Thermosetting Polymers Part 10 potx

insensitive to the large-scale structure; they depend essentially on the sive energy density, which is directly linked to the molecular scale structure.cohe-In contrast, the properties i

is heterogeneous, one also needs information describing this heterogeneity(supramolecular or morphological scale) The conceptual and experimentalinvestigation tools differ strongly from one scale to another(Table 10.1);it

is thus important to recognize that, to be fully efficient, an investigation onstructure–property relationships in the field of networks must be multidisci-plinary

Time and energy can be saved if one recognizes that there is only onequalitative difference between a linear and a tridimensional polymer: theexistence in the former and the absence in the latter of a liquid state (at amacroscopic scale) For the rest, both families display the same type ofboundaries in a time–temperature map(Fig 10.1).Three domains are char-acterized by (I) a glassy/brittle behavior (I), (II), a glassy/ductile behavior,and (III) a rubbery behavior The properties in domain I are practically

Basic Physical Properties of Networks 283

TABLE10.1 The three structural scales of polymer structure Conceptualand experimental tools

Structural scale Typical size Typical entities

Academicdomain Main toolsMolecular 0.1–1 nm Chemical

groups,monomerunits, CRU

Organicchemistry

IR, NMR

Macromolecular 10–100 nm Network

chains/strands,crosslinksnetwork defects(dangling chains)

molecularscience

Macro-Rubberelasticity,solventswellingSupramolecular 10–105nm Packing defects,

nodular/globularmorphology,multiphasestructure

Materialsscience

Microscopies,scatteringmethods,thermalanalysis

FIGURE10.1 Time–temperature map Shape of main boundaries for linear ornetwork polymers (I) Glassy brittle domain; B, ductile–brittle transition (II)Glassy ductile domain; G, glass transition (III) Rubbery domain The location

of the boundaries depends on the polymer structure but their shape is alwaysthe same Typical limits for coordinates are 0–700 K for temperature and

10
3s (fast impact) to 1010s; e.g., 30 years static loading in civil engineering

or building structures Fpr dynamic loading, t would be the reciprocal offrequency For monotone loading, it could be the reciprocal of strain rate _ee ¼dl= Idt:

insensitive to the large-scale structure; they depend essentially on the sive energy density, which is directly linked to the molecular scale structure.

cohe-In contrast, the properties in domain (III) are almost independent of themolecular scale structure; they depend essentially on the crosslink density(entanglement density in linear polymers) The intermediary domain II isespecially interesting for users because a relatively high stiffness (typicallyE1–3 GPa) can be combined with a high ductility/toughness/impact resis-tance Since there is a very abundant literature on structure–property rela-tionships of linear polymers (Van Krevelen, 1990; Porter, 1995; Mark,1996), we have chosen to recall very briefly the aspects of the physicalbehavior common to linear and tridimensional polymers and to insist onthe differences between both families Volumetric properties will be espe-cially developed because they are, often abusively in our opinion, used forthe interpretation of thermomechanical behavior

10.2.1 Volumetric Properties

a Density at Ambient Temperature

Density, r, depends on atomic composition, which can be represented by asingle quantity – the average atomic mass Ma– defined by

Ma¼ Molar mass of the CRU

Number of atoms of the CRU ð10:1ÞThe CRU (constitutive repeating unit) can be considered as the

‘‘monomer unit’’ of the network It has been defined inChapter 2

The density of many linear, as well as tridimensional, polymers hasbeen plotted against MainFig 10.2.One can estimate r from a power law:

where k  320, r is in kg m
3 and Main g mol
1

This empirical relationship remains valid for light inorganic materialssuch as calcium carbonate, silica, aluminium, etc., and allows the density to

be predicted with a maximum error of 10% in the range of most usualorganic network densities (1100  r  1400 kg m
3) In this range, densitycan be approximated by a linear relationship:

r ¼ 350 þ 120Ma (kg m
3Þ ð10:3ÞTypical density values for networks are 1150–1330 kg m
3 for amine-crosslinked epoxies (Ma7 1); 1120–1180 kg m
3for styrene-crosslinkedvinyl esters (M 6.3–7.1); and 1170–1220 kg m
3 for styrene-crosslinked

polyesters (Ma 8 for a maleate/phthalate (1/1) of propylene glycol linked by 35% of styrene) Many points corresponding to these systemshave been plotted in Fig 10.2 They reveal no systematic difference (withinthe observed scatter) between networks and linear polymers.

cross-In all structural series, the density is effectively an increasing function

of Ma, e.g., of the content of ‘‘heavy’’ atoms such as O or S

Let us consider the variation of the composition in a two-componentsystem (R+H), e.g., unsaturated polyester–styrene or epoxide–amine Aredensity measurements capable of detecting such a variation?

If y is the molar ratio H/R and the CRU contains 1 mole of R and ymoles of H, the molar mass M and the number of atoms N are

FIGURE10.2 Density (kg m
3) against average atomic mass Ma(g mol
1) forlinear polymers (~) and networks (^) Curve (full line), r ¼ 320 M2=3a ; straightline (dashed line), r ¼ 350 þ 120Ma

In a first approximation, one can use Eq (10.3) for dr/dMa, so thatdr

dy¼120

NRMH
NHMR

The sensitivity of the method is an increasing function of the difference

of average atomic mass (MaH
MaRÞ Let us consider four cases of trial networks (Table 10.2)

indus-Using simple laboratory equipment, it is possible to detect variations

of the density of the order of 1 kg m
3; therefore, density measurementswould be useless for diglycidyl ether of bisphenol A and diamino diphenylmethane (DGEBA–DDM), because both components have close Mavalues.They would be moderately sensitive for unsaturated polyesters of maleate/phthalate (1/1) of propylene glycol crosslinked by styrene (36 wt%), andrelatively sensitive for DGEBA cured by diamino diphenyl sulphone (DDS)

or phthalic anhydride (PA), for which relative variations of the molar ratio y

of about 5% could be detected In certain cases, where some comonomer(PA or styrene) can be lost by evaporation during the cure, density measure-ments can constitute a simple and efficient method of control way (e.g., fornon-filled materials)

b Packing Density at Ambient Temperature

InFig 10.2, the scattering is obviously linked to differences in molecularpacking In Eq (10.2), one could tentatively assume that the factor k is afunction of the packing density For instance, for the four materials of Table

TABLE10.2 Sensitivity of density to variations of the composition Thedefinitions of MR, NR, MH, NH, and y are given in the text

10.2, we would have the data of Table 10.3 The hierarchy of packingdensities would be :

DGEBA–DDS ¼ DGEBA-DDM > DGEBA–PA > UP–S

There is no clear correlation between k and density or Mavalues.The use of k values to represent packing density would be question-able, owing to the empirical character of Eq (10.2) The following definition

is better (Bondi, 1968; Van Krevelen, 1990):

in the same way as hydroxyl concentration with the composition which canlead to erroneous interpretations Other structural factors also probablyhave a small influence on packing density; e.g., aromatic networks seem

to be systematically less densely packed than comparable aliphatic ones,but this effect seems difficult to quantify owing to data scattering

TABLE10.3 Density, average atomic mass, and prefactor of Eq (10.2) forthe four networks ofTable 10.2

To appreciate an eventual effect of crosslink density on mass density,

we have to compare networks having the same hydrogen bond tion This is possible, for instance, in the series B ofTable 10.4(Morel et al.,1989)

concentra-One can see in series B that [OH] is constant within 1% and that thepacking density is almost constant or increases slightly with the crosslinkdensity In contrast, in series A, the packing density increases significantlyfrom A1 to A4, e.g., with the crosslink density, but the OH concentrationincreases at the same time To summarize: all the results of Table 10.4 are

3

kg mol
1 In fact, one can reasonably assume that all the intermolecularattractive forces participate in the increase of packing density The effect ofthese forces can be introduced in structure–property relationships throughthe concept of cohesive energy density (Sec 10.2.2)

c ‘‘Anomalous’’ Density Variations with the Structure in

Amine-Cured Epoxies

In many structural series based on a given epoxide–amine pair differing bythe amine/epoxide molar ratio (e.g., Won et al., 1991) or by the degree ofcure conversion (e.g., Venditti and Gillham, 1995), it has been observed thatthe density is a decreasing function of the crosslink density, and authorssuch as Venditti and Gillham (1995) have suggested that both phenomenawould be linked to the free volume fraction ‘‘trapped’’ in the polymer Infact, in series where the atomic composition and the cohesive energy density

FIGURE 10.3 Packing density, r* ¼ VW=V, versus hydroxyl concentration: V(~), vinylesters; E (*), amine crosslinked epoxies ; L (*), linear polymers

are almost constant, such as series B of Table 10.4, or tic amine systems studied by Won et al., (1991), or polyesters (Shibayamaand Suzuki 1965), the density appears to be rather an increasing function ofthe crosslink density In some of these series, the modulus tends to be adecreasing function of the crosslink density as found for instance in series B

DGEBA-cycloalipha-of Table 10.4 The tensile modulus is about 3.25 GPa for TGAP–DDM(crosslink density 5.88 mol kg
1) and 3.91 GPa for TGAP-aniline (AN)(crosslink density 2.40mol kg
1)

It thus seems that there is no direct link between volumetric and elasticproperties in the glassy state and that the ‘‘anomalous’’ density variationscannot be attributed to a crosslink density effect, either direct (on molecularpacking) or indirect (through internal antiplasticization as discussed below)

It seems reasonable to correlate this behavior with the presence of unreactedepoxides The density would be (in the systems under consideration) a con-tinuously increasing function of the amine/epoxide ratio, owing to the

TABLE10.4 Density, hydroxyl concentration, crosslink density, packingdensity, ultrasonic bulk modulus, average atomic mass (g mol
1, prefactor

of Eq (10.2) calculated from Eq (10.19), calculated density from Eq (10.21).Experimental data from Morel et al (1989)

Ma(gmol
1) k

rðcalcÞ(kg

m
3)A0 DGEBA-DDM 1200 4.56 2.28 0.678 6.31 6.913 330.7 1201A1 DGEBA–

AN (75–25)

1263 7.09 4.98 0.702 8.17 6.988 345.5 1275B3 TGAP–DDM/

AN (50–50)

1263 7.13 4.10 0.703 8.05 6.973 346 1268B4 TGAP–DDM

AN (25–75)

1259 7.19 3.24 0.702 7.62 6.957 345.5 1251B5 GAP–AN 100 1255 7.20 2.40 0.700 – 6.941 344.9 1255

strong contribution of OH and NH groups to cohesivity In the presence ofunreacted epoxide groups, this effect would be counterbalanced by a specificeffect – for instance, a decrease of Van der Waals volume linked to epoxidering opening (Bellenger et al., 1982) – or a hypothetical effect of danglingchains on packing (only present in networks containing unreacted epoxidesfor the systems displaying the ‘‘anomalous’’ effect).

d Thermal Expansion in Glassy State

When studied in a limited temperature interval around the ambient perature, expansion can be considered linear:

where Ar is the concentration of aromatic units in mol kg
1

Indeed, the linear expansion coefficient near ag temperature Ta, islinked to A by

vaporization corresponds to the rupture of intermolecular bonds Then, for

1 mole,

where Ec is the molar cohesive energy, Hv is the molar vaporizationenthalpy, and the term RT corresponds to the gas expansion – (pressure)(volume variation) ¼ RT

In polymers, it is usual to consider that Eccorresponds to 1 mole ofCRU Two materials can be compared through their cohesive energy density(CED), defined by CED ¼ Ec/V, where V ¼ M/r, is the molar volume of theCRU with molar mass M CED is expressed in J m
3; typical values for

J m
3 It is usual to convert thesevalues into MPa (200 MPa < CED < 800 MPa) A practically importantquantity is the solubility parameter, d, defined by d ¼ ðCEDÞ1=2 andexpressed in J1=2 m
3=2 or, better, in MPa1=2

In certain circumstances, it may be necessary to distinguish betweenthe different types of interactions This can be performed in several ways(Barton, 1983; Van Krevelen, 1990) The most usual method is to make adistinction between dispersion (London), dipolar (Debye–Keesom) andhydrogen-bonding components, each one being characterized by its contri-bution to CED and the corresponding solubility parameter, dd;dp;dh,respectively, such that d ¼ ðd2dþd2pþd2hÞ1=2

b Methods of Determining Cohesive Properties

The methods of determining d or CED are discussed below

(i) Calorimetric Measurement of Hv Unfortunately, this methoddoes not work in the case of polymers because they undergo thermal degra-dation long before vaporization

(ii) Polymer–Solvent Interactions From a thermodynamic approach(Hildebrand), it can be shown that, for a given polymer of solubility para-meter P, the miscibility with a solvent is a decreasing function of jdp
dSjwhere dSis the solvent’s solubility parameter For a thermoset that is totallyinsoluble in any solvent, the miscibility can be, easily quantified, in principle,

by the equilibrium concentration of the solvent determined, for instance,from weight uptake measurements in saturated vapor There are, however,many strong obstacles to such determinations (Bellenger et al., 1997), andliterature data are scarce This method cannot be reasonably recommended

as an efficient tool for determining cohesive properties Inverse gas tography can be also used to determine solubility parameters

(iii) Bulk Modulus at High Frequency/Low Temperature The laxed bulk modulus determined below the main secondary transition, e.g.almost free of viscoelastic effects, is almost proportional to CED (Morel etal., 1989; Bellenger et al., 1994):

This is probably the simplest and most accurate way of determiningthe cohesive energy density The measurements are easy to perform if homo-geneous, relatively thick (few mm) samples, with two parallel and smoothsurfaces, are available

(iv) Calculation from Molar Increments There are many ways tocalculate the cohesive energy of a given CRU (Van Krevelen, 1990) Thesimplest method is to consider Ecas an additive molar function:

Ec¼X

CRU

where Eciis the molar increment of the ith group of the CRU Indeed, Eci

depends on the polarity of the corresponding group, but also on its size.For very rough estimation, one can consider the cohesive energy/molar mass ratio as

Some typical values are

Very low polarity groups such as CF2, CF3: ei85 J g
1

Low polarity groups such as hydrocarbon groups, tertiary amines,chlorine, ether: ei¼200–400 J g
1

c Cohesion and Packing Density

From the results ofTable 10.4,one can write

Using the relationship between the bulk modulus and the cohesiveenergy density leads to

r* ¼ 0:17CED þ 0:584 ð10:18Þ

where CED is expressed in GPa (GJ m
3)

However, from a practical point of view, it is easier to start from Eq.(10.2) to predict density In this case, a relationship between the parameter kand a quantity characterizing the cohesivity, for instance the OH concentra-tion or CED, may be found

According to the data ofTable 10.4,k varies almost linearly with thesequantities:

d Data onThermosets

Data are available for unsaturated polyesters (Deslandes et al., 1998), linked- PMMA (Bellenger et al., 1997), vinyl esters (Bellenger et al., 1994),and amine-crosslinked epoxies (Bellenger et al 1989) Some results are sum-marized inTable 10.5

cross-From the point of view of cohesive properties, polymer networks: behave as linear polymers – the influence of crosslinking on cohe-sion can be considered negligible, at least in a first approximation; cannot be accurately characterized from solvent absorption experi-ments

can be easily (but only globally) characterized from unrelaxed ulus measurements (especially from measurements of the velocity ofultrasonic waves)

mod- can be characterized from calculations assuming the molar ity of group contributions

10.2.3 Local Mobility

a Principles of Measurement

Polymer glasses are characterized by a residual segmental mobility in a more

or less wide temperature interval below Tg This mobility is responsible forthe existence of relaxations often called secondary relaxations that are inopposition to the main (cooperative) relaxation occurring at T  Tg Thereare many types of localized motions that involve, for example, rotations oflateral groups, oscillations of aromatic rings, around the chain axis, chair–chair isomerization in cycloaliphatic rings, or crankshaft-like motions onshort-chain segments After a perturbation, the system returns to its initialequilibrium or to a new equilibrium at a rate which depends essentially onthe distance from equilibrium and on molecular mobility Molecular mobi-lity is represented by a characteristic time or relaxation time, or by anaverage relaxation time, or by a spectrum of relaxation times

Under a perturbation Po,the response (Q) rate, for the simplest case offirst-order relaxation kinetics, can be written as

CED(MPa)

CED (MPa)from bulkmodulus

[OH](mol kg
1)

t ¼ fðTÞ; and o
1 can be considered as the characteristic time of the turbation.

per-It may be concluded from the above results that the transition betweenunrelaxed and relaxed states occurs when the characteristic times of therelaxation process and of the experiment used to study the relaxation(here sinusoidal perturbation), are equal Therefore the following pointscan be made:

1 A given type of local motion in the glassy state must be sible for a characteristic transition This transition will be arbi-trarily called b; g; d, etc., in the order of decreasing temperatures

respon-or increasing frequencies; a is reserved in amrespon-orphous polymers tothe transition between glass (local motions) and rubbery (coop-erative motions) domains

2 A given transition is characterized by an inflection point in thecurve S0 (real part of S*) ¼ f (T) at o constant, or f(o) at Tconstant, and by a maximum on the curve S00¼f (T) or f(o).

3 A given transition is essentially characterized by two quantities:its amplitude S and its location in the frequency scale at a giventemperature, in the temperature scale at a given frequency or,better, in the frequency–temperature map

4 In the simple case of Fig 10.4, it can be easily demonstrated thatthe area under the curve S00¼f(o) is proportional to S Bothquantities characterize the ‘‘activity’’ of the relaxation in thephysical domain under consideration: for instance, S can cor-respond to the gap between two distinct conformations of thechain

When the perturbation is a stress, the response is a strain and S00 isthe complex compliance, J* ¼ J0
iJ00 (dynamic mechanical analysis:DMA)

When the perturbation is a strain, the response is a stress and S00 isthe complex modulus, E* ¼ E0þiE00 (dynamic mechanical analy-sis) The damping factor, tan d ¼ E00=E0, is a useful parameter

(Chapters 6and11)

When the perturbation is an electric field, the response is the rial polarization and one can define a complex dielectric constant,which is the electrical equivalent of a compliance: e* ¼ e0
ie00(dielectric spectroscopy; Chapter 6)

mate-FIGURE10.4 Real S0(*) and imaginary S’’ (~) part of S normed by ~S (seetext) versus log ot for a dynamic perturbation in the case of a first orderrelaxation process

When the perturbation is a magnetic field, the response is the rial magnetization related to a complex magnetic susceptibility: k ¼

mate-k0
ik00 (NMR and ESR spectroscopies)

All these methods can be used to study the glassy-phase transitions incomplementary ranges of frequency, typically 10
2–102Hz for DMA ; 10–

106 Hz for dielectric spectroscopy and 106–1010 Hz for NMR and ESRspectroscopies Photophysical measurements give access, eventually,tovery high frequencies/very short times Static mechanical testing (creep,stress, relaxation), or electrical testing (thermally stimulated depolarizationcurrents) give access to very low frequencies It must be noted that the

‘‘activity’’ of a given relaxation, as represented by S, depends on theselected property Certain relaxations display a strong mechanical activityand a low electrical activity, whereas others display the opposite behavior.Actually, relaxation processes do not follow a first-order kinetics In aglass there is a relatively wide variety of situations leading to the existence of

a spectrum of relaxation times A ‘‘static’’ relaxation kinetics can beapproximated by

S ¼ S0 1
exp
t

t

 b

(Kolrausch, Williams, Watt) ð10:29Þ

where the exponent bð< 0 < b  1Þ expresses the broadness of the tion spectrum Let us recall that b is temperature dependent Thus, a giventransition is essentially characterized by its amplitude S and its location in

relaxa-o and T scales, but it can alsrelaxa-o be characterized by its width, which isgenerally larger than the theoretical one for the first-order process(o ¼ 1:2 decades

Generally, the average relaxation times and the transition frequencies

in the glassy state obey the Arrhenius law:

t ¼ t0exp H

RT or o ¼ o0exp

H

b Sub-Glass Transitions in Thermosets

Some thermomechanical spectra of thermosets (at 1 Hz), are shown inFig.10.5.Some networks display at least two secondary transitions between 100

K and Tg Their assignment is often controversial and, to our knowledge,there is no widely accepted way of predicting their position (Tb) and theirintensity

From the practical point of view, it is interesting to make the followingdistinctions:

(i) Polymer networks with their b transition located above the ambienttemperature, such as unsaturated polyesters (Tb(1 Hz)  350K), or aro-matic polyimides (Tb(1 Hz)  330–400 K, activation energy Hb100–180

kJ mol
1), can be distinguished from networks having their b transitionbelow the ambient temperature, for instance amine-crosslinked epoxies:

T(1 Hz)  190–240 K, Hb70 30kJ mol
1, or phenol–formaldehydenetworks: Tb(1 Hz)  210K, Hb140kJ mol
1 As most of the mechan-ical tests are made at ambient temperature, the corresponding b relaxation isinactive for the former, in the most frequent use conditions, but is active forthe latter This has very important practical consequences

(ii) Networks with a highly active b transition – e.g., stoichiometric–epoxide–amine networks, in which the b relaxation is linked to crankshaftmotions of the CH2 CH CH2

j

OH segment, and which have an especially intense bdissipation peak – can be distinguished from networks having a low inten-sity b dissipation peak – e.g., styrene-crosslinked vinyl esters, polyimides,etc

The b relaxation can play a very important role in fracture properties.For comparable Tgvalues, for instance  1008C, amine-crosslinked epoxiesare considerably more ductile and tough than unsaturated polyesters.(iii) In networks with a highly active b transition, e.g essentiallyamine-cured epoxies, structural modifications leading to a decrease of theglass transition temperature (internal or external plasticization) lead to adecrease of the b-transition intensity, which leads to antiplasticization, asobserved in linear polymers such as poly(vinyl chloride) (Chapter 11).This

FIGURE10.5 Examples of thermomechanical spectra (tan d ¼ fðTÞ at 1 Hz quency) of some thermosets: PMR15 (polynadimide) ( ) ; EP ( ) ; VE(- - - -); and UP (———) The spectra are arbitrarily vertically shifted

fre-phenomenon has received various interpretations (Ngai et al., 1996; Heux etal., 1998).

The cohesive and volumetric properties (under the dependence of,essentially, atomic composition and CED), can be predicted or at leastestimated from the CRU structure

Parameters related to local mobility are more difficult to predict,although there have been significant advances in this field (Ngai, et al.1986) A complete understanding of these properties, including phenomenasuch as antiplasticization, will probably require the knowledge of factorssuch as the spatial free volume fluctuations

Certain important physical properties of polymer glasses depend also

on their nonequilibrium character (creep, yield stress, fracture properties)

RELATIONSHIPS

10.3.1 Introduction

The glass transition phenomenon has been presented in Chapter 4 Here,only structure–property relationships will be briefly examined

There is a relatively abundant literature on the relationships between

Tgand the network structure, but it is difficult to make a coherent synthesis

of the published data owing to the great diversity of empirical, rical, and physical approaches to the problem

It must be first remarked that structure–Tgrelationships can be sonably established on the basis of experimental data obtained on networks

rea-of well-defined structure Thus, it seemed interesting to us to distinguishthree cases:

Homogeneous ‘‘ideal’’ networks, also called ‘‘closed’’ networks,result from a single-step polymerization mechanism of a stoichio-metric mixture of monomers, reacted to full conversion Manyamine-crosslinked epoxies of Tg < 20 0 8C and polyurethanesobtained using a single isocyanate monomer and a single polyolbelong to this family

Homogeneous ‘‘nonideal,’’ e.g ‘‘open’’ networks, obtained fromthe same chemistry as the previous ones These networks containdangling chains as a result of incomplete cure, nonstoichiometriccomposition, or presence of monofunctional monomers

Inhomogeneous ‘‘open’’ networks characterized by spatial tions of the crosslink density (nodular/globular morphologies,microgels, seeChapter 7)

fluctua-10.3.2 ‘‘Ideal’’ Networks

a Epoxy Networks

Some important trends of the structure–property relationships in this fieldare well illustrated by a comparison of some stoichiometric, fully curedepoxide–amine networks(Table 10.6)

Three structural parameters have been calculated from the theoreticalCRU for these networks:

The crosslink density, n, expressed in terms of moles of crosslinkedmers (network crosslinks);

The chain stiffness expressed by the flex parameter F which, for agiven chain of molar mass Me, is given by : F ¼ Me/Ne, where Neisthe number of elementary (undeformable) segments F is essentially

an increasing function of the content of aromatic nuclei, Ar The concentration of hydroxyl groups [OH], which has a predomi-nant contribution to the CED It is clear that this latter variableplays a negligible role in the series under study: the highest cohesiveenergy density (A) corresponds to the lowest Tgvalue

The first two variables play an important role, especially chain ness, as shown by the comparison of systems DDM and DA12 based on thesame epoxide, which have the same crosslink density and very close OHconcentrations, e.g., close CED values Thus, it appears that any widely

stiff-applicable Tg–structure relationship must involve at least two variables thatexpress, respectively, the chain stiffness and the crosslink density To checkthis, it is convenient, in principle, to start from a structural series based on alinear polymer (glass transition Tgl), with variable but slight crosslink den-sities: in this case, crosslinking does not modify significantly the chain flex-ibility and one can establish the effect of the crosslink density on Tg.

A simplified version of the free volume theory, considering that Tgis

an iso-free volume point and that the free volume ‘‘absorbed’’ by one link is independent of the crosslink density, leads to (Fox et al., 1955):

where KFLis a constant for the structural series under consideration and ne

is the concentration of elastically active network chains (EANC)

One can try to derive such a law from a copolymer law of the Foxtype, considering a CRU containing crosslinks (of which the mass is negli-gible compared with the chain mass) Then,

M, since the crosslinks have a negligible mass Then,

TABLE10.6 Comparison of some stoichiometric, fully cured epoxide–aminenetworks: Tg, glass transition temperature; [Ar], aromatic group

concentration; F, flex parameter (see text); n, crosslink density; and [OH],hydroxyl concentration Note DGEBD ¼ diglycidyl ether of butane diol;DGEBA ¼ diglycidyl ether of bisphenol A (i ¼ degree of polymerization);and TGAP ¼ triglycidyl derivative of p-aminophenol;

DAj¼H2N
ðCH2Þj
NH2; DDM ¼ diamino diphenyl methane

[Ar]

(mol g
1Þ

F(g mol
1)