THERMODYNAMIC AND KINETIC ASPECTS OF REINFORCEMENT
4.1 THERMODYNAMIC INTERACTION BETWEEN POLYMER AND FILLER Reinforcement of polymers is accompanied by the formation of the region of
interfacial interaction between the polymer matrix and solid, which leads to the general change in thermodynamic properties of the system. Let us now consider a model which gives a thermodynamic description of the interaction between polymer and filler. The first attempts to estimate this interaction have been made using thermodynamic analysis of the sorption processes and applying thermodynamic cycles. To estimate the change in Gibbs free energy, ∆G, or enthalpy,∆H, of transition of a polymer from unfilled to filled state, Kwei1used the cycle which enables values∆G and∆H to be found from vapor sorption and its temperature dependence. This approach is possible only above a critical va-por concentration when the sorption is the same for filled and unfilled polymers.
It was found that unfilled polymer has higher free energy and enthalpy than filled polymer, and the transformation from unfilled to filled polymer state is ac-companied by decrease in both enthalpy and entropy. It is interesting to note that the values of∆H and∆S found1are considerably below the values observed in the phase transitions (e.g., during polymer melting), and are similar in order
of magnitude to the values found from the heats of sorption of loosely or densely packed polymers.
Let us now consider a model which offers a thermodynamic description of the interaction between polymer and filler. It was used to evaluate the sorption of molecules of solvent by a free polymer offering the conclusion that the solvent sorption was greater than that of filled polymer. This model is applicable when the increase in packing density in the boundary layer is a consequence of power-ful interaction of polymer and filler, leading to a reduction in the chemical poten-tial of the polymer. It is assumed that this interaction becomes negligibly small at infinite distance from the surface, decaying rapidly with the distance. At a distance greater than r* (from the center of the filler particle), the polymer may be regarded as free of interaction with solid. On the other hand, the polymer within the sphere of radius, r*, is regarded as bound. It is also assumed that the chemical potential of the polymer in the boundary layer increases with an in-crease in distance from the center of the particle. In this case:
µH(R*)≈µp [4.1]
whereµpis the chemical potential of the free polymer andµf(r*) the chemical po-tential of the bound polymer at the distance r* from the center of the filler parti-cle. The mixing of the free polymer with the molecules of solvent, the activity of which is a, reduces the chemical potential of the polymer toµp(a), whereas the chemical potential of the bound polymer at distance r* is higher thanµp(a). Con-sequently, a certain proportion of the bound polymer must dissolve and become free, until at a new boundary, r, there is established an equilibrium ofµf(r) and µp(a). Thus, in the presence of a filler, the boundary is shifted and the bound layer compressed. At a critical content of solvent, all bound polymer dissolves, and thenµf(r) =µp. From Henry’s law, for a small amount of sorbed solvent, n1, it fol-lows that:
µp=µop −RT(n n )1 / 2 [4.2]
where n2is the number of moles of chain segments. Then
∆µ µ= p −µf = RT(n n )1 / 2 [4.3]
In this equation, the following boundary conditions exist:µf(r) = (r*) =µf µp and r = rfat n1>>n1,crit;∆µ= 0 with n1= 0 (rfis the radius of the filler particle).
Since it is assumed that the sorption takes place only in the free polymer, its pro-portion vpin the total volume of the polymer at a given vapor pressure is deter-mined by the relationship:
vp= [(x/m)f]/[ (x/m)p [4.4]
where (x/m) is the gain in weight during vapor adsorption.
The fraction of bound polymer (1 - vp) may be found from
(r/rf)3 - 1 = (1 - vp)(vp,f/vp) [4.5]
where vp,fand vpare the volumes of polymer in filled and unfilled specimen, re-spectively. Eqs 4.2 and 4.5 allow us to find ∆µand r/rf from the values deter-mined experimentally. Under the boundary conditions,∆δ= 0. In this case, r = r*
and r*/rf= 2.36. When n1= n1,crit, the value of r/rf= 1 and∆δ=∆δf(r). The thickness of an unswollen boundary layer, equal to r* - rf, is approximately 1430 Å, the pro-portion of bound polymer (1 - vp) as calculated from Eq 4.5 is approximately 56%.
Having determined the values of∆µf(r)at various temperatures, we can also cal-culate:
∆S =f(r) − δ µ(∆ f(r)) / Tδ [4.6]
∆Hf(r)=∆µf(r) + T S∆ f(r) [4.7]
Thus, investigation of the processes occurring in the filled polymer systems from the thermodynamics standpoint makes it possible to draw conclusions as to the structure of the polymer in the boundary layer close to the interface. The influence of the filler is not restricted solely to the layers lying in the immediate proximity of the interface. It is substantiated by the data on thickness of adsorp-tion layers, obtained by various methods. The influence of the filler on the struc-ture formation is explained by the so-called “relay-race” mechanism. The use of thermodynamic methods for the investigation of sorption allows us to assess
other characteristics of filled systems. If the sorption of vapors by filled polymer is higher, compared with sorption of pure polymer, it is possible to assess the ad-hesion between polymer and filler (Figure 4.1).2
Let n1be the number of moles of polymer before the incorporation of filler, n3the number of moles of polymer which are adsorbed on the filler surface, and n2the number of moles of polymer which are not perturbed by the filler, i.e., n3= n1- n2(please note that Kwei1considered a different case, where n2=0 and n3=n1). If in the compounding of filled polymer there is no change in the Gibbs free energy, then n2= n1and n3= 0. For calculations of thermodynamic function the cycle was proposed in Figure 4.1.2This cycle refers to the changes in free en-ergy in the range of vapor pressure from (p1/po)=0 to (p1/po)crit. In this cycle,∆G1 and∆G2are the changes in free energy of independent sorption of solvent by free polymer and filler up to a given value of (p1/p0)>(p1/p0)crit.∆G is the change in a free energy of formation of filled polymer,∆G3the change in free energy of sorp-tion by unbound polymer, n2,∆G4is the change of free energy associated with re-duction of adhesion or break-up of the bonds and with independent sorption by
Figure 4.1. Thermodynamic cycle of sorption by a filled polymer. [Adapted by permission from B. Clark-Monks and B. Ellis, J. Polym. Sci.- Polym. Phys., 11, 2089 (1979)]
unbound polymer, n3, and filler. It may be seen that∆G3and∆G4determine the sorption and cannot be separated from each other. For the proposed cycle:
∆G = G + G∆ 1 ∆ 2 −∆G3 −∆G4 [4.8]
Moreover,
∆G = N G + n G1 1∆ 1,s 1∆ 1, pol [4.9]
∆G = N G + F G2 2∆ 2,s ∆ 2,fil [4.10]
where subscripts s, pol, and fil refer to the sorbate, polymer and filler, N1and N2
are the number of moles of sorbate, F is the number of moles of filler and n1is the number of moles of polymer. For sorption of vapor by unbound polymer, i.e., (p1/po)(p1/po)crit.:
∆G = N G + n G3 3∆ 1,s 2∆ 1,fil [4.11]
In the final stage, when the relative pressure (p/po)>(p/po)crit, the sorption obeys the equation
∆G = G4 ∆ 4,n3 + G∆ 2 [4.12]
where G2is determined from Eq 4.10 and characterizes the free energy of sorp-tion by filler, while
∆G4,n3 = N G + n G4∆ 4,s 3∆ 4,n3 [4.13]
Substituting Eqs 4.13 and 4.10 into Eq 4.12, we obtain:
∆G = N G + n G4 4∆ 4,s 3∆ 4,n3 + N G + F G2∆ 2,s ∆ 2,fil [4.14]
Then from Eqs 4.8-4.14, taking into account that n3= n1- n2, we obtain:
∆G(pol-fil) = [(N G + N G ) (N G + N G + n1∆ 1,s 2∆ 2,s − 3∆ 1,s 4∆ 4,s 2∆G )]2,s
+ [n G1∆ 1, pol −n G2∆ 1, pol −(n1 −n ) G2 ∆ 4,n3] [4.15]
or
∆G(pol-fil) = G *∆ (pol-fil) + (n1 −n )( G2 ∆ 1, pol −∆G4,n3) [4.16]
The term G* in Eq 4.16 can be determined from isotherms of sorption of free polymer, filler, and filled polymer. Based on the data on the sorption by free polymer, it is possible to find the magnitude of N1∆G1,s+ N2∆G2,s, and from the data on the sorption by filled polymer, N3∆G1,s+ N4∆G4,s+ N2∆Gscan be found, which allows one to calculate∆G*(pol,fill), because from the conditions of the cycle, it follows that N1= N3+ N4. This quantity can be a relative characteristic of ad-hesion, although it is equal to the change in free energy of adhesion∆G*(pol,fill) only in limited cases, namely, when (n1- n2)(∆G1,p-∆G4,n3)=0, i.e., with n1= n2
(absence of an adsorbed layer). In actual systems this case never occurs. Experi-mental determination of∆Gpol,fillis practically impossible, since the values of n1 and n3are unknown. It is worth noting that in equations presented above the number of filler moles is used. In order to obtain expected data, the number of (moles) active centers should be used instead, because it depends on the dispersity degree of the filler.
The changes in free energy and enthalpy due to the interaction between polymer and filler have been estimated3from the sorption and calorimetric data for filled plasticized poly(vinyl chloride). Free energies∆G and enthalpies∆H of interaction between polymer and filler were calculated from the corresponding data on∆G and∆H of interactions of filled and unfilled polymers with a solvent in equilibrium conditions. Thermodynamic cycles have been used for this pur-pose. The experimental data3show various influences of the filler nature on the thermodynamic functions (Figure 4.2). The differences are explained by the ef-fect of active and inactive fillers on the formation of the matrix structure. Using sorption method,4it was established that with increasing amount of the filler, partial specific entropy of polymer increases, which indicates structural and conformational rearrangements in filled system. Using thermodynamic cycles, one should have in mind that they are applicable only for the equilibrium
sys-tems and processes. Because in polymeric syssys-tems the establishing of thermody-namic equilibrium often takes a very long time, thermodythermody-namic cycles may give an incorrect result for the systems, if data do not regard the equilibrium state.
Thermodynamically, the interaction between filler and polymer can be esti-mated from the heats of dissolution of filled and unfilled polymers.5The heat of dissolution consists of two values:
∆Hs=∆Hg+∆Hl [4.17]
where∆Hgis the heat of polymer transition from the metastable state having higher enthalpy to the equilibrium state of lower energy in solution,∆Hlis the heat of interaction equilibrium of melt with solvent. For filled polymer, the value of∆Hg, calculated per g of polymer, is not the same as for unfilled polymer because of difference in packing density, and it depends on the filler concentra-tion. If this value, however, does not depend on filler amount, then the value of
∆Hlfor filled polymer can easily be calculated by subtracting∆Hg≈const from the heat of dissolution∆Hs, using Eq 3.1. It was established that with increasing filler concentration,ϕ, calculated values of isothermal heat effects of filled poly-mers∆H < 0′l monotonously decrease up to a definite value ofϕ,after which it remains the same. It means that the heat of interaction in the polymer-filler sys-tem reaches its limiting value, corresponding to the saturation of interaction at the interface with solid. The critical concentration,ϕcrit,depends on the molecu-lar mass and conditions of production of filled polymer.
Figure 4.2. Dependence of free energy∆G (a), enthalpy∆H (b), and entropy T∆S on the filler con-tent for poly(vinyl chloride) filled with fumed silica (1) and chalk (2).
Thermodynamic properties of filled melts can be found from the measure-ments of specific volumes of filled and unfilled specimens. Such investigation has been done for PS and PMMA melts containing fillers of various surface en-ergy.6-8The calculation of thermodynamic functions can be performed based on the Hirai-Eiring theory modified by Smith.9According to the model, the melt at equilibrium may be considered as the saturated mixture of No molecules (seg-ments) of volume, vo, with Nhholes at volume, vh, and energy,εh. The volume of the system is:
V = Novo+ Nhvh [4.18]
The change of outer conditions (temperature, T, and pressure, p) leads to an exponential decrease in the number of holes:
N v
(N v )= N
(nv )= exp - ( + pv ) kT
h h
o o
h o
-1 h h
σ ε
[4.19]
where n = vo/vh,σ= exp(1 - n-1- Sh/k), Shis the change in entropy relative to hole formation. Taking Vo= Novo, we obtain:
{ }
V = V 1+o σ-1exp[-(εh+ pv ) / kT]h [4.20]
or after some transformations:
-ln[(V - Vo)/Vo] = lnσ+ (εh+ pvh)/kT [4.21]
This equation shows the linear dependence of the left side on 1/T and p can be found from the intercept and slope of the experimental curve of temperature dependence of the system volume, V, the parametersε/K and lnσof Eq 4.20 and values vh/kT and lnσ+εh/kT of Eq 4.21. The parameters of Eq 4.20, found in this way, may be used to estimate the changes in enthalpy, H, entropy, S, and free energy, G, during the transition of polymer into a filled state. According to Smith,9∆H is determined as:
∆H = Nh(εh+ pvh) [4.22]
which in conjunction with Eq 4.19 may be presented in the following form:
∆H = (vokT/vhσ) x exp(-x) [4.23]
where x = (εh+ pvh)/kT.
The absolute values of∆H are changed withεh. The entropy changes are ex-pressed as:
∆S = (vok/vhσ)(1 + x) exp(-x) [4.24]
and free energy changes as:
∆G = -vokT/vhσ [4.25]
Substituting numerical data for vari-ous specimens into Eq 4.25, one can estimate the excess free energy during transition of polymer into filled state (or surface layer).
The difference between the values of the free energy of filled and unfilled polymer allows one to make some conclusions regard-ing the thermodynamic stability of the poly-mer in the presence of the filler. The formation of the surface polymer layer on a solid surface leading to its loosening has a significant influence also on the thermody-namics of polymer-solvent interaction (its dependence on the thickness of a film ap-plied to a solid surface). Such estimation was done for filled poly (ethylene glycol adipate) and PS,10using inverse gas chromatography.
Due to the peculiarities of the method, only a
Figure 4.3. Dependence of∆G on the filler content for annealed (1) and non-annealed (1’) films of poly-(ethyleneglycol adipate), poly(ethylene-glycol) (2) and PS (3) (1 and 2 for heptane, 3-for toluene).
small amount of filler was used. Filled films were applied on the solid sub-strate.11Figure 4.3 shows the results of determination of the free energy of mix-ing of solvent with the polymer,∆G.∆G decreases with increase in filler content.
The incorporation of filler leads to a loosening of the polymer, which eases the mixing of the polymer with the solvent. It must be noted that the value of∆G is considerably influenced by annealing. For filled PS, annealed at 443K (above the temperature of transition to the viscous-flow state), there is some increase in its density, as follows from the value of the retention volume. However, the value of∆G, for the annealed filled film, is lower, compared with non-annealed material. This observation indicates that annealing leads to compacting of the surface layer of the filled film at the interface: polymer-air. In fact, since, at Tg, the diffusion of the molecules of sorbate proceeds only to an insignificant depth of the film, the reduction in∆G, at this temperature, is evidence of a compacting of the top layer at the interface with air. Simultaneously, the density of lower boundary layers remains lower than for the free filled films, as shown by the val-ues of∆G determined in the region of equilibrium adsorption, i.e., under condi-tions where the molecules of sorbate penetrate through the entire polymer film and consequently enter the boundary layer. Such marked compacting of the filled annealed film apparently indicates that the influence of the filler (or inter-face) extends to a considerable depth.
The same method was used12to establish the influence of thickness of the polymer stationary phase on the interaction parameters between polymer and solvent. It was found that the adsorption layers at the interface with a solid af-fect the thermodynamic quantities such as excess free energy, enthalpy, and en-tropy of mixing, calculated from the general expressions of gas-liquid chromatography. Also, the polymer-solvent interaction parameter,χ, has been determined as a function of the film thickness. It was shown that all thermody-namic functions depend on the thickness of the polymer film at the solid surface.
Free energy of mixing is negative and it diminishes as film thickness decreases.
According to these data, the solubility of polymer increases with a decrease in film thickness. This can be connected with the formation of an adsorption layer and its influence on more distant layers. As a result of adsorption interaction with the surface and restriction of molecular mobility, the density of packing near the surface diminishes more rapidly when the film thickness is smaller.
This leads to the increased solubility.
The same conclusions have been ob-tained from the calculated values of the interaction parameter, χ, measured for PS films in the region of thicknesses of 200-600 nm, which diminishes with a de-crease in the film thickness.
The analysis of the thermodynam-ics of the polymer reinforcement allows for a general conclusion. The formation of filled polymer, e.g., the transition of polymer, in the state of thin polymer lay-ers at the interface with a solid, leads to a less stable state. During production of filled polymers, the system deviates from the state of thermodynamic equilibrium, and therefore filled polymers, as a rule, should be considered as non-equilibrium systems, due to the action of the surface on the equilibrium properties of macromolecules.