Microhardness additivity law

In conclusion, the deviation from the microhardness additivity law (Fig. 5.4, line 1) can be explained in terms of two distinct contributions (a) a crystallinity depression caused by the coexistence of the PE and t-PP phases, which yields curve 2, and (b) a substantial decrease of the crystal microhardness of the PE and t-PP components caused by an increase of the surface free energy erg with composition, which leads to curve 3. It is suggested that the rise in erg rise is a consequence of the increase in the level of surface defects including entanglements (see Section 4.3). 

By means of the microhardness additivity law (eq. (4.3)) one can attempt to evaluate the contribution of the soft-segment amorphous phase to the overall micro-hardness. For a four-phase system as in the present case (Fig. 6.7), (two crystalline modifications and two amorphous phases), eq. (4.3) can be written as ... 

It is important to note in these introductory remarks that, like many mechanical properties of solids, microhardness obeys the additivity law ... 

Figure 5.1. shows the linear increase of the microhardness of the low-density polymer with increasing concentration of the linear material, from 20 MPa up to about 70 MPa, for 100% of the HD component. A similar linear increase is obtained for the slowly crystallized materials (see arrows in Fig. 5.1.). The H values are, however, larger owing to the larger crystal thicknesses of both components (Baltd Calleja, 1976). Using the additivity law for a system comprising two types of crystals, LD and HD, then ...

We can now use the new composition values and Wg to derive the microhardness H for blends of recycled PE in terms of a composite comprising two populations of crystals (the thick ones and the thinner ones). According to the additivity law... 

Baltd Calleja, 1985). This implies that the microhardness in these materials is not substantially affected by the presence of microvoids a few micrometres in diameter (Matsuo et al, 1984). Furthermore, a very conspicuous deviation from the additivity law (straight line 1) ... 

Bearing in mind the outlined peculiarities of condensation polymer blends, and particularly when they consist of one component and one phase (this case is more the exception rather than the general rule since block copolymers usually consist of two, three, or more phases), the application of the additivity law for the evaluation of their characteristics does not seem to be completely justified. The observed good agreement between the measured microhardness values and the calculated ones (Fig. 5.7) allows one to make an important conclusion in this respect. 

According to the additivity law, eq. (1.5), one can calculate the microhardness H of any multicomponent and/or multiphase system provided the microhardness of each component and/or phase //, and its mass fraction u), are known. This relationship is of great value because it offers the opportunity to characterize micromechan-ically components of a system which are not accessible to direct measurement. 

What could be the reason for the failure of the additivity law Obviously one has to assume that, for multicomponent and/or multiphase systems, when one of the components (phases) is characterized by a viscosity at room temperature which is typical for low-molecular-weight liquids, the microhardness behaviour of the entire system should be different from the case in which all the components (phases) have TgS higher than room temperature because the mechanism of the response to the applied external mechanical field is different. In the latter case all the components (phases) plastically deform as a result of the applied external force. In the former... 

The advantage of the modified additivity law incorporating Tg (eq. (5.17)), is that it is possible to use it to account for the contribution of any amorphous phase and/or component to the overall microhardness of the system, provided the Tg of this phase and/or component is known. Hence, for systems which contain more than one crystalline and/or amorphous phases with glass transition temperamres and mass fractions Tgi and Wi, respectively, the additivity law can be presented in the following way ... 

In order to examine the relationship between the microhardness of the MFC and those of its constituents (PET and PA6) including the moiphological entities, the two constituents of the MFC were subjected to the same thermal and mechanical treatments as the MFC and characterized after each step. Further, to evaluate the microhardness of the reinforcing microfibrils the additivity law was applied and the effect of crystal size on the structure formation was taken into account. 

The fact that both the neat components and their blends are relatively well characterized with respect to their varying structures and morphologies as a result of the applied mechanical and thermal treatments, permits us to follow the gradual variation of microhardness as a function of structural parameters. In this way one can obtain the H values for material components which are not accessible to direct experimental determination. Furthermore, having the extrapolated values for completely amorphous and fully crystalline homopolymers and starting from a knowledge of the number of components (and/or phases) one can make use of the additivity law (eq. (1.5)) to evaluate the mechanical properties of components which cannot be isolated or do not exist as individual materials. A good example of this are the PET microfibrils studied here (Fig. 5.16(b)). 

As mentioned above, application of the additivity law (eq. (1.5)) supposes a knowledge of the number of the components (or phases) with given microhardnesses and weight fractions. What is not explicitly given in this equation is the type and the extent of mutual dispersion of the components as well as the quality of the adhesion on the contact surface boundary between the components (phases). We wish to stress here that this has an influence on the reliability of the H values derived from the additivity law. 

Equation (5.23) for the microhardness of the amorphous matrix of the MFC (Fig. 5.17(d)) is acceptable as it has been demonstrated (by Baltd Calleja et al, 1998) that the H values for completely amorphous copolymers (with random sequential order) obey the additivity law provided the Ha values for the respective homopolymers are used. In this way one obtains for the microhardness of... 

One can conclude that the microindentation technique allows the strain-induced polymorphic transition in PBT to be followed. The observed rather abrupt variation in H (within 2-4% of external deformation) makes the method competitive with respect to sensitivity to other commonly used techniques such as WAXS, infrared spectroscopy, Raman spectroscopy, etc. (Tashiro Tadokoro, 1987). Furthermore, by applying the additivity law it is possible to calculate the microhardness of completely crystalline PBT, comprising crystallites of the /6-type, as = 122 MPa. This technique can also be used to examine the stress-induced polymorphic behaviour of PBT in copolymers and blends as will be demonstrated in the following sections. 

Table 17.3 The Experimental Microhardness Values (7/exp). the Calculated Microhardness Values According to the Additivity Law (7/cal). PP Degree of Crystallinity Derived from WAXS (a), the Experimental Values of Melting Temperatures (Tin), the Values of b Parameter, and the Values of the Surface Free Energy of Neat PP and PP Component in Uncompatibilized and Compatibihzed Blends.

Based on the fact that all the samples (Table 13.4) have been prepared in the same manner [34], one can assume that the corresponding PVDF fraction in each blend is characterized by the same degree of crystallinity (25%). This finding allows us to consider formally the blend samples under investigation (Table 13.4) as two-phase systems. In the case of such blends the microhardness can be calculated by means of the additivity law as ... 

Using the repoited data on the experimentally derived values of glass transition temperature, Tg, degree of crystallinity, Vickers indentation microhardness, H, and blend compositions for homopolymers, block copolymers, blends of polyolefins, or of polycondensates, blends of miscible amorphous polymers and copolymers (some of them with rather complex molecular architecture), all of them containing a soft component and/or phase at room temperature, an attempt is undertaken to look for the reasons for the frequently reported drastic deviations of the experimentally derived H values from the calculated ones by means of the additivity law assuming that the contribution of the soft component and/or phase is negligibly small. 

---