Introduction

Recent years have seen the microindentation hardness technique gaining increasing application in the characterization of the structure and morphology of polymers [1–5]. The method uses a sharp indenter that penetrates the surface of the specimen upon application of a given load at a known rate. Pyramid indenters are best suited for indentation tests. Here the hardness, in principle, does not depend on the size of the indentation and the elastic recovery is minimized in comparison to other indenters. During an indentation test the response of a polymeric material is initially elastic. When the stresses exceed the elastic limit, plastic flow occurs and a permanent deformation arises. At this stage, the plastic yield stress and the elastic modulus govern the elasto–plastic response to indentation [6]. When the load is removed, the indentation depth recovers elastically while the diagonal of the impression remains nearly unaltered [7]. The complicating effects of viscoelastic relaxation are usually minimized by measuring the indentation diagonal immediately after the load release [8]. The size of the permanent area of impression has been shown to depend on the arrangement and structure of the microcrystals and the specific morphology of the polymeric material [8, 9]. From a mechanical point of view, the polymer may be regarded as a composite consisting of alternating crystalline and disordered elements [8]. An earlier study of the hardness dependence on the density, ρ, of melt crystallized polyethylene (PE), revealed that for crystallinities larger than 50%, the plastic strain is dominated by the deformation modes of the crystals [9].

It is important to note in these introductory remarks that, like many mechanical properties of solids, microhardness obeys the “rule of mixture”, frequently also called the “additivity law” (further referred to only as additivity law):

$$H=\Sigma H_{i}\varphi_{i}$$
(1)

where H i and φ i are the microhardness and mass fraction, respectively, of each component and/or phase. This law can be applied to multicomponent and/or multiphase systems provided each component and/or phase is characterized by its own H. Equation (1) is frequently used in semicrystalline polymers for one or other purpose operating with the microhardness values of the crystalline, H c, and amorphous, H a, phases, respectively. This relationship is of great value because it offers the opportunity to characterize micromechanically components and or phases of a system, which are not accessible to direct measurement.

Application of the additivity law (Eq. (1)) presumes a very important requirement—each component and/or phase of the complex system should have a T g or T m values above room temperature (at which the indentation is performed) and thus be capable of developing an indentation impression after the removal of the indenter. If this is not the case, the assumption H a = 0 for the soft component and/or phase does not seem to be the best solution, although it is frequently made.

The assumption H a = 0 would mean that the soft component and/or phase (with a T g or T m temperatures below room temperature and not displaying its own indentation impression) has only a “diluting effect”. The role of such a component and/or phase in the formation of the overall H of the complex system is not only in the “diluting effect”. It creates a completely different deformation mechanism (in addition to the plastic deformation of the solid component and/or phase) as compared to the deformation behavior of complex systems non-comprising a soft component and/or phase. The deformation mechanism of the entire system is changed in such a way that the system does not obey anymore the additivity law (Eq. (1)).

It has been demonstrated that many semicrystalline polymers, copolymers and blends obey the additivity law [1]. Exceptions, such as blends of high density (HD), polyethylene (HDPE) with polypropylene (PP) have been explained by a peculiarity in the morphology and characteristics of the crystallites formed (mostly related to the crystal sizes, surface free energy, and others) [1, 10].

An example of such an exception is illustrated in Fig. 1, where the experimentally obtained variation of H as a function of the weight fraction of the PE component, φ, (curve (3)) is plotted. It is seen that the H values for the initial PE and isotactic polypropylene (iPP) gel films (H PE = 105 MPa and H PP = 116 MPa) do not differ substantially from each other.

Fig. 1
figure 1

Microhardness, H, of PE/iPP blended gel films as a function of PE concentration, φ: additivity behavior from Eq. (1) using the w c values of the individual homopolymers (1); H values using \({w_{\rm c}^{\rm PE}}\) and \({w_{\rm c}^{\rm PP}}\) data (Table 1) (2); experimental data (3) [1, 10]

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One can see a very clear deviation (straight line (1) of Fig. 1) with increasing PE concentration, φ, from the additivity law:

$$H=\varphi H^{\rm PE}+(1-\varphi) H^{\rm PP}$$
(2)

Since the glass transition temperature of PE is much lower than the room temperature, the microhardness contribution of the amorphous phase has been accepted to be \({H_{\rm a}^{\rm PE}\sim}\) 0 [10]. Hence, for PE, with a degree of crystallinity, \({w_{\rm c}^{\rm PE}}\) , using the additivity model of Eq. (1) one may write:

$$H^{\rm PE}=w_{\rm c}^{\rm PE} H_{\rm c}^{\rm PE}$$
(3)

On the other hand, since for iPP \({H_{\rm a}^{\rm PP} \ne 0}\) , for iPP, with a degree of crystallinity, \({w_{\rm c}^{\rm PP}}\) :

$$H^{\rm PP}=w_{\rm c}^{\rm PP} H_{\rm c}^{\rm PP}+(1- w_{\rm c}^{\rm PP})H_{\rm a}^{\rm PP}$$
(4)

By combining Eqs. (3) and (4) one obtains for the overall microhardness of the blend, H, the expression:

$$H = \varphi w_{\rm c}^{\rm PE} H_{\rm c}^{\rm PE}+(1- \varphi) w_{\rm c}^{\rm PP} H_{\rm c}^{\rm PP}+(1-\varphi)(1- w_{\rm c}^{\rm PP}) H_{\rm a}^{\rm PP}$$
(5)

which describes the microhardness of the blended gel films in terms of the microhardnesses of the independent crystalline and amorphous components assuming H PE a ∼ 0. If one takes into account the crystallinity depression measured for the PE and iPP components in Equation 5, use H PEc  = 130 MPa and \({H_{\rm c}^{\rm PP}=145}\)  MPa, and let \({H_{\rm a}^{\rm PP}=90}\)  MPa [10] one obtains then curve (2) in Fig. 1 which is still far from the experimental values (Fig. 1, curve (3)).

It should be noted here that in the same paper [10] it is demonstrated that the differences between the experimentally measured and calculated H values disappear if the thermodynamically derived parameter b (accounting for the crystal surface free energy and the energy required to plastically deform the crystal [1]) is used for the calculations. However, it seems important to mention also that the values of the b parameter have been derived from the same H exp values and later used for the calculation of the H values (Eq. (1)).

The same approach (assuming H a = 0 for the soft component and/or phase) applied to thermoplastic elastomers of poly(ether ester) (PEE) type fails also to explain the large discrepancy (up to 100 MPa when the measured H values are in the range 20–40 MPa) between the experimental values and those calculated according to Eq. (1) [11, 12]. For this reason, one has to look for other factors, which may be responsible for such a discrepancy. Before discussing them let us recall briefly some of the characteristic features of the structure and morphology of thermoplastic elastomers of PEE-type, which illustrate in the best way the concept disclosed in the present study.

Thermoplastic elastomers of PEE type represent polyblock copolymers comprising poly(butylene terephthalate) (PBT) as the “hard” segments and poly(glycols) as “soft” segments, both of them forming “hard” and “soft” domains, respectively. Since the soft domains are characterized by T g values around −50 °C, they are in a liquid state at room temperature and are distinguished by a viscosity being much closer to those of low-molecular-weight liquids rather than to that of a solid amorphous polymer. In this respect it seems useful to recall that the molecular weights of the poly(tetramethylene glycol) (PTMG) and poly(ethylene glycol) (PEG) used are around 1,000, i.e. one deals with typical oligomeric materials. For this reason, it looks reasonable to accept that such a liquid (soft phase) will be characterized by a negligibly small microhardness, H s, in the equation for the overall microhardness of such a copolymer, H:

$$H=\varphi[w_{\rm c}H_{\rm c}^{\rm h}+(1 - w_{\rm c}) H_{\rm a}^{\rm h}]+(1-\varphi)H^{\rm s}$$
(6)

where φ is the mass fraction of hard segments (PBT in the present case), H hc and H ha are the microhardnesses of the crystalline and amorphous phases of the same hard domains, respectively, and w c is the degree of crystallinity of PBT.

Assuming, as in the case of the PE/iPP blend, the microhardness of the soft domains being H s = 0, the calculations of H according to Eq. (1) for a series of PEEs lead to a discrepancy between the measured, H exp, and calculated (using again Eq. (1)) \({H^\prime_{\rm cal}}\) amounting to 40–64 MPa, depending on the soft-segment composition, as will be discussed below.

A question arises about the reason for the failure of the additivity law in the above-mentioned systems, both, the polyolefins blends [10] and the multiblockcopolymer. 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 close to those of the low-molecular weight liquids, the mechanism of the response to the applied external mechanical field is different from that when all the components (phases) have T g and T m higher than room temperature. In the latter case all the components (phases) plastically deform as a result of the applied external force. In the former case, in addition to the plastic deformation of the harder components (phases), they are also displaced within the soft (liquid) matrix in which they are “floating”. The extent of this displacement depends on the viscosity of the matrix (the softer component and/or phase). This is the reason why the harder components cannot display their inherent microhardness. The microhardness is reduced by the ability of the harder components to move. This situation is illustrated in Fig. 2.

Fig. 2
figure 2

Schematic the indentation mechanism for three types of samples: (A) With T g and T m above the test temperature (room temperatute, RT), microhardness H ≈ 1/h, (a); (B) With T g and T m below RT, H ≈ 0, (b); (C) Complex system with matrix B and a “floating” dispersed phase A, H ≈ 1/(h A + h f), (c). In all the cases the elastic recovery of the samples is not taken into account

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How can one account for this microhardness depression, i.e. for the “floating effect”? As demonstrated above, the simple assumption that the soft phases have H s∼ 0, i.e. if one accounts only for its “diluting effect”, does not solve the problem. It is necessary to characterize the ability of the harder phase to move about within the soft matrix, and this will depend on the viscosity of the matrix, i.e. the soft phase. Since T g and viscosity are closely related to each other, it is possible to look for an analytical relationship between microhardness of the amorphous polymers and their T g.

The main goal of this study is, by means of larger number of studied systems, to show that the assumption H s = 0 for the soft component and/or phase being dominating in complex polymer systems leads to drastic deviations from the additivity law (Eq. (1)). The contribution of the soft component and/or phase to the overall microhardness can be much more reliably accounted for using the relationship between T g and the microhardness of the amorphous component and/or phase, particularly for systems characterized by dominating amorphous component and/or phase. A secondary goal of the work is to demonstrate, that the experimentally derived relationship between H and T g can be applied for evaluation of the T g value of a practically non-accessible component and/or phase.

Experimental

No experimental work was undertaken for this study since a good deal of data have already been reported in the literature in this respect. One needs only to evaluate the contribution of the soft component and/or phase through their glass transition temperature to the overall microhardness. Usually, samples differing in their crystallinity are used, which are prepared by means of various techniques indicated to the respective system below. What deserves to be mentioned here is that in all the cases the microhardness measurements have been carried out at room temperature employing a Vickers pyramid diamond. The microhardness value, H, has been always derived from:

$$H=1.854 P/d^{2}$$
(7)

where P is the applied load and d the diagonal of the residual impression. The permanent deformation has been measured immediately after load release to avoid long delayed recovery. Loads of a couple hundreds of mN have been used, for an indentation time of 0.1 min, in order to correct for the instant elastic recovery. This correction has been then applied to derive the H values using indentation times typically in the range 0.1–21 min and loads of around 150 mN.

Results

In accordance with the main goal of this study, i.e. recalculations of the overall microhardness by means of additivity law (Eq. (1)), however, assuming (in contrast to the approach used for the already published data), that the microhardness of the soft component and/or phase is not zero. Their contribution to the overall microhardness of the complex system is evaluated by means of the recently derived relationship between H and T g for a series of amorphous homo- and copolymers [13]:

$$H=1.97T_{\rm g}-571\;(H \hbox{in MPa},T_{\rm g} \hbox{in K})$$
(8)

Subsequently, examples for comparison of the measured, H exp, and calculated (with \({H^{\rm s}\ne 0}\)) microhardness values, \({H^{\prime \prime}_{\rm cal}}\) will be considered below for two-component multiphase systems comprising soft phases (blends of polyolefins), multiblock copolymers of thermoplastic elastomers (TPE) of condensation type, blends of amorphous miscible polymers, blends of copolymers with different molecular architecture, and finally, homopolymers comprising amorphous phase with very low T g (PE).

Two-component multiphase systems comprizing soft phase(s) (blends of semicrystalline homopolymers)

The best studied system in this respect is the afore-mentioned blend PE/iPP [10]. As already commented (Fig. 1), a significant discrepancy between H exp and \({H^\prime_{\rm cal}}\) (assuming H PEa  = 0) can be observed (compare the values of H exp with H′cal in Table 1). If one tries to account for the contribution of the PE amorphous phase, (which amounts up to 30%), to the overall microhardness of the blend by means of its glass transition temperature T PEg using Eq. (8), for some of the samples, characterized by lower content in the blend (50 and 25 wt%) of the high crystalline PE (w c between 70 and 80%), one obtains \({H^{\prime \prime}_{\rm cal}}\) values rather close to the measured ones (differences of 2–3%), as can be concluded from Table 1. For these calculations a value of \({T_{\rm g}^{\rm PE}}\)= −80 °C is assumed being the most frequently used for the amorphous unbranched PE [14–19]. It is worth noticing here that much higher values for T g of PE (around −30 °C) are also reported [20–25], however, they are derived from branched PE [20–24] or samples characterized by highly strained amorphous chains [25].

Table 1 Composition, crystallinity, \({w_{\rm c}^{\rm PE}}\), and \({w_{\rm c}^{\rm PP}}\), measured microhardness, H exp, calculated microhardness(according to Eq. (1), with H a = 0), \({H^\prime_{\rm cal}}\), and according to Equation 8, (with \({H^{s}\ne}\) 0), \({H^{\prime \prime}_{\rm cal}}\), and the differences between the measured and calculated values, \({\Delta H'}\) and \({\Delta H^{\prime \prime}}\), respectively, for PE/iPP blends
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Basically, the unsatisfactory agreement between \({H^{\prime \prime}_{\rm cal}}\) (assuming \({H^{\rm s}\ne}\) 0) and H exp for the blend with the highest content of PE (75 wt%) could have another, additional origin. In order to apply Eq. (8) correctly, one needs to know the T g values of the amorphous phase of the particular sample under investigation. This was not the case for the discussed system PE/iPP for which the common value of T g = −80 °C was used, although the real value can be higher, particularly in the cases with high degree of crystallinity. What is more, a higher T g value would lead to a smaller difference \({\Delta H^{\prime \prime}}\) for the same sample. Therefore, for the next systems only the experimentally measured T g values of the respective soft component and/or phase will be used for the similar calculations.

One-component multiphase systems containing soft phase(s) (polyblock copolymers)

Micromechanical studies have also been carried out on thermoplastic elastomers. The latter represents a special class of multiphase systems (block copolymers) exhibiting an unusual combination of properties: they are elastic and at the same time tough and they show low-temperature flexibility and also strength at relatively high temperatures (frequently ca. 150 °C) [26, 27].

In addition to the thermoplastic elastomers of PEE type, also a series of new poly(ester ether carbonate) (PEEC)) multiblock terpolymers with varying amount of ether and carbonate soft-segment content will be considered. Dielectric relaxation experiments on the same PEEC revealed the existence of two relaxation processes [28]. The dielectric loss values show a relaxation maximum appearing at about 0 °C for 10 kHz (β relaxation) accompanied by a lower temperature relaxation (γ relaxation) that appears at about −50 °C.

The microhardness of films of thermoplastic elastomers based on PBT–cycloaliphatic carbonate (PBT–PCc) block coplymers has also been studied [29]. The microhardness of their amorphous films has been discussed in terms of a model given by the additivity values of the single components H PBTa and H PCca . In the case of semicrystalline copolymers, the authors related the observed deviation from the additivity law as mainly due to the depression of the crystal microhardness of the PBT crystals and partly due to a decrease in crystallinity of the PBT phase [29]. The measured H values for the terpolymers studied [29] are very low in comparison to the values known for common synthetic polymers even those in the amorphous state [30]. What is more, the H exp values are more than three times smaller than the calculated ones. For the purpose of such calculations, the authors [29] present the additivity law (Eq. (1)) in the following form:

$$H =\varphi H^{\rm PBT}+(1- \varphi)H^{\rm s}$$
(9)

where φ is the weight fraction of the hard PBT segments and H s the microhardness of the soft domains. Further, taking into account the fact that the T g of the soft-segment amorphous phase lies between −50 and 0 °C, (depending on the PCc content [29]), the authors assumed again that H s ∼0. As a result, H will be depressed with decreasing values of φ according to the simple expression [29]:

$$H=\varphi H^{\rm PBT}=\varphi[w_{\rm c}H_{\rm c}^{\rm PBT}+(1- w_{\rm c})H_{\rm a}^{\rm PBT}]$$
(10)

Again, in this case also the assumption H s = 0 means to take into account only the “diluting effect” of the soft phase and no attempt to be made for considering the possibility for another deformation mechanism. So, by applying the numerical values φ = 0.6, \({H_{\rm c}^{\rm PBT}=287}\)  MPa, and \({H_{\rm a}^{\rm PBT}=54}\)  MPa [29] one can derive the calculated values H′cal for the terpolymers depending on their crystallinity w c (Table 2).

Table 2 Composition, degree of crystallinity of PBT, w c, glass transition temperatures of the soft, \({T_{\rm g}^{\rm s}}\) , and the hard, \({T_{\rm g}^{\rm h}}\) , domains, measured microhardness, H exp, calculated microhardness (according to Eq. (1) with H s = 0),H′cal, and according to Eq. (8) (with \({H^{\rm s} \ne 0}\)), H″cal, and the differences between measured and calculated values, \({\Delta H^\prime}\), and \({\Delta H^{\prime \prime},}\) for PEEC block terpolymers
Full size table

The differences between H exp and H′cal are quite obvious—the calculated values are two to three times higher than the measured ones (Table 2).

For a quantitative evaluation of the microhardness depression effect of the soft phase, one has to replace H s in Eq. (9) with Eq. (8) using for T g the experimentally measured values of the soft-segment phase \({T_{\rm g}^{\rm s}}\) . This leads to the expression:

$$H=\varphi[w_{\rm c} H_{\rm c}^{\rm h}+(1- w_{\rm c})H_{\rm a}^{\rm h}]+(1- \varphi)(1.97T_{\rm g}^{\rm s}- 571)$$
(11)

Calculation of H for PEE and PEEC by means of Eq. (11) offers data that are in a very good agreement with the measured H exp values as shown in Table 2 for PEEC and Table 3 for PEE and PEEC (samples 1–6).

Table 3 Composition, annealing temperature (T) of 5 × drawn samples, degree of crystallinity of PBT, w c, glass transition temperature of the soft domains, \({T_{\rm g}^{\rm s}}\) , measured microhardness, H exp, calculated microhardness,\({H^\prime_{\rm cal}}\) , (according to Eq. (1), with H s = 0) and \({H^{\prime \prime}_{\rm cal}}\) (according to Eq. (8), with \({H^{\rm s} \ne 0}\)), and the difference between measured and calculated values, \({\Delta H^\prime}\) and \({\Delta H^{\prime \prime}}\), respectively, for thermoplastic elastomers of PEE- or PEEC-type [1]
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Another system of interest demonstrating the limits of the additivity law (Eq. (1)) (with the assumption H s = 0) is again a thermoplastic elastomer of novel type—multiblock polyester-amide copolymers, synthesized recently [31]. These materials, similarly to PEE, possess a hetero-phase structure, with two T g values and only one melting temperature above room temperature, which corresponds to the fusion of PBT crystals. The diamide segments are chosen to mainly contribute to the amorphous domains and confer to the material an elastomeric character.

Data on the molecular weight of the used oligotetrahydrofuran, blocks fractions (in mol%), temperature transitions, degree of crystallinity, and density, together with the measured and calculated H exp and H′cal (with H s = 0) values are reported [32]. Hardness is shown to drastically decrease with increasing etherdiamide content. The experimentally measured hardness values of the copolymers clearly deviate from the linear additivity law, where the authors [32] assume H s∼ 0 for the poly(etherdiamide) homopolymer, as its T g is far below the room temperature. Following the same logic as in the case of PEE, PEEC and PE, they have used the following equation, which formally describes the hardness of a two-component system in terms of the H values of the individual constituents [32]:

$$H=(1- \varphi^{\rm DA})H^{\rm PBT}+\varphi^{\rm DA}H^{\rm DA}$$
(12)

Here, H PBT and H DA are the microhardness values of the PBT and etherdiamide domains, respectively, (H DA∼ 0) and φDA is, in this particular case, the mole fraction of the soft segments (etherdiamide component). In analogy with the previous cases, the value of H PBT has been expressed in terms of the crystal hardness, \({H_{\rm c}^{\rm PBT}}\) , the hardness of the PBT amorphous regions, \({H_{\rm a}^{\rm PBT}}\) , and as a function of the volume degree of crystallinity of PBT referred to the volume fraction of PBT “component” in the sample, \({v_{\rm c}^{\rm PBT}}\) . Therefore, one can rewrite Eq. (12) to yield [32]:

$$H=(1- \varphi^{\rm DA})[v_{\rm c}^{\rm PBT} H_{\rm c}^{\rm PBT}+(1- v_{\rm c}^{PBT})H_{\rm a}^{\rm PBT}]$$
(13)

Using the volume fraction crystallinity,\({ v_{\rm c}=v_{\rm c}^{\rm PBT}}\) (1- φDA), Eq. (13) finally reads [32]:

$$H=v_{\rm c}^{\rm PBT}H_{\rm c}^{\rm PBT}+(1- v_{\rm c}^{\rm PBT}- \varphi^{\rm DA})H_{\rm a}^{\rm PBT}$$
(14)

\({H_{\rm a}^{\rm PBT}}\) has been reported to be of 54 MPa [29, 33].

It has been found [32] that the calculated by means of Eq. (14) values \({H^\prime_{\rm cal}}\) , i.e. assuming again H s = 0, are 8–10 time larger than the measured H exp values for the two-third of the samples under investigation. For this reason, one has to look again for another reason for the observed discrepancy. In order to apply again the “floating effect” concept, one has to replace in Eq. (12) the H DA with \({H^{\rm DA}=1.97T_{\rm g}^{\rm DA}-571}\) and combining further with Equation 13 to obtain for the overall microhardness:

$$H=(1- \varphi^{\rm DA})[w_{\rm c}^{\rm PBT} H_{\rm c}^{\rm PBT}+(1 - w_{\rm c}^{\rm PBT})H_{\rm a}^{\rm PBT}] +\varphi^{\rm DA}(1.97 T_{\rm g}^{\rm DA}-571)$$
(15)

where w c is the weight fraction crystallinity of PBT.

The calculated values for \({H^{\prime \prime}_{\rm cal}}\) according to Eq. (15) also differ from the measured one, and what is more, they are not consistent—only one third of them show small differences (around 3–8%) from the H exp values, while the rest scatter in a large interval contrasting the results from the thermoplastic elastomers of PEE type (Tables 2 and 3). An attempt to explain these differences will be undertaken later, when discussing the obtained results.

Two-component one-phase systems (miscible blends of amorphous polymers)

In addition to the studies on blends of polyolefins, as well as on multiblock copolymers described above, in which some of the components and/or phases are crystallizable, investigations have also been carried out on blends of non-crystallizable components.

Amorphous films of poly(methylmethacrylate)/poly(vinylidenefluoride) (PMMA/PVDF) blends have been prepared by initial precipitation from a solvent and rapid solidification at ∼15°C from the molten state [34]. Moreover, these two constituents are considered as miscible [35, 36]. The PMMA/PVDF compositions studied have 25/75, 45/55, 50/50, 55/45, 60/40 and 75/25 ratios (by weight). The presence of a single X-ray halo as well as a single T g value for all the blends, in the above composition range, favored the view that these materials are composed of homogeneous mixtures at molecular level [34]. For this reason, the authors assumed [34] that the parallel decrease of the microhardness obeys a simple expression for the overall microhardness of the blend, H:

$$H=H^{\rm PMMA}(1- \varphi^{\rm PVDF})$$
(16)

where φPVDF is the mass fraction of PVDF. Since the T g value of PVDF is known to be − 40 °C [35], the authors have applied the common approach to that time, assuming that the PVDF molecules do not offer any mechanical contribution to the yield behavior of the blend [34].

If one follows this logic and calculates the H values from Eq. (16) taking into account only \({H_{\rm a}^{\rm PMMA}}\) value and PMMA mass fraction, one obtains values being quite different from the experimental ones, as can be concluded from Table 4, where the measured values of T g and of the density for the blends are given, as reported in [34].

Table 4 Composition of PMMA/PVDF blends, their glass transition temperatures, \({T_{\rm g}^{\rm B}}\) , density, \({\rho}\) , the experimentally measured microhardness, values, H exp, calculated by means of Eq. (1) (with H s = 0), H′cal, or calculated with \({H^{\rm s} \ne }\) 0 values, \({H^{\prime \prime}_{\rm cal}}\) and \({H^{\prime \prime \prime}_{\rm cal}}\) , as well as the differences between the calculated and the measured values, \({\Delta H^\prime, \Delta H^{\prime \prime} }\) and \({\Delta H^{\prime \prime \prime}}\), respectively
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If one applies the other approach for accounting the contribution of the soft component, as in the two cases described in the previous sections, i.e. via the T g value, the results look quite differently, as demonstrated below.

Formally, the blend PMMA/PVDF can be considered as a three-phase one, because the reported density, ρ, values (Table 4, [34]) differ from the commonly accepted for \({\rho_{\rm a}}\) one. For the neat PVDF a density value of 1,740 kg m−3 is reported [34] while in the literature a value for the fully amorphous PVDF (ρa) of 1,680 kg m−3 can be found [14, 36]. This difference in ρa suggests that some “ordering” in the system may have taken place during the sample preparation. Using the value of ρc = 1,930 kg m−3 for the completely crystalline PVDF [36] (which corresponds to the α-, also called type I-modification, i.e. crystallization from melt), one could estimate an apparent “degree of crystallinity” \({w_{\rm c}= \rho_{\rm c}/\rho}[(\rho-{\rho_{\rm a})/(\rho_{\rm c}-\rho_{\rm a})]}\) for the sample with ρ = 1,740 kg m−3, leading to w c = 0.25.

Based on the fact that all the samples (Table 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 4) as two-phase systems, In case of such blends the microhardness can be calculated by means of the additivity law as:

$$H= \varphi[w_{\rm c}H_{\rm c}^{\rm PVDF}+(1- w_{\rm c})H_{\rm a}^{\rm PDVF}]+(1- \varphi)H_{\rm a}^{\rm PMMA}$$
(17)

where φ is the mass fraction of PVDF in the blend, and H c and H a—the microhardness values for the completely crystalline and fully amorphous samples, respectively.

By combination of Eqs. (17) and (8) one obtains [37]:

$$H=\varphi[w_{\rm c}H_{\rm c}^{\rm PVDF}+(1- w_{\rm c})(1.97T_{\rm g}^{\rm PVDF}-571)] +(1- \varphi) (1.97T_{\rm g}^{\rm PMMA}-571)$$
(18)

\({H_{\rm c}^{\rm PVDF}}\) can be easily evaluated using the extrapolated microhardness value for the neat PVDF (with wc = 25%) of H = 0 MPa and the value of \({H_{\rm c}^{\rm PVDF}=336}\)  MPa is obtained. Using this value of \({H_{\rm c}^{\rm PVDF}}\) and letting \({T_{\rm g}^{\rm PVDF}}=233\) K and \({T_{\rm g}^{\rm PMMA}} =393\) K (Table 4), one can calculate by means of Eq. (18) the microhardness of the studied blends. The values obtained are summarized in Table 4 as \({H^{\prime \prime}_{\rm cal}}\) from Eq. (18).

After taking into account the fraction of the densified PVDF one observes now a better agreement between the experimental and calculated results.

The suggested treatment (Eq. (17)) considers formally the PMMA/PVDF blends as a three-phase system—two amorphous and one “crystalline” (PVDF). Now, let try to calculate the H values taking into account the real situation, i.e. (i) the two components are completely miscible in the amorphous state, and (ii) this amorphous phase displays only one T g, that of the blend, \({T_{\rm g}^{\rm B}}\) . Then H will be [37]:

$$H=\varphi_{\rm c} H_{\rm c}^{\rm PVDF}+(1- \varphi_{\rm c})(1.97T_{\rm g}^{\rm B}-571)$$
(19)

where φc is the mass fraction of the “crystalline” PVDF phase and (1 − φc) of the amorphous two-component blend.

Using the values for \({T_{\rm g}^{\rm B}}\) as derived from Gordon and Taylor equation [38] one obtains by means of Eq. (19), the values of H listed also in Table 4 on its last column as \({H^{\prime \prime \prime}_{\rm cal}}\). A relatively good agreement with the reported [34] H exp values can be observed (Table 4). In specific cases the agreement is even better than in the previous three-phase model treatment (according to Eq. (17)).

The most serious exception in this respect is the blend 75/25, for which much better value (see the value in parentheses, Table 4) is obtained if one uses the measured \({T_{\rm g}^{\rm B}=359}\)  K (Table 4) instead of the calculated by means of Gordon and Taylor equation value. This difference can be explained by a possible higher “densification” of this particular sample, which results in a T g- increase as reported for many polymers [39, 40].

Two-component two-phase amorphous systems containing a soft phase

In recent publications on blends of novel copolymers of polystyrene (PS) with polybutadiene (PB), with well-defined linear block- or star block architecture, detailed morphological and mechanical investigations have been performed [41–43]. Studying the microhardness behavior of these copolymer blends, the authors, in analogy with previous cases discussed above, have drawn the following conclusion regarding the contribution of the soft phase to the overal microhardness: “since the T g of the phase in the present case is always well below the test temperature (i.e., the room temperature, 23 °C), which may be regarded as being at liquid-like state, it does not affect the measured H values. Therefore, there is no correlation between the soft phase glass transition temperature and the microhardness of the polymer blends discussed in the present study” [41].

At the same time, for the blends of the star block and a linear triblock copolymers, both consisting of PS and PB, has been found that the experimental hardness of the blends show much lower H values than those predicted from the additivity law (again assuming H s = 0), as shown in Fig. 3. This finding is similar to the results obtained for microphase separated blends styrene/butadiene block copolymers [42].

Fig. 3
figure 3

Microhardness of the blend of linear and star-like copolymers of PS and PB, H, as a function of the total PS content (assuming volume fraction ≈ weight fraction), φPB; dashed stright line represents the additivity law (Eq. (1)) [41]

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To what extent the cited assumption of the authors [41] regarding the lack of influence of the liquid-like phase on the H values is correct and if the observed deviation from the additivity law is not due to the neglecting of this influence?

For the studied two-component system [41] consisting of two copolymers (linear and star-like) based on the same two monomers (styrene and butadiene) the additivity law (Eq. (1)) can be presented in the following way, accounting also for the contribution of the soft phases to the overall microhardness, H:

$$H=\varphi_{\rm block}[w_{\rm PS}H^{\rm PS}+(1- w_{\rm PS})H^{\rm PB}]+(1-\varphi_{\rm block})[w^\prime_{\rm PS}H^{\rm PS}+(1 - w^\prime_{\rm PS}) H^{\rm PB}]$$
(20)

where H PS and H PB are the microhardness of the respective homopolymers, w PS, \({w^\prime_{\rm PS}}\) and (1 − w PS), (\({1- w^\prime _{\rm PS}}\)), are mass fractions of PS and PB in each copolymer, respectively, and φblock and (1 −φblock) are the mass fraction of the copolymers in the blend. Taking into account the important fact that the two copolymers (as well as their blocks) are completely amorphous, one can express their microhardness by means of Eq. (8) and obtain for the overall H of the blend the following expression:

$$\begin{aligned}H=&\varphi_{\rm block}[w_{\rm PS}(1.97T_{\rm g}^{\rm PS}-571)+(1-w_{\rm PS})(1.97T_{\rm g}^{\rm PB} - 571)] \\&+(1- \varphi_{\rm block})[w^\prime_{\rm PS}(1.97T_{\rm g}^{\rm PS}-571)+(1 - w^\prime_{\rm PS})(1.97T_{\rm g}^{\rm PB}-571)]\end{aligned}$$
(21)

Equation (21) reflects analytically the contribution to the overall microhardness H of each constituent of the blends (copolymers and their blocks). At the same time, one should not forget a very important fact—experimentally are observed only two glass transition temperatures (that of PS and that of PB), regardless of the fact that blocks of them are incorporated in the two type of copolymers [41]. For this reason, it seems justified to use for the calculation of the overal microhardness of the copolymer blend H (assuming again \({H^{\rm s}\ne 0}\)) a more simple equation accounting only for the total mass fraction in the blend of the two species, PS and PB, and their T g values:

$$H=\varphi_{\rm PS}(1.97T_{\rm g}^{\rm PS}-571)+(1-\varphi_{\rm PS})(1.97T_{\rm g}^{\rm PB}-571)$$
(22)

where φPS and (1 − φPS) are the total mass fractions in the blends of PS and PB, respectively.

Using the reported data [41] about the total mass fractions of PS and PB in the blends (φPS and (1 −φPS), respectively), and the values of T g for both type of phases (as summarized in Table 5), one can calculate the respective H values, \({H^{\prime \prime}_{\rm cal}}\) , of the studied blends by means of Eq. (22), and compare them with the experimentally measured ones, H exp, as well as with the calculated values using Eq. (1) and assuming H s = 0, \({H^\prime_{\rm cal}}\), (given also in Table 5).

Table 5 Content of the linear block copolymer, LN4, in the copolymer blends, total polybutadiene content, φPB, (assuming volume fraction ∼ weight fraction), glass transition temperatures of PS, \({T_{\rm g}^{\rm PS}}\) , and of PB, \({T_{\rm g}^{\rm PB}}\) , phases, measured microhardness of the blend, H exp, calculated microhardness (according to Eq. (1), with H s = 0), \({H^\prime_{\rm cal}}\) , and according to Equation 8 (with \({H^{\rm s}\ne 0}\)), \({H^{\prime \prime}_{\rm cal}}\) , as well as the differences between the measured and the calculated values, \({\Delta H'}\) and \({\Delta H"}\), respectively
Full size table

One can see that the difference between the measured H values and the calculated ones, however neglecting the contribution of the soft phase (H s = 0, i.e. using a relationship similar to Eq. (16)), is quite large (H exp values are between 2 and 5 time smaller that those of \({H^\prime_{\rm cal}}\)). Even so, when accounting for the contribution of the soft phase (\({H^{\rm s}\ne 0}\) , i.e. by means of Eq. (22)), the two types of values are much closer to each other (Table 5).

What is more, since the system under investigation is completely amorphous, it is possible to predict its microhardness for various compositions having only the T g and weight fraction of each phase, as described in Eq. (21). The good agreement between calculated and the measured H values demonstrate again the validity of the analytical relationship between H and T g (Eq. (8)).

Quite similar is the situation regarding the microhardness behavior of binary blend comprising the polystyrene homopolymer (hPS) and a star block copolymer of styrene with butadiene over a wide composition range. As in the case of separated block copolymers and binary block copolymer blend, as described above [41, 42], a clear deviation of the microhardness behavior from the additivity law has been also observed [43].

Using an equation similar to the last one (Eq. (22)), however, accounting for the fact that one of components of the blends is a homopolymer and applying the respective values for T g, and mass fractions ([43], Table 6), the H values of the blends are calculated assuming that \({H^{\rm s} \ne 0}\). The results are presented in Table 6.

Table 6 Content of the homo PS in the blends, hPS, total polystyrene content in the blends, φPS, (assuming volume fraction ∼ weight fraction), measured microhardness of the blends, H exp, calculated microhardness (according to Eq. (1), with H s = 0), \({H^\prime_{\rm cal}}\), and according to Equation 8 (with \({H^{\rm s}\ne 0}\), \({H^{\prime \prime}_{\rm cal}}\), as well as the differences between the measured and the calculated values, \({\Delta H^\prime}\) and \({\Delta H^{\prime \prime}}\), respectively
Full size table

Again a much better agreement between H exp and \({H^{\prime \prime}_{\rm cal}}\) values is found if one accounts for the contribution of the soft phase (\({H^{\rm s} \ne 0}\)), in contrast to the opposite case (H s = 0), (Table 6), however, the differences \({\Delta H^{\prime \prime}}\) are still quite large, as compared with the case of PEE (Tables 2 and 3).

One-component two-phase systems (semicrystalline polymers with T g below room temperature)

As demonstrated above, there is hardly any doubt regarding existence of a linear relationship between T g and the microhardness H of the amorphous polymers characterized by dominating single, mostly C–C bonds in the main chain. This empirically derived analytical relationship (Eq. (8)) makes it possible to account quantitatively for the contribution of the soft component and/or phase to the overall microhardness of multicomponent and/or multiphase systems as demonstrated above.

The same relationship offers another challenging opportunity—to evaluate the T g of amorphous phase(s), which are not commonly accessible, as for example, the wholly amorphous PE. Basically, this can be done starting again from the additivity law, and more specifically, by the fact that the overall microhardness depends linearly on those of its constituent components and/or phases and their respective weight fractions. For a semicrystalline polymer one can extrapolate the dependence H c versus degree of crystallinity, w c, (or density) to w c = 0 (or to the ρ a value) and thus evaluate H a for this polymer. Further, exploring the relationship between H and T g (Eq. (8)) it is possible to get an idea about T g of an inaccessible practically amorphous phase. This approach will be illustrated below using as example PE.

Polyethylene is still nowadays one of the most common and most studied polymers. However, there is not yet full consensus among researchers about such a basic property as the value of its glass transition temperature T g. Values as different as around −25 and −120 °C are reported [14]. These discrepancies can be found also in more recent publications: T g = −35 °C [44], and T g = −125 °C [45]. The lack of agreement is related to the fact that PE is not commonly accessible in amorphous state (below its melting temperature) due to its extremely high crystallization rate originating from the perfect chain structure. Even the preparation of samples with different degrees of crystallinity is not a routine task. The frequently used approach by varying the crystallization temperature and/or crystallization time is not applicable as for many other polymers. Better results can be obtained by using PE samples with different degree of branching. By introducing various amounts of chain defects in the main chain, it is possible to control the degree of crystallinity. In this way, even at constant crystallization conditions (temperature and time), one is capable to prepare a series of samples with a systematic variation in the structural parameters such as degree of crystallinity, crystal size, long spacing, density, paracrystalline lattice distortions, melting temperature, etc. [46–48]. Following a given property of such a series and extrapolating to the density of completely amorphous sample, one can find support in favor of one of the two rather different values of T g for PE.

It has been found that H increases linearly with the rise of crystallinity, as reported for many polymers [1, 9]. From the straight line of the plot of H versus density, ρ, for differently branched PE samples, the H value for the completely amorphous PE (ρ = 855 kg m−3 [14]) has been evaluated [49]. The obtained H value has been further used for the evaluation of T g of PE by means of the linear relationship between H and T g for completely amorphous polymers (Eq. (8)), and a value of T g = −23 °C has been obtained [50].

Taking into account the findings of Geil et al. [15–19] based on the direct study of wholly amorphous linear PE, that the T g of PE is −80 °C from one hand, and from the other, that branched PE is supposed to have a much higher T g value (corresponding to the β-relaxation peak), one can consider the extrapolated T g value of −23 °C from branched PE samples as the T g of completely amorphous branched PE [50].

Noteworthy in this respect is also the report of Perena et al. [20] who studied microhardness using dynamic mechanical (DMTA) measurements at low temperatures (between −60 and 25 °C) with five commercial samples PE, two of them of HD and another three of low density (LD). The experimental data for H show clear transition around −30 °C (for LD samples) and around −10 °C (for HD) samples). The data from DMTA show this transition only for the LD samples in agreement with the observation [21] that β-relaxation is clearly detected by DMTA only in branched PE and has been not detected at all in linear PE of medium molecular weight. Having in mind the fact that the PE used for the application of Eq. (8) is also branched one, it should be emphasized that there is very good agreement between the experimentally observed transition temperatures (between −20 and −30 °C) [20] and those predicted by means of Eq. (8) (−23 and −25 °C) [24].

The same approach was very recently applied to PE samples characterized by chain-folded and chain-extended crystals [25]. Considering the fact that in these samples the amorphous phase amounts low percentage (in the chain-extended samples around 5  wt%) and because of the highly strained chains in the amorphous areas, one can expect relatively high values for T g of these amorphous phases. The calculations lead to T g = −1 °C (for the chain-folded samples) and T g = 10 °C (for the chain-extended samples) [25].

Summarizing this section, we have to note that thanks to the empirical linear relationship between H and T g in a rather broad range of T g (−50 °C up to 250 °C), which covers most commonly used polymers of the polyolefin type and also polyesters and polyamides [13], it is possible to calculate not only the microhardness value of many amorphous polymer provided its T g is known (H = 1.97T g−571) or to account for the contribution of soft components and/or phases to the microhardness of the entire system, but also, to evaluate the T g values of practically inaccessible amorphous phases in semicrystalline polymers.

Discussion

Importance of the ratio hard/soft components (or phases)

Analyzing the results summarized in Table 1 one can conclude that the agreement between the H exp values and the calculated H cal ones (assuming \({H_{\rm a}\ne 0}\)) is not as good as expected for all the samples. Quite close to each other are the values for the samples characterized by lower w c values (below 80%) and higher amount of PP in the blends. This could mean that for the cases where the crystalline phase (or component) dominates (80% or more) the overall microhardness is determined by the “diluting effect” of the amorphous phase, i.e. the amount of the amorphous phase is not enough in order to be considered as a matrix in which the crystallites are immersed. As a matter of fact, in such cases the matrix is represented by the crystalline phase in which is dispersed the much smaller in amount (30% or less) amorphous phase. It is quite obvious that in such a situation one cannot apply the concept of the “floating effect” for explanation of the mechanical behavior of the complex system and one has to accept the plastic deformation mechanism of the solid component and/or phase as dominating.

The above considerations are supported by the results of the thermoplastic elastomers (PEE) with various compositions (hard/soft segments ratio). For example, in Table 3 data for other two PEEs (samples 7 and 8) with not such a good agreement between H exp and \({H^{\prime \prime}_{\rm cal}}\) (according to Eq. (11)) values are presented. A possible explanation for the different behavior of these two PEE samples is their composition. Samples 1–6 are characterized by hard/soft segments ratios of roughly 60/40 while in the samples 7 and 8 this ratio is 75/25. The fact that in the second case the hard PBT segments dominate, suggests another response mechanism to the mechanical field—the PBT hard segments are no longer “floating” in the liquid-like matrix of soft domains.

Further support in favor of the importance of the hard/soft components ratio can be found in the blend hPS/star block copolymers of PS and PB (Table 6). A more precise inspection of Table 6 shows that the total amount hard, glassy homopolymer PS in the blends varies between 75 and 95 wt%. In other words, we are coming to the same conclusion as for the above two systems (PE/iPP and PEE), where for the cases when the hard, (crystalline) phase dominates (for example, amounting 80% and more) the concept of “floating effect” cannot be applied as successfully as for the cases when the hard component or phase does not dominate.

Crystalline or amorphous solids

In order to illustrate to what extent the presence or absence of order (crystalline) in the hard component and/or phase is important, let us come back first to the system polyester-polyamide copolymers for which serious deviations from the additivity law (assuming H s = 0) have been reported [32]. Before suggesting some possible reasons for the failure of the calculations using Eq. (15) (assuming \({H^{\rm s} \ne 0}\)), one has to stress the fact that the paper under consideration [32] is distinguished by a couple of peculiarities being rather important for this discussion.

In this study [32] a relatively large number of interesting samples, differing in their hard/soft segments ratio as well as in the molecular weight of the segments, is investigated. The measured H exp values differ drastically from the \({H^\prime_{\rm cal}}\) values (assuming H s = 0), the difference being up to ten times (while 9 from totally 12 samples have H exp values between 10 and 30 MPa, the \({H^\prime_{\rm cal}}\) for all the samples vary between 70 and 110 MPa). What is more striking, is the fact that the degree of crystallinity of PBT domains w c (determined by wide-angle X-ray scattering, WAXS) for all the samples studied varies between 10 and 30% (for 9 of the samples—between 10 and 20%). Nevertheless, the drastic differences between H exp and \({H^\prime_{\rm cal}}\) values (assuming H s = 0) are explained exclusively on the basis of the changes in the crystal’s characteristics as crystal sizes, crystal surface free energy, the energy needed for plastic deformation of crystals (usually approximated to the enthalpy of fusion of crystals [1, 32]).

At this stage important questions could arize: how it would be possible to explain (or even to predict) the mechanical behavior of a complex system consisting of 80 wt% (and more) soft, liquid-like (T g between −40 and −70 °C) matrix in which are dispersed not more than 20 wt% crystallites only accounting for the properties of crystallites? Does it look reasonable to neglect the contribution of the prevailing (up to 5 time) soft phase in the formation of the mechanical response of the system to the external load?

The samples studied [32] represent an excellent example for the case when the solid particles (crystallites in this case) are “floating” in the dominating soft matrix. As noted above, our attempt to recalculate the \({H^{\prime \prime}_{\rm cal}}\) values taking into account the “floating effect” (\({H^{\rm s} \ne 0}\)) failed. The data obtained are inconsistent, possibly because, for the fraction of the soft segments the molar concentration has been used [32].

It should be noted that there are not obvious reasons, except the misleading fact that the soft liquid-like substances do not produce Vickers indentation impressions, to accept that the soft component and/or phase does not contribute to the overall microhardness, i.e. H s = 0. Only taking into account the fact that the soft component changes dramatically the deformation mechanism of the complex system, one is able to avoid more “sophisticated” explanations for the observed discrepancies between the H exp and H cal values (assuming H s = 0). For example, in the case of the blends of PS and PB copolymers, as well as with various partners, these differences are explained by different origins: the molecular architecture which modifies the effective phase ratio, the presence of a microphase separated morphology and some specific effects such as yielding of thin layers, etc. [43] or by the assumption that the volume fraction of styrene and butadiene phases in the block copolymer blends does not reflect the effective hard/soft phase volume ratio owing to the modified copolymer architecture and microphase separated morphologies [41].

Quite similar explanations of the observed deviations from the additivity law dealing with semicrystalline polymers and their blends involving peculiarity in the crystalline characteristics (crystal sizes, crystal surface free energy, etc.) are offered, as mentioned in the previous sections. The most serious drawback of such explanations, even if they indicate on some of the possible reasons for the observed deviations from the additivity law (Eq. (1)), is that they cannot account quantitatively for the observed differences as, for example, Eq. (8) does.

Let consider now the blend of homo PS with star block copolymer of PS and PB, where the agreement between the values of H exp and \({H^{\prime \prime}_{\rm cal}}\) (even assuming \({H^{\rm s}\ne 0}\)) was not as good as expected (Table 6).

It seems important to stress here on a peculiarity of this system, namely, its similarity with the blend PE/iPP, with respect to their microhardness behavior. Regardless of the seemingly important facts that the blends of hPS with copolymers are completely amorphous, non-crystallizable ones, and that the blends of iPP/PE consist of semicrystalline homopolymers, to both of them one cannot apply successfully the concept of the “floating effect” for explanation of the deviations from the additivity law (Eq. (1)) with the assumption H s = 0. The concept of the “diluting effect”, i.e. the domination of the plastic deformation mechanism of the solid particles, seems to be more appropriate. The reason for this is, as stated in the preceding section, the fact that the dominating (in amount) solid particles (crystalline or amorphous ones) form the matrix, in which is dispersed the minor in amount soft component and/or phase.

From the comparison of these two different (with respect of crystallinity) systems, one can conclude that for the explanation (and overcoming) the frequently observed deviations from the additivity law (Eq. (1) assuming H s = 0), the crystal characteristics (crystal sizes, crystal free surface energy, etc.) are not of basic importance. What counts in this case is the mechanism of deformation of the solid particles under the indenter, i.e. if one deals only with their plastic deformation under the indenter (the case of “diluting effect”) or with the same mechanism, however, paralleled by a displacement of the solid particles in the soft matrix (the case of “floating effect”). The domination of one or the other deformation mechanism depends exclusively on the ratio solid/soft (liquid-like) components and/or phases and not by the fact if the solids particles are amorphous or crystalline.

Finally, the fact that the analytical relationship between T g and H (Eq. (8)) helps to solve “the problem” regarding the deviations from the additivity law (Eq. (1)) means that the very basic starting assumption regarding the deformation mechanism, i.e. the “ floating effect” of the solid particles in the dominating soft component and/or phase, is quite reasonable. This fact should be always taken into account when the mechanical behavior of such systems is considered.

Copolymers versus polymer blends

In the present section are analyzed various amorphous systems—homopolymers, their miscible blends, copolymers and their blends. This variety makes it possible to follow the effect of presence or lack of chemical linkages between the constituent monomers, i.e. if one deals with a blend of two homopolymers or with a copolymer of the same monomers. Interesting conclusions in this respect can be drawn from the miscible blend PMMA/PVDF (Table 4). The results obtained allows one to conclude that the amorphous blends of miscible partners can be treated in the same way as the amorphous neat homo- and copolymers regarding the relationship between their glass transition temperature and microhardness. What is more, it is not necessary to measure not only their microhardness but also their T g values for any blend composition because these values can be evaluated by means of the Gordon and Taylor equation [38].

Particularly striking in the present results (Table 4) is the observation that the calculated H data for the discussed blends do not significantly depend on the model applied, i.e. if the amorphous blend is considered as a two-phase one (or even as a three-phase one) or as one-phase two-component system. The only parameter that counts is the mass fraction of each component and/or phase and the respective T g value. It is noteworthy that a similar behavior has been observed between blends of homopolymers and copolymers prepared from the same monomers [1, 50].

Important support of this conclusion can be found in the microhardness behavior of the glassy block copolymers of PS and PB with various architectures, their blends, as well as their blends with amorphous homopolymers (hPS).

As shown, Eq. (21) considers the blend of star-like and linear type block copolymers of PS and PB as comprising four components, the two types of blocks in the two different with respect of the molecular architecture block copolymers. The use of this equation presumes the knowledge of four T g values (for each of the four blocks). Experimentally have been detected only two T g, that of PS and that of PB [41]. For this reason, Eq. (21) was modified into Eq. (22) that accounted only for two glassy phases, PS and PB. The data obtained by means of Eq. (22) are satisfactory, particularly for the samples with dominating soft phase (PB), (Table 5).

Quite similar is the situation with the amorphous blends of hPS and star block copolymers of PS and PB (Table 6). This two-component system consists of homo PS and a block copolymer of PS and PB, however, again only two T g have been experimentally revealed [42].

The cases described lead to the important conclusion that the microhardness of a complex multicomponent and/or multiphase system depends only on the number of the actually observed components and/or phases (on their individual microhardness values and their respective mass fraction), regardless of whether these components and/or phases consist of homopolymer(s), their blends or parts (blocks) of copolymers. It seems that for amorphous polymer systems the microhardness depends exclusively on the chemical composition and structure of a specific monomer, but not on the type of chemical linkages (homo- or copolymers) in agreement with our former observations [1, 50].

New data on the relationship between H and T g of amorphous polymers

Since the publication of the equation relating H and T g of amorphous polymers in 1999 [13], new data about the microhardness values were published. This include results on PS materials with various molecular architectures (highly branched) displaying some higher T g values as compared to the linear PS [51].

Samples of glassy PS with different amounts of long branches have been investigated (PS-165 is a linear PS, and the amount of branches for samples PS-174, PS-177, and PS-179 increases with the number code) [51]. The molecular weight, polydispersity, glass transition temperature, and microhardness data of these materials are listed in Table 7.

Table 7 Molecular weight, M w, polydispersity, Mw/M n, glass transition temperature, T g, experimentally measured microhardness, H exp, calculated microhardness (according to Eq. (8)), H cal, and the difference between the calculated and the measured values, \({\Delta H}\), of various types of PS
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A fair agreement between the measured, H exp, values [51] and those calculated by means of Eq. (8) ones for H cal can be observed (Table 7). This fact confirms the validity of Eq. (8) for the group of amorphous polymers (characterized by dominating single, mostly C–C bonds). In addition, one can again conclude from the results in Table 7 that the molecular architecture does not affect the validity of Eq. (8).

With regard to the future development of the present study it seems important to mention that Eq. (8) was recently modified in such a way that it accounts also for the temperature dependence of H for amorphous polymers [52]:

$$H^{\rm T}= 1.97T_{\rm g}-0.6T- 395(\hbox{MPa}),(T_{\rm g}\hbox{and}T \hbox{in K})$$
(23)

where H T is the microhardness value at the test temperature T.

In other words, the microhardness of the same group of amorphous polymers, covered by Eq. (8), can be calculated for any temperature T below T g if T g is known. The experimentally derived Eq. (23) based on results for 4 amorphous polymers, (PS, PMMA, PEN and PET) was recently verified by the reported data on the temperature dependence of H for PMMA synthesized by radiation polymerisation [53]. In Table 8 the experimentally measured \({H^{\rm T}_{\rm exp}}\) values at various temperatures, (ranging between 15 and 78 °C) are compared with the calculated ones, \({H^{\rm T}_{\rm cal}}\), applying Eq. (23).

Table 8 Measured microhardness values, \({H^{\rm T}_{\rm exp}}\), and the calculated ones, \({H^{\rm T}_{\rm cal}}\), according to Eq. (23), as well as the difference, \({\Delta H^{\rm T}=H^{\rm T}_{\rm cal}-H^{\rm T}_{\rm exp}}\), for various test temperatures, T, of glassy PMMA
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A quite good agreement between the two types of values can be found. What is striking in this case, is the observation that the difference between \({H^{\rm T}_{\rm cal}}\) and H T exp tends to zero with increasing temperature of measurements, T. This could mean that T g, being indicative for the viscosity of the amorphous material, is getting more sensitive with this respect when the test temperature T approaches the softening point, T g.

In the same paper [53] careful measurements of H T have been performed in the same temperature interval for the blend PMMA/natural rubber (NR), the latter being in amount of up to 5 wt%. The blend films have been prepared from a common solvent [53]. In the next Table 9 are presented the data on \({H^{\rm T}_{\rm exp}}\) (as reported in [53]), and the 23 \({H^{\rm T}_{\rm cal}}\) values calculated by means of Equation.

Table 9 Measured microhardness values, \({H^{\rm T}_{\rm exp}}\) and the calculated ones, H T cal according to Eq. (23), as well as the difference \({\Delta H^{\rm T}=H^{\rm T}_{\rm cal }-H^{\rm T}_{\rm exp}}\) for various test temperatures, T, of blend of glassy PMMA with natural rubber (up to 5 wt%). The measured by DSC T g value of PMMA is reported to be 354 K [54]
Full size table

Surprizingly, again a fair well agreement between H T cal and H T exp values can be seen, particularly with the progress of the test temperature (for two-thirds of a total of 12 measurements, the difference \({\Delta H^{\rm T}=H^{\rm T}_{\rm cal}- H^{\rm T}_{\rm exp}}\) amounts to between 2 and 6 MPa), which, occasionally, correspond to the same percentage of deviations (Table 9).

Taking into account the results presented in the last two Tables 8 and 9, it sounds challenging to study to what extent Eq. (23) can be applied to complex polymer systems comprising dominating soft component and/or phase, i.e. if it would be possible to predict the overall microhardness of such systems for any temperature below T g and/or T m of the solid component and/or phase without experimental measurements. Such an expectation seems feasible, as long as the T g value of the soft component and/or phase account for the changes in their viscosity according to Eq. (23), as demonstrated by the data summarized in Tables 9.

As a matter of fact, the verification of this expectation is the target of the next step of this study.

Modified additivity law for systems containing soft component and/or phase

Coming back to the main goal of the present study, the application of the additivity law (Eq. (1)) to complex polymer systems containing soft component and/or phase, we would like to suggest its modification by incorporating the relationship between H and T g (Eq. (8)). The advantage of this modification consists in the possibility to use it for accounting for the contribution of any amorphous phase and/or component to the overall microhardness of the system, provided the T g value of this phase and/or component is known, regardless of their value. Hence, for systems, which contain more than one crystalline and/or amorphous phases with crystalline microhardness, glass transition temperatures and mass fractions H ci , T gi and φ i , respectively, the additivity law can be presented in the following way:

$$H=\Sigma \varphi_{i} w_{ci}H_{ci}+\Sigma\varphi_{i}(1-w_{ci})(1.97T_{gi}-571)$$
(24)

For systems where the solid (hard) components and/or phases are not crystallizable materials, Eq. (24) can be simplified as:

$$H=\Sigma \varphi_{i}(1.97T_{gi}-571)$$
(25)

In these two last forms, contrasting to the traditional one (Eq. (1)), the additivity law is applicable to multicomponent or multiphase systems comprising soft components or phases displaying a more complex deformation mechanism than the case in which all the components and/or phases have T m and T g above room temperature.

Conclusions

Using the reported data on the experimentally derived values of the glass transition temperature, T g, degree of crystallinity, w c, 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 at room temperature component and/or phase, 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.

In contrast to this commonly used approach, it is suggested in the present study that the soft component and/or phase can dramatically change the deformation mechanism under the indentor, and thus to contributor significantly to the formation of the overall H value. It is demonstrated that this contribution can be quantitatively accounted for via the empirically derived relationship between H and T g. The above disclosed results allows one to derive the following conclusions:

  1. 1.

    The additivity law can be successfully applied also to complex polymer systems comprising soft component(s) and/or phase(s) if one accounts for their contribution to the overall microhardness of the system via their T g and applying the linear relationship between H and T g derived from solid amorphous polymers [13]. In this way one takes into account the fact that the deformation mechanism under the indenter is rather different compared to the systems distinguished by T g and T m values being above the ambient temperature.

  2. 2.

    This approach allows one to overcome to some extent the main disadvantage of the indentation technique for measuring H, the necessity to obtain observable, well defined indentation impressions on the sample surface.

  3. 3.

    The microhardness behavior of blends of completely amorphous homo- and copolymers supports our previous conclusion [1] that the contribution of a component and/or phase to the overall microhardness depends mostly on the chemical nature of the respective monomers. Whether the monomers are chemically linked giving a homopolymer, a copolymer (even with a complex molecular architecture) or one deals with blends of them does not play any significant role.

  4. 4.

    In contrast to the “floating effect” concept, the application of the thermodynamic approach (accounting for the crystal sizes, crystal surface free energy, etc.) for the explanation of the deviations from the additivity law is possible only for systems comprising crystalline component and/or phase; what is more, even for the last systems, the calculation of H requires the knowledge of parameters which are not easily accessible.

  5. 5.

    A modified additivity law is suggested which contains a term accounting for the contribution of the soft component and/or phase to the overall microhardness via the relationship between H and T g; its application results in much smaller differences between the measured and calculated H values.