GLASS TRANSITIONS IN FILLED POLYMERS

4.2.1 INFLUENCE OF FILLER ON THE GLASS TRANSITION OF FILLED POLYMERS

It is known that the glass transition temperature, T_g, corresponds to the temperature at which mobility of segments of the polymer begins to occur. Con-sequently, the adsorption and adhesion are to be reflected in Tg.

There is now a large amount of experimental data on the change in Tgof polymer under the influence of the solid surface. The comprehensive reviews were given earlier.^13,14Therefore, there is no need to analyze all data and we con-fine ourselves only to describing the general picture. The data on the change in T_g of filled polymers have been obtained by various methods, including dilatometry, measurement of heat capacities, from dielectric and mechanical

re-Figure 4.4. Dependence of the changes in Tgfor PMMA vs. the fumed silica content, in measurements of Tgby various meth-ods: 1-calorimetry, 2-dilatometry, 3-dy-namic, 4-dielectric, 5-NMR.

laxation, NMR, RTL, etc. Each method has its own limitations and therefore the results of various methods do not always coincide. However, the general picture remains unchanged, namely, under the solid surface influence, the glass transi-tion of a polymer shifts to the higher temperatures. For example, Figure 4.4 demonstrates the shift in Tgfor PMMA filled with fumed silica, measured by various methods. While the general tendency is the same, the degree of the sur-face influence on glass transition depends on the method used. This effect is con-nected with varying frequency of action on the polymer by different methods.

The effect is greatest when methods of low frequency are used (calorimetry, dilatometry). A great discrepancy was observed between the values of T_g for elastomers.¹⁵However, in the majority of cases with increasing filler content, Tg

grows. The effect depends, at the same filler content, on specific properties of surface and conditions of production. Increase in T_gis determined by the nature of the filler surface and surface energy of a polymer (see Chapter 3). It is impor-tant that experimental data show that changes in glass transition temperature have a macroscopic character, i.e., are typical for the whole volume of filled poly-mer. Indeed, if the effects were to involve only the restrictions in molecular mo-bility in the surface layers, it would be impossible to detect any change in glass temperature.

It is known that processes taking place during glass transition are coopera-tive. Therefore, changes in Tgreflect the restrictions of mobility, not only those macromolecules which have direct contact with the surface, but molecules re-mote from the surface due to the relay-race mechanism and formation of various supermolecular structures near the surface. Aggregates are one example. At the same time, the increase in Tg with filler amount has, as a rule, some limits, which seem to correspond to the transition of all polymer molecules into the sur-face layer (see Chapter 3). Here the question may arise why we cannot distin-guish between glass transitions in the surface layer and in polymer bulk not subjected to the action of the surface. The resolution of two possible maxima is a very rare case (as an example, see Figure 3.9). The reasons for a lack of resolu-tion in most cases are discussed in Chapter 5.

The dependence of Tgon filler content may be expressed on the same basis of the changes in the properties of the polymer phase in the boundary layer in yet a different way.¹⁶Let the volume of polymer, which has been under the

influ-ence of the surface, be vp=VϕSδ, where V is the total volume of the system,ϕis volume fraction of filler, S is specific surface, andδis the thickness of the bound-ary layer. The whole polymer phase is under the influence of the filler when con-ditions are: vp = (1 - ϕ)V or ϕ = 1/(1 + δ). In spite of simplifications, the experimental data are satisfactorily expressed by the relationship:

T_g- T_g,o= ∆T_g,∞[1 - exp(-Bϕ)] [4.26]

where∆T_g∞is the maximum shift in T_gand B is a constant. A simplified compu-tation of the thickness,δ, leads to a value of 35-100Å.

4.2.2 THEORETICAL APPROACH TO GLASS TRANSITION PHENOMENA IN FILLED POLYMERS

The main difficulty in developing the theory of glass transition in filled polymer consists of insufficient understanding of the physical reasons that de-termine the long-range effect of surface on the properties of polymer. There are two glass transitions of surface layers and polymer in bulk, especially when the amount of the filler is low. The glass transitions in filled polymers are thus not only determined by the existence of an interphase but also by other factors. The traditional approach to the glass transition phenomenon consists of application of the concept of the iso-free volume.^17,18This concept postulates that the frac-tion of free volume at Tgis constant for all polymers.

According to definition:

fg= vf/vg = 1 - vo/vg [4.27]

where vf= vg- vois a free volume, voand vgare the values of occupied and real vol-ume of liquid at Tg. Value vo cannot be found experimentally. There are some methods of determination of fg. In accordance with the model proposed by Simha and Boyer,¹⁹the following equations can be used as criteria for constancy of f_g:

fg,1=α_lTg= 0.160 [4.28]

fg,2=∆αT_g= 0.113 [4.29]

where∆α α= l −α αg, l andαg are the coefficients of thermal expansion above and below Tg. According to the “hole” model developed by Hirai-Eiring-Frenkel²⁰ the equilibrium value of a specific volume of liquid vlis given by:

v = v [1+l -1exp (-x)]

∞ σ [4.30]

(see Eq 4.23). Here v_∞is the value of v corresponding to the most dense packing, ε_his the energy of hole formation, vhis molar hole volume,σis model parameter, and P is pressure. Substituting v = v∞ _o in Eq 4.27, we obtain value fg,3.

It was shown¹⁸that, for filled and unfilled PMMA, the straight lines de-scribing the dependence of vgon T_gand v on T_{β β} (where v and T_{β β} are specific volume and temperature of the secondary transition) have an intercept at the point with coordinates v and T∞ ∞. The last values are limiting values of the spe-cific volume of a melt and temperature, at which, due to the high hydrostatic pressure, the segmental mobility is fully suppressed and free volume becomes zero. Using v = v∞ _o for Eq 4.27, one can find fg,4.

In agreement with the Williams-Landell-Ferry (WLF) theory,²⁰ the translational mobility of macromolecular segments fully disappears, due to di-minishing free volume which reaches zero at temperature To< Tg. For this the-ory, the following expression is valid:

fg,5= (Tg- To)/B [4.31]

where B is a parameter of the Vogel-Tamman equation accounting for the tem-perature dependence of the Newtonian viscosity

η= A exp([B/(T_g- T_o)] [4.32]

In Table 4.1, the values of fgare given for various filled systems.¹⁸

The data presented in Table 4.1 demonstrate not only a sharp difference in values of fg, depending on the method of calculation, but also varying magnitude for the same system having different filler concentrations. This fact may be re-lated to changes in vgand voduring filler introduction.^21-23Another reason may

be the appearance of structural defects, on the molecular or submolecular level, during filling.⁷

These results confirm the idea put forward previously^13,17that the concept of the iso-free volume cannot be applied to filled polymers. Therefore, the empir-ical equations cannot be used as a basis to describe properties of filled polymers.

The limitations of the iso-free volume concept may also be related to the fact that value f_gcannot be universal value, even for unfilled polymers, because it de-pends on molecular parameters of chains — specifically, on their flexibility.^24-26

Another approach can still be taken to describe the glass transitions in filled polymers on the basis of a criterion of the constancy of excess entropy at T_g. In accordance with thermodynamic theory of glass transitions,²⁷it is supposed that in polymers with flexible chains, the excess (in crystalline state) entropy of polymer liquid during its cooling monotonously diminishes to zero. The effect is due to the decrease in the relative amount of rotational isomers, having high en-ergy, up to some critical minimum value at a hypothetical temperature of transi-tion of the second order T2. The temperature dependence of the excess entropy is described as:^28,29

∆S = S₁ ∆ _m ∆C dlnT = ∆C dlnT

T T

p

T T

p

m

2

−

∫ ∫

^[4.33]

Table 4.1: Calculated values of fgfor various amounts of filler

System fg Filler amount, wt%

0 1 5 10 20 50

PS+glass

fg,1 0.237 0.211 0.194 - 0.239 0.959

fg,2 0.115 0.093 0.091 - 0.152 0.151

fg,3 0.085 0.041 0.099 - 0.127 0.255

PMMA+glass fg,3 0.102 0.125 0.120 - 0.110 0.138

PMMA+Aerosil fg,4 0.124 - 0.138 0.158 0.192

-where∆S_mis entropy of melting,∆C_p =Cl - Cgheat capacities in the melt and glass, and Tmis a melting point. It is supposed³⁰that the value of excess entropy,

∆S (calculated for single bond), which is frozen at T_g, should have a universal value for all polymers capable of vitrification. Neglecting the temperature de-pendence of∆C_p, in the interval from Tgto T2, and accepting that T2/Tg≤1, from Eq 4.33, one may obtain:

∆S = C ln∆ T

T = const

g p

g 2



 

 [4.34]

The changes in molecular packing due to filling may influence the value of

∆C_ponly through the value of C_lbecause the mobility in glassy state does not de-pend on packing.

In the majority of cases of filled polymer systems (especially filled with fine disperse active fillers such as fumed silica), the values of C_land∆C_pdiminish with an increase in filler concentration.¹⁸Because the decrease in∆C_pis accom-panied, as a rule, by increasing Tg, from Eq 4.34 it follows that values of∆S_g, for filled specimens are comparable to∆S_gof pure polymer only when parameter T₂ remains constant or slightly decreases. Unfortunately, one cannot use the stan-dard method of evaluating T2because of a lack of direct calorimetric measure-ments. This value, however, may be found from indirect data.

It was assumed^31-33that the free volume and excess entropy of a liquid be-comes zero at the same temperature, i.e., T2coincides with Tofrom Eq 4.32. The measurements of the temperature dependence of Newtonian viscosity of melts in the system ethylene glycol-adipate-fumed silica have shown that introduc-tion of 10 wt% of fumed silica leads to decreasing To from 175 to 155 K. In-creasing the ratio of Tg/To, due to decrease of To, at constant Tg, is compensated by the decrease in∆C_p. As a result, the calculated values of∆S_gremain practi-cally the same.³⁴

It was also shown by Miller³⁵that the value T2may be determined by ex-trapolation of a linear part of temperature dependence of (∆S_l)²to∆S_l=0. Values of T2for PS-glass beads system, found from the temperature dependence of the calculated values of the excess entropy, decrease from 285 K for pure PS to 130 K at filler concentration of 50 wt%. Taking into account that for this system, Tgand

∆C_pare constant, then, from Eq 4.34, the conclusion can be drawn that value of

∆S_gincreases due to transition from unfilled to filled material. At the same time, for the system PMMA-glass beads,²⁴constant values of T2were obtained which, together with∆C_p≈const, indicates approximate constancy of∆S_gfor all sam-ples. From this discussion, it follows that the application of the iso-entropy con-cept to describe the glass transitions in filled polymers needs additional development.

Let us analyze another criterion of glass transition, namely, criterion of the viscosity constancy. It is known that with decreasing temperature, the shear viscosity of Newtonian liquid,η, monotonously increases, reaching values of the order of 10^-10Pas in the region of the glassy state. According to the empirical cri-terion proposed by Tamman,³⁶the transition of a liquid into solid glassy state proceeds at the temperature T_g, at which the viscosity reaches its universal magnitude ofη=10¹²Pas. For unfilled polymers, this criterion meets experimen-tal data. However, there is no data till now on viscosity of filled polymers near glass transition temperature, and therefore to investigate the applicability of the Tamman criterion to filled systems one has to use indirect data. In some works, the systematic increase of the viscosity of melts of filled polymers at tem-peratures T >> Tgwas found.^37-39The analysis of such data allows for the follow-ing conclusion to be drawn: If the conditionη= const = 10¹²Pas is valid for filled polymers, then introduction of fillers should lead to the sharp increase in Tg. The experimental data confirm this statement. However, increasing amount of filler also leads to the change of flow from Newtonian behavior, due to elimination of structures formed by the filler particles in viscous media (see Chapter 5). There-fore, the application of the Tamman criterion needs the following conditions to be met:

• measurements should be done in the stress range which corresponds to the Newtonian behavior of a filled melt

• temperature range of viscosity should be rather broad to find reliable data from Eq 4.32.

Experimental data collected¹⁸qualitatively meet the concept of the viscos-ity constancy at Tgfor filled polymers.

For theoretical description of the behavior of filled polymer it is important to analyze the processes of structural relaxations near Tg.Some experimental data concerning the volume relaxation in filled polymers were presented.¹³ These data show that the average relaxation time increases with increasing amount of filler at constant temperature. This is very significant for selecting conditions of processing of filled polymers, since optimum properties of material depend on the schedule of its processing (temperature, time, pressure), bound to differ from that for unfilled polymer. However, in order to understand the mech-anism of the processes taking place at the polymer-filler interface it is desirable to compare the relaxation times, not at identical temperatures, but at tempera-tures at equal distance from the glass transition temperature (considering that it increases with filling). In such comparison, the relaxation times for filled sys-tems are shorter.

Two effects should be taken into account to explain these results. The rise in Tgas a result of restriction of molecular mobility leads to the increase in the relaxation time, whereas diminishing packing density, accompanied by an in-crease in free volume, causes the shortening of the relaxation times. The volume relaxation is described by the relationship:

V = (v - v_∞)/(v_o- v_∞) = exp[-(t - t_o)τ] [4.35]

where v is specific volume at time t, v∞is equilibrium volume, t is current time,τ is average relaxation time. For filled polymers, experimental dependence of lnV on t is usually nonlinear. It is connected with the existence of the relaxation spectrum. The latter may be formally taken into account using the empirical re-lation:⁴⁰

V = exp[-(t - t_o)/τ]^β [4.36]

where 0 <β< 1 is a parameter characterizing the width of the relaxation spec-trum (atβ=1 we arrive at Eq 4.35).The relaxation spectra of filled polymers are discussed in Chapter 5.

The influence of the fillers on the relaxation, during transition from glassy to liquid (rubber-like) state, may be estimated from analysis of the relaxation

4.2.3 STRUCTURAL RELAXATION IN FILLED POLYMERS NEAR Tg

enthalpy.^41-43Theoretical basis for the analysis is as follows: It is supposed that during cooling of melt at equilibrium from T1 >> Tg at the constant rate (g=dT/dt), there is a certain point at which the rate of structural rearrange-ments in the melt, determined by the heat mobility of the chain segrearrange-ments, is lower than g and further cooling leads to larger deviations of the “instant” melt structure from the equilibrium state. In other words, at rather low temperature, T₂< T_g, the structure or a set of structures is frozen in a specimen which would have had equilibrium properties corresponding to some conditional tempera-ture T2<Tf≈Tg.⁴⁴Temperature dependence of the structural parameter, Tf, is determined from the equation:^45,46

τ is the relaxation time,∆E is activation energy, 0 < X < 1 is the parameter of non-linearity of the relaxation processes, 0 <β< 1 is phenomenological measure of the width of the spectrum of relaxation time,τ_ois coefficient, and dT′and dT′′

are the magnitudes of increments determined by the necessary correctness of the numerical solution. Parameters of Eq 4.35 are determined by any struc-ture-sensitive property of amorphous polymer during its cooling, heating or transition through the glass transition temperature point. If we choose as a phenomenological characteristic of the structural state the value of the relative enthalpy,∆H, the value T_fat T2can be found from the expression:

∆H(T ) = H(T )∆ d H∆

which, taking into account the standard definition of the heat capacity, C = d H / dT∆ , may be transformed to:

T T

1 g

T T

g

1 f

1

(C C )dT = (C C )dT2

∫

^{− ′}

∫

^{− [4.40]}

The differentiation of the last equation gives the temperature coefficient, Tf: (dTf/dT)T= (C - Cg)T/(C₁ −C )_{g Tf} [4.41]

where C is the relaxation part of the heat capacity in the glass transition inter-val. The experimental study of the relaxation enthalpy^41-43for PS having high content of filler allowed us to establish the kinetic parameters of the relaxation enthalpy for PS filled with fumed silica (0-35 wt%). Figure 4.5 shows experimen-tal, and calculated according to Eq 4.37, temperature dependence of dTf/dT (so-called reduced heat capacity; Eq 4.41). Good agreement between theory and experiment is pertinent. The numerical values of the parameter

Figure 4.5. Calculated (solid lines) and experimental (points) temperature dependencies of the re-duced heat capacity dTf/dT for PS at heating rate 16 grad/min after preliminary cooling with vari-ous rates (grad/min) 1-16, 2-4, 3-0.5.

lnτ_oproportionally changes ∆E/R, and in-versely proportionally to parameterβ. Incor-poration of fumed silica is accompanied by a regular growth of both ∆E/R and lnτ_oand de-crease of X and β. Diminishing b may be ex-plained by a broadening of the relaxation spectrum of filled PS, due to increasing microheterogeneity as a result of the forma-tion of the surface layers. At the same time, the increase in E/R and decrease of X show the retardation of the structural relaxation in boundary layers. This is confirmed by the cor-relation between parameters X andβ, on one hand, and parameters found from independ-ent experimindepend-ents on the other. It is interesting that the transition of all polymer into the sur-face layers does not suppress segmental mo-bility and manifests itself only in the change of the kinetic parameters of vitrification. It is possible that this effect is related to enlargement of the border layer (the order of inertia radius of the coils, which is larger as compared with the size of segment). Theocaris^47,48proposed another approach for determination of the glass transition temperature in filled poly-mers. Its advantage is in the introduction to calculations of the characteristics of the transition layer between filler and matrix. This method is based on determi-nation of three thermal expansion coefficients, which correspond to matrix, transition layer, and filler. Correspondingly, the changes in these coefficients are analyzed (Figure 4.6).

Let the volume fraction of filler beϕ_f, transition layer -ϕ_iand matrix -ϕ_m which have the corresponding expansion coefficients:α α_f, _i,andα_m. The tran-sition region has its own glass trantran-sition temperature, Tgi. In accordance with Figure 4.6, the expansion coefficient of the filled polymer at T < Tgis given by:

α_c1=ϕ α_{f f} +ϕ α_{m m1}+ϕ α_{i i1} [4.42]

For T_gi< T < T_gm

Figure 4.6. Scheme of volume changes for composite (c), filler (f), transition interphase layer (i) and matrix (m). [Adapted by permis-sion from P. S. Theocaris, Adv.

Polym. Sci., 66, 156 (1985)]

α_c2=ϕ α_{f f} +ϕ α_{m m1}+ϕ α_{i i2} [4.43]

and for T > T_m

α_c3 =ϕ α_{f f} +ϕ α_{m m2}+ϕ α_{i i2} [4.44]

The glass transition temperature of filled polymers corresponds to the in-tersection of the first and the last part of the curve (Figure 4.6), and can be calcu-lated from:

T = ( )T + ( )T

( ) + (

gc

i i2 i1 gi m m2 m1 gm

i i2 i1 m m

ϕ α α ϕ α α

ϕ α α ϕ α

− −

− ₂ − α_m1) [4.45]

This equation correlates Tgc with the thermal properties of matrix and transition layer.

In document Copyright <9 1995 by ChemTec Publishing ISBN 1-895198-08-9 (pagina 171-182)