1.1 Polymer Dynamics
1.1.1 Dynamics of bulk polymers
1.1.1.2 Empirical models
Molecular theories were not successful in fully describing and predicting the existence and the features of the glass transition in glass formers and poly-meric materials and their main relaxation (the α-relaxation), fit equations and empirical models were proposed in order to describe this process in a systematic way.
Photon correlation techniques observe the relatively slow relaxational components of a system in the spectrum of scattered light [49]. Extremely fast processes including thermal diffusion of polymer chains and Brillouin modes cannot be detected [49]. By collecting the whole scattered light from polystyrene in the range of DLS, a third of the radiation is due to isotropic scattering and two thirds are related to anisotropic scattering [49, 50]. The isotropic scattering has been linked to a longitudinal stress relaxational ex-periment at constant longitudinal strain [49, 51–53], and the dynamic prop-erties of this mode are determined by the relaxational components of the compressional or bulk modulus (K) and by the shear modulus (G) [49].
A main feature of the relaxation behavior of the α-relaxation is that a stretched exponential (also called KWW after Kohlrausch-Williams-Watts–
similar to equation 1.19) with 0<βKW W <1 is needed to describe the auto-correlation functionC obtained in DLS [54]:
C(t) =a⋅e(−t/τ)βKW W
(1.8) whereC(t)is the autocorrelation function,ais the contrast,tis the real time, τ is the relaxation time and βKW W is the stretching parameter (also called Kohlrausch parameter). βKW W describes a distribution of relaxation times.
For polymers as polystyrene and cis-polyisoprene, βKW W ≈ 0.4 [28, 49, 54]
and it is always independent of temperature [28].
The α-relaxation was long ago measured in bulk polystyrene by Depolar-ized Dynamic Light Scattering (DDLS) by Patterson [54] (what corresponds to the above mentioned two thirds of the overall scattered light, i.e., the anisotropic scattering). The autocorrelation functions obtained were not single exponentials, i.e., they did not show a single decay time of a single exponential function, but the α-relaxation appeared to have a distribution of relaxation times. A stretched exponential having stretching parameter βKW W ≈ 0.4 was needed to fit the experimental autocorrelation functions.
This distribution of relaxation times delivering βKW W ≈ 0.4 is experimen-tally observed for a variety of systems and can be measured by different experimental techniques as DLS, BDS, etc. Other values for the stretching parameter are possible tough.
Another feature of theα-relaxation is that its relaxation times are, in the range accessible to DLS, always q-independent at a given temperature, i.e., it is not related to diffusional or translational processes within that range of q-vectors observed in those DLS experiments (a more detailed view of q-dependence is given in section 1.2 and Chapter 2.1). This relaxation was then related to the orientation fluctuations of chain units.
Finally, a plot of the relaxation times, τα, as a function of temperature, displayed as -log(τα)vs. 1000/T (normally called Arrhenius plot or activation plot) delivers a curve that can be conveniently fitted by the so-called Vogel-Fulcher-Tamman-Hesse equation (VFTH or just VFT) instead of the linear1 Arrhenius type. The Arrhenius equation is given by:
τ(T) =τ0⋅exp(−Ea
RT ) (1.9)
or alternatively the equation may be expressed as:
τ(T) =τ0⋅exp(−Ea
kBT) (1.10)
where Ea is the activation energy, R is the universal gas constant and kB is the Boltzmann constant.
The VFT has several equivalent expressions [29]:
τ(T) =τ0⋅exp( B
T −T0) (1.11)
1The Arrhenius curve is just linear in an activation plot. Otherwise it is an exponential function of temperature.
τ(T) =τ0⋅exp( DT0
T −T0) (1.12)
τ(T) =τ0⋅exp( f
T −T0) (1.13)
where, τ0, B, D, f and T0 are constants; f is called the “fragility” of the liquid, and it is a measure for the deviation from the Arrhenius law, i.e., the bending of the observed experimental curves.
This kind of behavior of the relaxation times from the α-relaxation with temperature is also found for the viscosity as a function of temperature in rheology experiments, where the shift factors, aT, also follow the same type of curve, but there the fit equation is normally called Williams-Landel-Ferry (WLF):
log(aT) =
−C1⋅ (T −Tr)
C2+ (T −Tr) (1.14)
whereTris the reference temperature at which the master curve is taken and C1 andC2 are universal constants ifTr=Tg. Their values areC1g=17.44 and C2g=51.6 [25]. Such equations were found to apply in the temperature range Tg <T <Tg+100 ○C [25].
In 1965 Adam and Gibbs developed a theory that relates the relaxation times with the configurational entropy (Sc) and the temperature of viscous liquids [55]. They suggested that for densely packed liquids the conventional transition state theory for liquids, which is based on the notion of single molecules passing over energy barriers established by their neighbors, was not adequate to describe these materials. They proposed that, viscous flow occurs by increasingly cooperative rearrangements of groups of particles called CRRs (cooperative rearrangement regions). It was supposed that each CRR was acting independently of other such groups, but the minimum size of such a group was temperature dependent. The relationship between the minimum sized group and the total configurational entropy of the liquid delivers the Adam-Gibbs equation (AG):
τ(T) =τ0⋅exp( C
T Sc) (1.15)
It has been shown that the VFT equation is mathematically equivalent to the WLF equation [56] and to the Adam-Gibbs model (AG) [57] if the
temperature dependence of the configurational entropy (Sc(T)) is taken into account. Other equations are also equivalent to those ones as the equation proposed by Cohen and Turnbull [58] that is equivalent to the empirical for-mula based on free volume proposed by Doolittle in 1951 [59]. This means all these equations are mathematically equivalent and are a trial to describe the features of viscous liquids in the same way,i.e., they can all be derived based on the empirical free volume equation [25] primary proposed by Doolittle:
η(T) =a⋅exp( b
fv) (1.16)
where a and b are constants and fv =Vf/V, the fractional free volume.
The real behavior of viscous and viscoelastic liquids deviates from the VFT/WLF universal formulas in the range of extremely high or extremely low temperatures as summarized by McKenna [60]. Actually, the widespread belief that glasses cease to flow under Tg seems to be equivocal. In this work McKenna collected data for polystyrene in a broad range of temper-atures from several groups and, it turned out that the actual behavior of viscous materials diverges from the VFT for extremely high and extremely low temperatures (in other words, the VFT cannot “bend” enough to fit the α-relaxation times or the viscosity on the long range temperature scale).
This means that the nature of the glass transition is not that predicted by the VFT, i.e., there is no T0 temperature, or “Vogel temperature” at which the material freezes completely. Consequently, as the VFT, WLF, Adam Gibbs and some other less used fit equations are all equivalent to each other, all these models fail to fully explain the nature of glasses in a broad range of temperatures. Hence, these equations are just convenient fit functions that approximately describe the behavior of the α-relaxation or the viscosity in glasses within a certain range of temperatures.
As pointed out before, theα-relaxation shows a distribution of relaxation times characterized by 0 < βKW W ≤ 1 ( being βKW W ≈ 0.4 for a variety of polymers). A reasonable explanation for this characteristic of the glass transition is given by the main idea behind what exactly happens at the glass transition, and the nature of the α-relaxation. It has been thought for many years that the α-relaxation is a large scale conformational rearrange-ment of the main chain backbone [15, 25] involving long range cooperative thermal motions of individual chain segments [25]. The hindrance of these
“micro-Brownian” motions can be described in terms of frictional or viscous forces, resulting from the interactions of the moving chain segments with their neighboring molecules and between segments of the same molecule [25].
These molecules have an intrinsic relaxation, i.e., the chains change their conformations in given rates that are temperature and probing rate depen-dent. It is important to note here that external forces applied to probe the system should be small enough in order to rely on being within the linear response regime.
Theories describing the α-relaxation as being the relaxation of segmen-tal conformations involve the solution of diffusion equations in multidimen-sional chain space (fractal dimensions) and yield results involving distribu-tions of relaxation times instead of a single relaxation time. For the same reason the stretching exponential parameter is expected to be in the range 0 < βKW W ≤ 1, more precisely βKW W ≈ 0.399 as normally found in ac-curate experiments [28, 54] (βKW W = 1 indicates a unique relaxation time, while 0 < βKW W < 1 stands for a distribution of relaxation times). The nature and physical meaning of the stretching parameter,βKW W, also called Kohlrausch parameter is a 165 year old problem, considered by some sci-entists as one of the most intriguing and important problems in contempo-raneous physics [61, 62]. Before introducing this problem, it is of valuable interest to show the simple derivation of the KWW equation, and the nature of the phenomena behind it. While treating the problem of decay of residual charge in Leyden jars, Kohlrausch found that this decay was not single expo-nential, and that the relaxation rate decreases with time, following a power law function:
dQ(t)
dt = −γ(t) ⋅Q(t) (1.17)
where γ(t) is the relaxation rate and decreases with time following a power law:
γ(t) ∝t−(1−β)= 1
t1−β (1.18)
by integrating equation 1.17 one finds:
Q(t) =Q(0) ⋅e(−t/τ)β
(1.19) that is exactly the form of the well known stretched exponential or KWW, equation 1.8. Kohlrausch found the valueβ=0.43 for this specific problem.
Phillips [61,62] found that the physical meaning of the stretching param-eter is related to the fractal dimension at which the relaxation process is taking place and, is simply given by:
β = d
d+2 (1.20)
whered is the dimensionality of the configuration space at which the micro-Brownian motion occurs [61,62].
The following question arises: Why does the α-relaxation of the glass transition show relaxation decays with a stretching parameterβKW W =0.399 for a variety of polymers? What is the dimensionality, or in other words, the fractal dimension of the glass transition? Why do different kinds of polymers, show the same stretching parameter, i.e., the same segmental relaxation behavior? What do these materials have in common? Why polymers such as polystyrene or cis-polyisoprene have the sameα-relaxation characteristics, even though their chemical structure is completely different?
A non-understood characteristic of the α-relaxation in glass for-mers: when studying the α-relaxation in different glass forming systems, where distributions of relaxation times are obtained, experimental relaxation data can be well described by a correlation function with the form of equation 1.19:
C(t) =a⋅e(−t/τ)βKW W
(1.21) Some specific values of stretching parameter, βKW W, repeat for materials of completely different chemical nature. One of these values, that is the same for polystyrene, cis-polyisoprene and others, isβKW W = 0.4, and is indepen-dent of temperature [28]. Why this happens? These questions remained unsolved so far, and their answer would bring a deeper understanding about the α-relaxation and, therefore, the glass transition itself.
The nature of the phenomena described by stretched exponential func-tions was introduced while deriving the stretched exponential function. The stretching parameterβKW W is related tod, where d is the dimensionality of the configuration space in which the micro-Brownian motion occurs [61,62].
Considering that one of the common features of glasses of completely differ-ent chemical nature is that they are formed by random walks, let us take this as a starting point. The mathematician Benoit Mandelbrot showed that in average a 2-dimensional random walk has the fractal dimension d≈ 1.33 2. By substituting this value in equation 1.20, one finds:
2Note that R ≈ N1/2b, where D = 1/2 is the one-dimensional fractal dimension of random walks [33], whileD=1.33 is the two-dimensional fractal dimension of a random
β= 1.33
1.33+2≈0.399 (1.22)
that is exactly the searched stretching parameter for theα-relaxation.
Could it be the reason for the well known value found for the stretching parameter in several glass former materials? Is the tetrahedral symmetry of these systems the reason for these relaxation characteristics? If so, then more attention should be brought to this issue since the glass transition in itself is a topic of major importance in our times.
Systems where other stretching parameters are involved should show dif-ferent types of dimensionality for the micro-Brownian motions, likely because different numbers of chain segments are involved in the relaxation process.
The normal mode however, shows stretching parameter larger than 0.5 and increases with temperature [36]. Does it mean that the chain conformation, or more precisely the number of segments participating in this relaxation is dependent on temperature? If so, this changes our classical view of the rep-tation motion itself and brings another view to the chain self-diffusion within the polymeric mass.
1.1.1.3 Open questions concerning the glass transition in bulk