and
the temperature, Tα(-6) , at which the primary α relaxation time, τa , reaches just 10-6s for a series
of small molecular and polymer glass-formers.
Explanation of this feature in the o-Ps response is the subject of a
continuing interest. The two possible-not necessary excluding-explanations
based on either structural or dynamical hypothesis have recently been suggested
by us5. Here, we discuss the latter possibility consisting in the influence of some
more rapid local mobility, generally named as secondary relaxations, on a
redistribution of the free volume hole population. Figure 4 presents the
temperature dependences of relaxation time for both the primary α relaxation
and the secondary β process for glycerol. Their values were obtained by a
FREE VOLUME FROM PALS AND MODELING
115
spectral deconvolution of the dielectric spectra in terms of the Cole – Davidson
(CD) function for the α process and the Cole - Cole (CC) function for the β
one14. In particular, the closseness between the o-Ps lifetime, τ3, and the mean
secondary β relaxation time, τβ , at temperature 253K quite close to TLb1 = 241K
suggests that the local rapid β relaxators might contribute to the effective free
volume distribution accesible for o-Ps probe. According to the general length –
frequency principle of the dynamic events15, presumably smaller free volume
holes requesting the smaller amplitude local motion might be eliminated from
detection by an o-Ps probe. This elimination may result into the dominance of
the larger free volume holes above TLb116.
4
log τ (s) , -
2
Glycerol
0
-2
DS data (Lunkenheimer 2002)
α - process
-4
-6
-8
-10
β - process
τα(Tb1)
τα=τα(T
L
b2
τβ=τ3
Tg=190K
-12
)
T
L
b1
=241K
T(τ3=τβ)
T
L
b2
=290K
180 200 220 240 260 280 300 320 340 360 380 400 420
T,K
Figure 4. Relationship between the o-Ps lifetime, τ3 , and the secondary βJG relaxation time, τβ
for GL.
On increasing further the temperature, a dramatic discontinuity on τ3 - T plot
occurs, at 1.53 TgPALS in GL and 1.54 TgPALS in PG, characterized by a crossover
from the steeply increasing linear trend in the weakly cold liquid to a quasi plateau level in the warm liquid. The onset of the plateau effect correlates with
several dynamic features of the primary α process as observed in dielectric
spectroscopy studies17,18. First, Figure 5 presents the so-called Stickel’s plot for
DS data on GL taken from Ref. 17. Here, the two linear regions indicate a
crossover between the two different motional regimes at the so-called Stickel
temperature, TBST = 287K. Similar type of analysis for PG gives TBST = 270 K17.
In both the cases, good coincidence between the onset of plateau region at TLb2
and this characteristic DS temperature can be found for GL (Ref. 6) and PG.
Secondly, Figure 6 shows the so-called Schönhals’ plot for the relaxation
116
strength of the α process for PG, ∆εα , vs. logarithm of the α relaxation time,
log τα18. Evidently, a dramatic change at the characteristic DS temperature,
TBSCH = 268K can be found being in the vicinity of the TLb2 = 265K. Good
Φ = [ ∆ log τα / ∆(1000/T) ] , -
0.9
0.8
Glycerol
0.7
DS data
(Stickel 1995)
0.6
0.5
0.4
ST
TB
= 287K
0.3
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
1000/T , 1/K
Figure 5 Stickel’s plot for the primary α relaxation time, τα for GL demonstrating the change in
the dynamical regime at the characteristic DS (Stickel) temperature, TBST correlating with the
characteristic PALS temperature TLb2 .
agreement between TBST and TBSCH for GL and PG and its correlation with the
dramatic change in the PALS response at TLb2 indicates the common origin of
these significant changes in the relaxation time,τα ,the relaxation strength, ∆εα,
and the o-Ps lifetime,τ3. This feature will be addressed further below.
Finally, in addition to the above-mentioned coincidence between the
characteristic PALS and DS temperatures reflecting the changes in both the
annihilation and relaxation quantities, more direct relationship between the
corresponding time scales of both the methods exists. In other words, a
comparison of the τ3 value at the onset of the quasi-plateau region in Figure 1
with the mean relaxation times of the α process, τα , from Figure 2 for PG
confirms our so far findings of the equality between both the time parameters
for other small molecular and polymer systems8. The same has already been
found to be valid for GL6. This fact indicates very close relationship between
the progressively fluctuating matrix via the less cooperative α dynamics and the
mean lifetime of the o-Ps probe8 at the relatively higher temperatures, and at
higher macroscopic volumes, V(T), and then, at higher static empty free
volumes Vfe(T) = V(T) - VW, where VW is the van der Waals volume of the
molecules. The continuously
FREE VOLUME FROM PALS AND MODELING
117
∆ εα , -
70
60
50
40
PG
II+III
DS data
(Schönhals 2001)
low T region
∆εα = 3.59log τα+ 62.19
30
20
high-T region
∆εα = 11.66log τα+131
IV
SCH
log τα,B
10
SCH
= - 8.56 => τα,B
-9
= 2.7 x10 s
0
-12 -11 -10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
log τα(s) , Figure 6 Schönhals’ plot for the primary α relaxation time, τα , for PG showing the change in the
α relaxation strength at the characteristic DS (Schönhals) temperature, TBSCH correlating with the
characteristic PALS temperature TLb2.
increasing macroscopic volume of the system is accompanied by a continuously
increasing empty free volume and the still less cooperative α process
characterized by the more intense cage reorganization. Both these aspects
together result into a quasi-plateau feature in the PALS response above TLb2 ,
where o-Ps seems to be formed and localized in the sufficiently long living
dynamic free volume holes only.
2.2 FREE VOLUME INTERPRETATION OF THE O-PS RESPONSE USING
SEMIEMPIRICAL MODELS
The o-Ps response can be transformed into free volume information by means
of existing physically plausible models which address two basic aspects of free
volume characterization of any disordered material, i.e., free volume hole size,
Vh, and free volume hole fraction, fh .The former quantity is obtained using
a standard quantum-mechanical model of o-Ps in a spherical hole which
provides the following relationship between the o-Ps lifetime, τ3, and the free
volume hole radius, Rh19:
⎡
⎛ Rh ⎞ ⎤
Rh
1
⎟⎟⎥
sin ⎜⎜
τ 3 = τ 3,0 ⎢1 −
+
2
π
R
+
∆
R
R
+
∆
R
h
⎝ h
⎠⎦
⎣
−1
(1)
118
where τ3,0 is 0.5ns and ∆R is the thickness of electron layer about hole obtained
as a calibration parameter from fitting the observed o-Ps lifetimes to known
vacancy or free volume hole sizes in molecular crystals or zeolites. In fact, the
form of free volume entities is not strictly spherical because of the nonspherical form of constituent’s particles so that the standard model is used
usually in the sense of a volume equivalent spherical hole size Vh = (4π/3)Rh3.
Other geometrical forms of holes such as ellipsoid, cuboid and cylinder have
been considered20. However, as it was recently shown some parameters derived
from PALS such as the free volume fraction are not strongly influenced by the
choice of the free volume hole geometry21.
The mean free volume hole sizes in GL and PG at the above-mentioned
characteristic temperatures are summarized in Table I. In the glassy state as
well as in the supercooled liquid one below the second liquid PALS
temperature, TLb2 , the mean hole sizes represent a part of the van der Waals
volume of the corresponding molecules, while above TLb2 , the corresponding
mean hole sizes overcome the own molecular size apparently in consistency
with the low viscosity normal liquid character of the systems.
TABLE 1. The mean free volume hole characterization of GL and PG according to Eq.1
at the characteristic PALS temperatures.
System
GL
PG
TgPALS
Vh
Vh V W
L
Tb1
Vh
Vh V W
L
Tb2
Vh
Vh V W
K
189
172
Å
20
30
0.23
0.38
K
241
220
Å
47
61
0.54
0.78
K
290
265
Å
105
165
1.2
2.1
3. Extended free volume (EFV) model and its relationship to the free
volume data from PALS
Some microscopic models based on free volume concept can be tested by using
the above-mentioned free volume information. Here, we present an application
of the extended free volume (EVF) model formulated by Cohen – Grest22.
Briefly, this microscopic model of the liquid state accounts for a liquid as being
formed by solid-like and liquid-like cells, only the latter ones contain a free
volume. Using the percolation ideas, the specific mean free volume within the
liquid-like clusters is given by the four-parameter equation:
FREE VOLUME FROM PALS AND MODELING
119
) [(
(
⎧
vf = A' ⎨ T − T0CG + T − T0CG
⎩
)
2
+ C' T
]
1
2
⎫
⎬
⎭
(2)
where A’= (vmlne/B), and B, C’ and T0CG are material – dependent coefficients;
here vm is the molecular volume and T0CG represents the temperature at which
percolated liquid cluster should occur.
120
Vh , A
3
IV
Glycerol
100
80
III
Data: PALS+QM data 190-280K
Model: EFV model
60
2
χ
2
R
= 2.64298
= 0.99694
A
1.17874
B=T0 249.42666
C
14.18715
40
±0.18173
±9.2479
±0.82608
II
I
20
T0=249K
L
T b1=241K TLb2=290K
PALS
Tg
=189K
0
0
50
100
150
200
250
300
T,K
Figure 7 Application of the EFV model on the free volume data for glycerol.
Figure 7 shows the result of fitting Eq. 2 to the mean free volume hole sizes
being identified with vf from the EFV model. As can be seen, the EFV model is
able to describe the free volume hole data over the whole range of the
equilibrated supercooled liquid from the high temperature limit at TLb2 down to
TgPALS. Interestingly, the characteristic temperature of the EVF model, T0CG, is
quite close to the first characteristic PALS temperature. In particular, for GL we
find T0CG(PALS) = 249±9K close to TLb = 241K and for PG T0CG(PALS) =
237±7K not too distant from TLb1 = 220K. According to the EFV model, this
coincidence of the temperature parameters could be interpreted as the
occurrence of the free volume percolation at around TLb1 being responsible for
the steeper slope in slightly supercooled liquid. However, the situation is not so
simple as it will be discussed below.
120
log τmax (s) , -
4
2
Glycerol
0
Data: DS data
Model: EFVmodel - dynamics
-2
2
χ
-4
R
= 0.0041
= 0.99962
-6
A
B
CG
T0
C
-13.29825
1150.91832
182.96187
26.19047
2
±0.08562
±49.61164
±4.00845
±1.29087
-8
-10
DS data Wang 2002
-12
160 180 200 220 240 260 280 300 320 340 360 380 400 420 440
T,K
Figure 8. Application of the EFV model on the primary α relaxation data for glycerol
4. Atomistic modeling of free volume microstructure of supercooled
liquids
The significance of free volume information, as obtained from the FVA of the
o-Ps response can be evaluated by means of atomistic modeling. In general, this
approach consists in a combination of the generation of disordered systems by
using, e.g., molecular dynamics (MD) simulation followed by an appropriate
free volume analysis of the generated microstructures. Here, we demonstrate
such an atomistic modeling approach for both the model systems. The first step
of modeling has been performed by detailed MD simulations by means of the
ORAC program23. The next step consists in sampling of the free space between
the molecular bodies in the simulated microstructures by inserting the rigid
probe of a given geometry and size and the volume integration of the inserted
probes. In the case of our cavity analysis (CAVA) program24, the o-Ps probe is
modeled by the hard sphere of diameter 1.06 Å and the molecular bodies,
formed by atoms with van der Waals radii employed, represent the matrix
particles. The CAVA code provides a set of the cavity parameters which can be
compared with the free volume hole characteristics derived from the abovementioned free volume analysis (FVA).
The results of our MD plus CAVA investigations for GL are presented in Ref.
24 and for PG here. Figure 9 compares the calculated volume from our very
FREE VOLUME FROM PALS AND MODELING
121
extended MD simulations of GL and PG with the experimental macroscopic
volume from dilatometry (DIL)9,10 in terms of the volume of simulation box
containing the sufficient amount of molecules. In general, satisfactory
agreement between both the macroscopic volume quantities can be found,
especially at higher temperatures.
3
V , 10 A
3
40
38
36
GL
MD
Tg
34
Tg
PG
DIL
32
30
100
150
200
250
300
350
400
T,K
Figure 9. The simulated (points) and experimental (lines) box volumes of 300 molecules for
glycerol (GL) and propylene glycol (PG).
In particular, for GL the relative difference between the simulated volume
data and the measured ones reaches 0.3 % at higher temperature regions and 1.4
% at lower ones. Evidently, the larger deviations for the lower temperature
regions are connected with the effectively very high cooling rate in MD
simulation25 and the associated easier onset of the glass transition phenomenon.
The onset of deviations defines the simulated glass transition temperature, TgMD,
which can be related to the experimental glass transition temperature, Tg, from
dilatometry via fragility, mg, by the following Donth relation15:
Tg
mg
=−
Tg − TgDIL
∆logφ c
(3)
where ∆log Φc is the difference between the simulated and experimental
cooling rates. Table II summarizes the the dilatometric, TgDIL, the estimated, Tg,
and the simulated, TgMD, glass transition temperatures for both the systems.
Good agreement between the estimated and simulated values indicates that
a linear approximation for cooling rate dependence of glass temperature is valid
122
even for the 14 orders of magnitude change in Φc , so that the simulated
microstructures should correspond to the real physical ones as obtained with
very rapid cooling.
TABLE 2. The experimental, Tg DIL, estimated, Tg* , and simulated, TgMD, glass
temperatures for GL and PG according to Eq.3 for ∆ log Φc = 14. The mg values from
Beiner, M. (2001) J.Non-Cryst.Solids 279,126.
System
TgDIL
mg
Tg*
TgDIL
GL
PG
K
189
172
49
48
K
240
220
K
250
230
The obtained static microstructures of GL and PG have been analysed by
using the CAVA program24. On the basis of determination of the sizes
and the number of the individual cavities Figure 10a and 10b show for
the case of GL and PG a comparison between the mean free volume hole,
Vh , from Eq. 1 and the mean cavity volume,Vcav, as calculated from the
following equation26:
Vcav =
Vcav, max
∑Vcav,i
Vcav, min
N cav
∑N
cav,i
(4)
i
As can be seen, the mean cavity volumes are smaller than the mean free volume
ones over the whole temperature range predominantly due to a large amount of
he small cavities sampled by very small o-Ps probe (Vo-Ps = 0.64Å3). Having on
mind that the simulated microstructures are the static ones while PALS is
performed on the fluctuating matrix, we introduce within the framework of
general length scale - frequency principle15 the dynamic aspect via a concept of
the limiting cavity size.24 This is expressed by a set of curves for different
Vcav,lim’s. As the result, a good agreement for the equilibrated liquid above TLb1
between both the mean free volume quantities is achieved with Vcav,lim ≈ 20 30Å3. The physical backround for this concept accounts for a possible blocking
effect of some small scale mobility on a part of the total free volume
distribution. Experimental support for this concept in terms of the secondary β
relaxation has been reported for GL elsewhere16. In fact, from Figure 4 it
follows that the time scale of the β process above 250K is shorter than the mean
o-Ps lifetime so that a part of the free volume hole population is fluctuating too
FREE VOLUME FROM PALS AND MODELING
123
rapidly to be accessible for the o-Ps probe localization. The finding about the
size of Vcav,lim is consistent with other estimations as obtained by other ways27.
200
140
180
Vh and Vcav , A
3
Vcav and Vh , A
100
80
60
40
20
Vlim,A
30
25
20
15
10
5
0
PG
3
GL
120
3
160
140
PALS + QM data
120
100
W
V
40
L
PALS
SIM
Tg
Tb2
20
0
=78.5A
3
25
20
15
10
5
60
Tg
PG
80
T
L
Tg
0
T
PALS
L
b2
b1
0
150
200
250
300
T,K
350
400
100
150
200
250
300
350
400
T,K
Figure 10. Mean cavity volumes, calculated from equation 4 for several cases of Vlim, together
with the experimental mean hole volumes (points) from PALS for GL (a) and PG (b).
Figure 11 presents the integral volume distributions of all the cavities at
several temperatures in PG. Two qualitatively different temperature regions can
be distinguished. At lower temperatures below 260K, a continuously increasing
integral distribution is observed which changes into a discontinuous bimodal
one in the higher temperature region above 270K.The detailed visual inspection
of the static cavity configurations reveals an increasing contribution of the
larger cavities which results into the fully percolated cavities over the whole
simulation box. This is evident from the temperature evolution of threedimensional cavity microstructures in GL Figure 12. Interestingly, the the
presence of percolated cavity phenomenon from our MD + CAVA approach in
between 260 and 270K is commensurable with the characteristic PALS
temperature of the onset of the plateau region at TLb2 = 265K as well as with the
characteristic DS temperatures TBST and TBSCH as mentioned above. Moreover,
the equality between τ3 ≡ τα at TLb2 ≅ TB’s is observed – Figure 2 – so that for T
< TLb2 the τ3 < τα relationship is valid, while for T > TLb2 the τ3 > τα
relationship is found. All these facts allow us to formulate a concept of the
dynamic free volume facility, i.e., the presence of the static fully percolated
cavity volume facilitates for a qualitative change in the primary α relaxation.
Thus, according to our proposal, the crossover effects in PALS and DS data
result from the mutual interplay between the increasing empty free space
between particles with the progressively increasing appearance of percolating
cavities and the associated intense cooperative rearrangements of particle
constituents of the matrix.
124
Icav , -
1.0
≤ 250
0.8
PG
270
300
310
0.6
330
0.4
0.2
350
0.0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
3
Vcav , A
Figure 11. Integral cavity distribution in PG for a series of temperatures exhibiting the change to
the bimodal cavity distribution in between 250K and 270K coinciding with TLb2.
5. Crossover phenomenon from a viewpoint of the present microscopic
models and atomistic modeling
Finally, it is of interest to compare the explanations of the crossover
phenomenon in the primary α relaxation properties at TB proposed so far with
the results of our atomistic modeling studies on GL and PG. Several
interpretations of this important aspect of the supercooled liquid dynamics in
terms of various microscopic models have been offered in the literature. They
are based either on a coincidence between the crossover temperature, TB, and a
FREE VOLUME FROM PALS AND MODELING
125
150K
220K
270K
300K
320K
350K
0.64 Å3
2500 Å3
Vcav
Figure 12. Temperature evolution of the cavity microstructure in 3D space for GL. Yellow color
indicates the presence of the percolated cavity volume over the whole simulation box.
certain characteristic temperature corresponding to the specific microscopic
model or on the ability to describe the cooperative primary α relaxation via the
preceeding simpler process the so-caled primitive, i.e. independent relaxation,
after extraction of many-body aspect of the cooperative α dynamics. The
former group is represented by two cases: i) the characteristic free volume
temperature, T0CG, from the extended free volume (EFV) model22 at which the
onset of the free volume percolation should take place or ii) the critical
temperature, TcMCT, of the mode coupling theory (MCT)28 where a critical
slowing down of the α process should occur.
As for the EFV model, the relaxation time is expressed in the following
form22 :
(
) [(
⎧
log τ (T ) = A + B ⎨ T − T0CG + T − T0CG
⎩
)
2
+ CT
]
1
2
⎫
⎬
⎭
(5)
where A, B and C are materials coefficients and T0CG is the temperature at
which a continuity of the liquid-like cells is attained. The closeness between the
crossover temperature at TB ≅ 1.2Tg from DS data and the characteristic
temperature T0CG(DS) from application of Eq.5 on the DS data has been
observed for a series of six van der Waals systems29. Consequently, the
crossover at TB is interpreted as the result of free volume percolation leading to
the change in the primary or α dynamics. Very recently, one of us has
demonstrated for the case of DEP a good coincidence between TBST from DS
and TLb1 from PALS with the T0CG value determined not only from DS data
alone T0CG(DS) but also from the free volume ones from PALS T0CG(PALS)13 .
On the other hand, as mentioned above, the EFV model can describe the PALS
data of GL and PG with a reasonable coincidence between T0CG(PALS) and the
126
first bend temperature TLb1, but not between T0CG(PALS) and the second
characteristic PALS temperature TLb2 ≅ TB as indicated by atomistic modeling.
Moreover, for our hydrogen-bonded liquids GL and PG Eqs.2 and 5 are not
able to account for both the free volume and the relaxation data simultaneously,
with the same T0CG(PALS) = T0CG(DS) parameter as predicted by the EFV
model22. In fact, the T0CG(DS) values for GL (Figure 8) and PG (not shown) are
178.5±7K or 162±6K, being close to the respective Tg’s, are not consistent with
the above-mentioned T0CG(PALS) ones: 249±9K and 237±7K, respectively.
Thus, the unability to describe both PALS and DS data simultaneously as well
as the discrepancy with the results of atomistic modeling allow us to conclude
that the EFV model appears to be irrelevant for strongly hydrogen-bonded
glass-forming liquids such as GL and PG, at least. On the other hand, the
validity of the EFV model for slightly hydrogen-bonded and van der Waals‘
systems remains to be further tested.
Regarding the interpretation within the mode coupling theory (I-MCT),
althought TB ≈ Tc is often observed30,31, the predicted divergency of the α
relaxation time is not actually observed so that the ideal version of MCT needs
some modification. The extended mode coupling theory (E-MCT) when applied
on a few van der Waals systems such as ortho-terphenyl, gives Tc(I-MCT) ≅
Tc(E-MCT)32 so that TB ≈ Tc(I-MCT) should imply an onset of the thermally
activated dynamics.Very recent I-MCT analysis of GL provided Tc=288K31 in
agreement with the above-mentioned coincidence: TBST≅TBSCH≅TLb2 as well as
close to the onset temperature of the percolated free volume from our MD and
CAVA approach. Unfortunately, the E-MCT has not been applied so far on our
more complicated hydrogen-bonded model systems so that the range of its
validity remains unknown.
Finally, the latter class is represented by the coupling model (CM) of Ngai et
al.33. For a series of five van der Waals liquids, the crossover at TBST is
explained in terms of the large change in the heterogeneous character of the
primary dynamics via the change in the strength of the intermolecular
cooperativity on the preceeding primitive relaxation process from strongly
cooperative process below TB to less cooperative one at higher temperatures34.
In the last years, a growing amount of the evidencies that the primitive process
may be identified with the secondary Johari-Goldstein β process has
appeared35,36.The original set of the five compounds does not include strongly
intermolecularly hydrogen-bonded systems34. In the case of GL, the closeness
of the β relaxation time, τβ, as obtained from a deconvolution of the DS spectra,
with the primitive relaxation time, τ0 , calculated from the primary α relaxation
time, τα, via τ0 = tcn.τα1-n, where n is a measure of cooperativity connected with
the Kohlrausch-Williams-Watts (KWW) exponent, βKWW, by relation n = 1βKWW as well as the mutual merging of α and β processes at around 300K,
127
FREE VOLUME FROM PALS AND MODELING
close to TLb2 = 290K,37 seem to suggest some important change in the
microstructure of the GL matrix, which may change the character of the α
dynamics. This quasi-microscopic CM explanation together with the fact of the
equality of the α relaxation time, τα , and the mean o-Ps lifetime, τ3 , appears to
be qualitatively consistent with our MD and CAVA results. Thus, the onset of
the fully percolated static free volume in the MD microstructures close to TLb2
might imply some significant influence on the character of the cooperative α
dynamics. As mentioned above, one possibility may be that the transition to the
percolated free volume facilitates the crossover from a strongly cooperative
mobility below TBST to a slightly cooperative one above TBST together with a
full diminishing the more local β process just above TBST.
Acknowledgment
The authors would like to express their thanks to the VEGA Agency, Slovakia
for the financial support by grants 2/3026/23 (JB) and 2/4103/24 (JK) as well as
the APVT Agency,Slovakia for grant APVT 51-045302 (JK&JB).
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