Relaxations in the Glassy State. Short-Range Dynamics TUTURIAL 8

INTRODUCTION Although molecular mobility is severely restricted below the glass transition temperature, a cascade of subglass relaxation phenomena may occur below T g. That reveal different modes of mobility. These processes, called secondary relaxations, are labeled β, γ, etc., in order of decreasing temperatures. The temperature at which a given group of segments undergoes a relaxation usually depends on both: the polymer structure, and the local molecular environment.

It is widely accepted that relaxations in the glassy state basically imply intramolecular processes, and perhaps only the β-relaxation (the highest temperature secondary relaxation) is also related to intermolecular effects. In fact, some authors consider the β-relaxation as the precursor of the glass transitions, and in this sense it would be a "universal feature" of the glass-forming materials, including low molecular weight compounds. As a general picture, it is considered that molecular units associated with a dipole tend to rotate within a cage, cluster, or mobile islands by the effect of the electric field.

The motion is restricted by limits of the cluster (island of mobility) made up of other polymer atoms acting as the boundaries of the cage. The motion of the unit is described in terms of the coordinates fixed with respect to the coordinates of the cavity, and an intramolecular potential barrier expressed as a function of these coordinates controls it. Consequently, subglass relaxation processes can be envisaged as a thermally activated motion between two potential wells separated by a potential barrier.

The probability of location of molecules at each side of the barrier is determined by the Boltzmann distribution. The application of an external electric field alters the equilibrium distribution by changing the relative depths of the minima, thus causing a redistribution, the rate of which is controlled by the activation energy barrier or activation enthalpy

PHENOMENOLOGICAL CLASSIFICATION OF SECONDARY RELAXATION PROCESSES There are two competing theories concerning the origin of the β-relaxation peak. According to Johari and Goldstein, the appearance of the β- peak does not require any specific molecular motion. The occurrence of the β-peak would be due to the so-called "clusters" or "islands of mobility" present in glass-forming materials.

Another authors interprets the presence of all the observed secondary losses as the result of some specific molecular motion. It’s also possible to think that these theories are not mutually exclusive; in fact, it is possible to consider specific molecular motions taking place in the above mentioned "islands of mobility“.

Several cases of molecular motions causing dielectric relaxations could be considered: – –1 In polymers without prominent lateral chains, such as polyvinyl chloride (PVC) or polycarbonates, local main chain motions can give rise to secondary dielectric relaxations – –2 Motions of side groups about the bonds linking them to the main chain, as in the case of poly-n-alkyl methacrylates, are probably the best studied examples of the polymers containing dipolar groups in their lateral chains – –3 Internal motions of the side chain groups without cooperation of the main chain are typical of polymers containing flexible units or polar final groups – –4 Another possibility concerns the motions of small molecules, as water, for example, embedded in a polymer matrix. – –5 Secondary relaxations in semicrystalline polymers – –6 Secondary relaxations in liquid crystalline polymers.

In any case, the relevant parameters concerning the secondary relaxations are: the relaxation strength, , the frequency of the maximum, and the shape and broadness of the relaxation peak. Unfortunately, in many cases, secondary relaxations mutually overlap or appear as a shoulder of the prominent  -relaxation The determination of the relaxation strength and the maximum of the peak may be subjected to some degree of uncertainty owing to the errors involved in the deconvolution of the overlapping peaks.

Local Main Chain Motions Polymers such as polyvinyl chloride (PVC), polycarbonates (PCs), and aromatic polyesters derived from terephthalic acid (PET) and similar polymers or isophthalic (PEIT) acid show secondary peaks which in some cases can be depressed by the effect of additives that increase the modulus and decrease the damping (antiplasticizers).

Journal of Non-Crystalline Solids (1998)

Macromolecules, Vol. 19, No. 8, 1986

Effect of the crystallinity in the relaxation strength

The higher the temperature, the narrower the distribution

Motions of Side Groups About Their Link to the Main Backbone In this category are included the β-relaxations of poly(alkyl methacrylates) and poly(itaconates). Experimental data show that the position of the β-relaxation in polyalkyl methacrylates is insensitive to the length of the alkyl group. However, as the temperature of the  -relaxation decreases by the effect of adding successive methylene groups in the lateral chain, the  -relaxation tends to overlap with the β-relaxation, giving rise to the  β-peak

The intensity of the β-relaxation in PMMA is higher than that of the  -relaxation, an uncommon fact in polymers.

Macromolecules, 29 (1), , 1996

Motions within Side Groups There are a number of examples corresponding to this category, such as the polymers in which one or more hydrogen atoms of alkyl groups of polyalkyl methacrylates, polyalkylitaconates, etc., are substituted for halogen atoms. This category also includes polymers containing flexible rings as side groups. These polymers may present ostensible β-relaxations

Polymer (2004) 1845–1855  ” max =0,06  ” max =0,18

(a)Loss factor and (b) Electric loss modulus as a function of the temperature at a frequency of 1Hz for PCHMA(  ), P4THPMA(  ) and PDMA(  ). Arrows show the calorimetric glass transition temperature (T g ), measured by DSC γ γ β β  

Figure 4. Loss factor data (symbols) and their deconvolution in two FK functions (lines) as function of temperature, at 10.3 Hz for P4THPMA

Figure 3. Dependence of log f max with the inverse of temperature in the range of α (  P4THPMA),  (  P4THPMA),  (  P4THPMA,  PCHMA,  PDMA) and  (  P4THPMA) relaxations. PDMA, 43.8 kJ/mol PCHMA, 43.1 kJ/mol P4THPMA, 48.0 kJ/mol 68.2 kJ  mol-1

Figure 6. Potential energy (kJ mol -1 ) profile for rotation of O-C bond for (  1 : -180º to +180º) for one-unit model compounds of (  ) PCHMA, (  ) P4THPMA and (  ) PDMA.  -relaxation: parcial rotation of the cyclohexyl ring as a whole

γ-relaxation: Chair to inverse-chair interconversion of the cyclohexyl ring

Chair  Boat  half-chair  chair Energy Barrier (kJ  mol -1 ) P4THPMA MMX SE (48.0)  PDMA MMX SE (43.8)  PCHMA MMX SE (43.1) 

Motions due to the Presence of Small Molecules in the Polymer Matrix Low molecular weight compounds not only act as plasticizers depressing the glass transition temperature of polymers but also interact with the motions that cause subglass activity A typical case is the effect of water on the secondary relaxations of polymers containing hydrophilic groups, such as hydroxylic or amide groups

Secondary Relaxations in Semicrystalline Polymers The analysis of the dielectric spectroscopy of semicrystalline polymers is more complicated than that of amorphous polymers. The reason is that, in addition to the difficulties concerning the molecular assignation of the observed peaks, the phase in which the relaxations occur must also be elucidated. To start with, the existence of a double  -process in many semicrystalline polymers is well known.

The lower-temperature process, called the  a -peak, is usually related to the cooperative relaxation of the amorphous phase. This fraction is sometimes called "intercrystalline", because it refers to the fringing material existing between lamellar structures. Typical WLF behavior corresponding to an amorphous polymer is expected for this peak. At slightly higher temperatures, another narrower peak appears that is related to some sort of mechanism in which crystalline entities of the material are implied.

The dielectric spectrum of PE displays three characteristic relaxational zones conventionally designated as γ, β, and  in order of increasing temperatures Semicrystalline halogen-polymers such as polyvinyl fluoride (PVF), polytetrafluorethylene (PTFE), and related polymers also show considerable sub-T g activity. The low-temperature relaxations in these polymers are understood as "cooperative'', local mode, main chain motions.

Eyring equation   H and the activation energy Ea given by the Arrhenius equation are related by Ea=  H + RT.  The values of  H and  S can directly determined from ln (f/T) versus 1/T plots.  Low values of the activation entropy suggests, that the processes are a simple secondary relaxation.  High values of entropies indicate some degree of cooperativity.  Secondary relaxations near to the  relaxation sometimes shows high entropy.

Eyring

Summary Secondary relaxation in polymers revels molecular mobility in the glassy state. Secondary relaxation in polymers revels molecular mobility in the glassy state. Multiple causes could origin secondary relaxation: motions of the main chain, motion of side chain, motion of part of the side chain. Multiple causes could origin secondary relaxation: motions of the main chain, motion of side chain, motion of part of the side chain. To characterize secondary relaxation it is necessary to know the evolution of the relaxation strength, the relaxation map, and the broadness of the peak. To characterize secondary relaxation it is necessary to know the evolution of the relaxation strength, the relaxation map, and the broadness of the peak.