The mechanics of network polymers with thermally reversible linkages

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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 63 (2014) 386–411

0022-50http://d

n Tel.:E-m

journal homepage: www.elsevier.com/locate/jmps

The mechanics of network polymers with thermallyreversible linkages

Kevin N. Long n

Solid Mechanics Department, Sandia National Laboratories, Albuquerque, NM, United States

a r t i c l e i n f o

Article history:Received 2 December 2012Received in revised form30 June 2013Accepted 24 August 2013Available online 21 September 2013

Keywords:Active polymerDynamic bondEncapsulationDiels–Alder chemistryConstitutive model

96/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.jmps.2013.08.017

þ1 505 844 5452.ail address: [email protected]

a b s t r a c t

Network polymers with thermally reversible linkages include functionalities that con-tinuously break and form covalent bonds. These processes dynamically change thenetwork connectivity, which produces three distinct behaviors compared with conven-tional thermosetting polymers (in which the network connectivity is static): permanentshape evolution in the rubbery state; dependence of the number density of chains andassociated thermal and mechanical properties on temperature and chemical composition;and a gel-point transition temperature above which the connectivity of the network fallsbelow the percolation threshold, and the material response changes from a solid to liquid.This last property allows such materials to be non-mechanically removed, which is anattractive material capability for encapsulation in specialized electronics packagingapplications wherein system maintenance is required. Given their complex, multi-physics behavior, appropriate simulation tools are needed to aid in their use.

To meet this need, a thermodynamically consistent constitutive model is developedthat accounts for the thermal–chemical–mechanical behavior of such materials. Thismodel includes a representation for the permanent shape evolution that accompanies thedynamic network connectivity as well as non-equilibrium viscoelastic behavior torepresent the material's glassy response. Analytic solutions in the rubbery state arederived to show the effects of competing time scales in the material, and the model iscalibrated and validated against experimental data published in the literature. Finally,simple encapsulation scenarios are examined that demonstrate a substantial difference inbehavior between conventional polymer networks and those with thermally reversiblelinkages under thermal–mechanical cycling.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Network polymers with reversible chemical functionalities are atypical thermosets. A snapshot of the macromoleculewould appear normal; chains are joined together via covalent, chemical cross-links as well as physical entanglements thatarise from Van-der-Waals interactions between monomers. However, in these materials, specific functionalities have beenincluded along polymer chains and at cross-link sites, and these functionalities undergo reversible chemical reactions thatconnect or disconnect polymer chains and cross-links from the network (Kloxin et al., 2010; Wojtecki et al., 2011; Tasdelen,2011; Montarnal et al., 2011). These types of reversible linkages have been labeled, “dynamic covalent bonds”, in theliterature (Wojtecki et al., 2011) to describe the fact that bond formation and destruction occur with associated kinetics that

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K.N. Long / J. Mech. Phys. Solids 63 (2014) 386–411 387

depend on the state of the material and possibly the presence of an external stimulus (if required by the specific reversiblechemistry).

The chemical literature has seen a considerable activity in this area associated with“clique chemistry” (Kloxin et al.,2010). Dynamic bonds can be stimulated by temperature (Adzima et al., 2010; Chen et al., 2003; Montarnal et al., 2011), light(Scott et al., 2005; Lendlein et al., 2005), electric fields (Wojtecki et al., 2011), and mechanical deformation (Martin andBeyer, 2005). In contrast to other stimuli, temperature as a stimulus cannot be “turned-off” (such as “turning off” the lightsource), and so for network polymers with thermally reversible linkages, chains and cross-links are connected anddisconnected from the network continuously in time. This dynamic rearrangement of the chain connectivity provides thenetwork an intrinsic relaxation mechanism and is responsible for three distinct phenomena that are not observed inconventional thermosets, which we define as network polymers with permanent covalent cross-links and physicalentanglements.

First, under conditions of thermal, chemical and mechanical equilibrium, the aggregate connectivity of the network isstatic—for example, the macroscopic number density of chains and rubbery shear modulus are constant—even while thechains themselves dynamically break and reform. However, over time, the network topology evolves as the chainconnectivities are rearranged. If the material is subjected to mechanical stresses, the network topology evolves to relaxthem, which changes its permanent shape (McElhanon et al., 2002; Chen et al., 2003; Adzima et al., 2008; Sheridan et al.,2011). A representative thermal chemistry that has been actively explored in the chemical literature integrates furan andmaleimide functionalities into the network as both chain extending and cross-linking monomers (McElhanon et al., 2002;Szalai et al., 2007; Adzima et al., 2008; Polaske et al., 2010; Sheridan et al., 2011). These functionalities undergo thereversible, Diels–Alder (DA) cycloaddition reaction which connects and disconnects chains in the forward and retroreactions respectively, which is schematically shown in Fig. 1. To narrow the scope of this work, we will focus on thischemistry as a model system in this paper.

The second distinct phenomenon of networks with thermally reversible linkages compared with conventional networksis that their mechanical and thermal properties depend on the network connectivity such as the number density chains,which is itself a function of the dynamic covalent bond's chemical equilibrium. Temperature change alters this chemicalequilibrium, which changes the network connectivity and therefore the thermal and mechanical properties of the networksuch as the shear moduli and heat capacities. Therefore above the glass transition, we anticipate that these properties ofsuch materials will be more sensitive to temperature changes than that for the same properties in conventional thermosets.

The third distinction of network polymers with thermally reversible linkages compared with conventional thermosets isthat they may exhibit a reversible gel-point transition. If the temperature is changed so that a sufficient number of chainsare broken such that no unbroken chain spans the material, then the polymer network has crossed the percolation threshold(gel-point). In this state, the material behaves as a viscous liquid and can be removed. Because of this capability, we labelnetwork polymers with thermally reversible linkages as RNPs for removable network polymers. A schematic of networkconnectivities is presented in the glassy state, rubbery state, and liquid state (beyond the gel-point) in Fig. 2.

A variety of applications have been proposed for RNPs such as thermoset recycling (Montarnal et al., 2011), self healingmaterials (Wojtecki et al., 2011; Mauldin and Kessler, 2010; Long et al., 2010a; Adzima et al., 2010; Chen et al., 2003), novelsensors and actuators (Kloxin et al., 2011; Ryu et al., 2012; Long et al., 2011), and removable encapsulation materials forelectronics packaging (James et al., 2010; McElhanon et al., 2002). This last application motivates this work as such materialswere developed at Sandia National Laboratories to provide non-destructive, removable thermosetting encapsulates withtailored thermal and mechanical properties in both the glassy and rubbery regimes. For example, McElhanon and co-workers have demonstrated the non-destructive (non-contact) removal of a conforming RNP epoxy foam about electricalcomponents (James et al., 2010; McElhanon et al., 2002). Such demonstrations summarize the state-of-the-art with respectto RNP product realization. That is, RNP thermoset removability is demonstrated, but the interplay between permanentshape evolution and glassy polymer behavior on the in service component performance is lacking. These phenomenabecome especially important in encapsulation scenarios with complex geometries and thermal–mechanical histories sinceexperimental demonstrations of reliability may be inadequate. Hence, theoretical and computational design tools areneeded to support reliable RNP product realization.

Fig. 1. (a) Schematic of a dynamic covalent bond that connects and disconnects a chain through the forward and reverse reactions. This concept has beendemonstrated for a variety of stimuli (temperature, light, deformation, etc.). (b) A representative thermally reversible chemistry based on the Diels–Aldercycloaddition reaction. Here, furan and maleimide functionalities bond together to form an adduct that connects chain segments R’ and R”. When multiplereactive functionalities are incorporated in one or both of the monomers, then polymer networks are formed exclusively with reversible cross-links andfunctionalities along chains (Adzima et al., 2008). Permanent cross-links can also be incorporated into the network (McElhanon et al., 2002) in addition toreversible functionalities.

Fig. 2. Schematic of a representative volume element of a network polymer with thermally reversible and permanent covalent cross-links in three states.Here, the reversible linkages are assumed to be tetra-functional and serve both as monomers and cross-links. In the glassy state, physical entanglements,permanent covalent cross-links, and reversible cross-links connect chains. In the rubbery state, physical entanglements no longer play a role, and some ofthe thermally reversible linkages dynamically break and reform, which rearranges the connectivity of the network over time even as the permanentlinkages remain unchanged. In the viscous liquid state (above the gel-point), a large fraction of the reversible linkages are broken such that no single chainspans the representative volume element. In this state, the material may be removed.

K.N. Long / J. Mech. Phys. Solids 63 (2014) 386–411388

The objective of this paper is to meet this engineering need through the development of a thermal–chemical–mechanicalconstitutive model useful to describe the in-service behavior of RNP encapsulation that is subjected to a large range ofthermal and mechanical environments. We assume that the removability process above the gel point transition temperaturecan be sufficiently described by experiments such that computational tools are not required in this material regime. Instead,we focus on modeling the solid mechanics response of RNP materials to environments below their gel point temperatures.We are particularly interested in distinguishing between RNP and conventional (non-removable) thermoset encapsulation,which is a standard class of materials used in electronics packaging. Such differences may be especially important whenconsidering thermal mismatch strains between the polymer encapsulation and other components on printed circuit boardssubjected to thermal cycling.

While RNPs are relatively new materials, there have been several efforts to model thermosets with dynamic bonds. Invulcanized rubber systems, Tobolsky and co-workers identified a threshold temperature above which the permanent shapeof the network evolves as cross-links break and reform (Andrews et al., 1946). Tobolsky proposed a two-networkrepresentation by which the permanent shape evolution was accommodated through the parallel decomposition of thepolymer network into two volume fractions stress free two different configurations. Flory's (1960) statistical mechanicsapproach, which assumed Gaussian chain statistics, confirmed the validity of the two network hypothesis when cross-linksare added but not removed while the deformation applied to the network is fixed. If linkages are also removed, Florypreserved the two-network approach via a volume fraction transfer function between the networks formed in the twostates of deformation. Molecular dynamics simulations (Rottach et al., 2004) confirmed Flory's analytic work and examinednon-Gaussian chain statistics scenarios as well though again for a fixed state of network deformation. Several continuumconstitutive modeling efforts (Wineman and Shaw, 2002; Septanika and Ernst, 1998; Shaw et al., 2005; Rajagopal andSrinivasa, 1998) followed Flory's approach of multiple parallel networks based on multiplicative splits of the deformationgradient to account for different stress-free configurations of each network. Typically, these models focused on two states ofdeformation, an undeformed and a fixed deformed state. However, these approaches are computationally difficult to pursuewhen continuous deformation and network evolution occur. Moreover, none of the previous work considered the interplaybetween glassy network behavior and relaxation processes due to network evolution, and these previous works do notconsidered details of the Diels–Alder chemical equilibrium, kinetics, and coupling with the material state. Hence, we pursuea different modeling approach based on the recently developed polymer curing work (Adolf and Chambers, 2007).

There are several novel features of the model developed in this work. The thermal, chemical, and mechanical responsesare derived from a Helmholtz functional consistent with equilibrium and non-equilibrium thermodynamic requirements,which allows for the essential coupling between the Diels–Alder chemistry, the permanent shape evolution, thermal andmechanical properties, and glassy viscoelastic behavior. Furthermore, the permanent shape evolution and mechanical freeenergy are related to an additive split of the Hencky strain tensor into stress producing and stress free parts, which isdistinct from the polymer curing work of Adolf and Chambers through the reversible chemistry and its coupling with thematerial state. It is also quite simple compared with models based on the multiplicative split of the deformation gradientinto stress producing and stress free parts (Long et al., 2013, 2009; Andrews et al., 1946; Flory, 1960; Shaw et al., 2005; Simo,1988). The approach taken in this work is simple, effective, and in fact reproduces a multiplicative split of the deformationgradient under principal motions.


The paper is laid out as follows. First, we select a model experimental system from the literature and discuss basicexperimental observations in the context of the Diels–Alder chemistry, network topology evolution, and associated stressrelaxation. Next, we develop the thermal–chemical–mechanical constitutive model and the associated kinetics of thechemistry, glassy behavior, and evolution of the permanent shape due to changes in the network topology. The specificforms of the balance laws that govern its behavior are also presented. We then examine special cases of the model behaviorthat admit semi-analytic solutions, present validation results of the model against limited experimental data, and finallycompare the behavior of removable vs. conventional encapsulation materials under simple thermal–mechanical cycling.

2. Experimental observations

We briefly discuss experimental results developed by Bowman and co-workers (Adzima et al., 2008) that guidetheoretical and validation efforts in this paper. They form a network which is cross-linked with tri-furan and bismaleimidefunctionalities capable of undergoing the (retro-)Diels–Alder (DA) reaction as shown in Fig. 1b. Specifically, they polymerizepentaerythritol propoxylate tris(3-(furfurylthiol)-propionate) (PPTF) and 1,1'-(methylenedi-4,1-phenylene)bismaleimide(DPBM) by mixing them in a 1:1 furan-to-maleimide molar ratio, heating the mixture for 5 min at 155 1C to completethe step-growth reaction, and then cooling the material to room temperature. They observed a solid material that visiblyreverted to a liquid above 110 1C and vitrified below 45 1C. The furan and maleimide functionalities are analogous to the twodifferent geometric symbols in Fig. 2 and may reversibly bond together to form a Diels–Alder adduct.

Using Fourier Transform Infrared Spectroscopy (FTIR), Bowman and co-workers, measured the equilibrium extent ofreaction as a function of temperature, which is defined as the adduct concentration normalized to its maximum possibleconcentration (assuming all species are bonded). The equilibrium extent of reaction vs. temperature is reproduced in Fig. 3a.They additionally measured both the forward and reverse reaction kinetics and fit their respective behaviors following asimple, single reaction step, thermal–chemical reaction kinetics. Using their calibrated thermal–chemical kinetic results, wecalculate the half-life of a DA adduct in Fig. 3b, which characterizes both the equilibrium and the non-equilibrium rate atwhich DA linkages in the network break (and, in equilibrium, reform). Hence, this chemical half-life controls the rate atwhich the permanent shape of the material evolves in the absence of vitrification. The advantage of using this system overmore complicated DA thermoset chemistries formulated with both reversible and permanent cross-linking species is thatthe polymer structure and dynamics are simpler. The key experimental observations that we wish to capture through thedevelopment of a thermal–chemical–mechanical constitutive equation are the following:

�

Filinw

The dependence of the material's permanent shape change (network topology evolution) on the DA chemical kinetics.
� The relationship between the rubbery shear modulus and the extent of the DA reaction. � Thermodynamic properties. � Non-equilibrium (Glassy) behavior.
20 40 60 80 1000.5

0.6

0.7

0.8

0.9

1

Temperature (C)

Exte

nt o

f Rea

ctio

n

Rubbery Regime

Vis

cous

Liq

uid

Gla

ssy

Reg

ime

20 40 60 80 100100

101

102

103

104

Temperature (C)

Tim

e (m

in)

1 day~1,440 min1 week~10,080 min

Gla

ssy

Reg

ime

Rubbery Regime

Vis

cous

Liq

uid

g. 3. The equilibrium extent of reaction (with the exception of points below the glass transition) (a) and the chemical half-life of a bonded Diels–Alderkage (b). The data is partially reproduced from Adzima et al. (2008). Note that even at room temperature, half of the linkages will break and reformithin a week!


3. Constitutive model development

We seek to develop a constitutive model for network polymers with thermally reversible linkages in a continuumsetting. The purpose of this model is to inform the use of such materials in electronics packaging. Since the networktopological rearrangement occurs at a length scale smaller than that of encapsulation, which is our main application driver,we consider our continuum approach reasonable and model the bulk behavior of the material. We develop the model in thefollowing order. We briefly discuss kinematics including non-traditional invariants later used in the free energydevelopments, and then we summarize the mass, species, and momenta balance laws applied to this material. We thenexamine the first and second thermodynamic laws and derive quantities and constraints for the model. Next, we model theDiels–Alder chemistry and thermal–chemical kinetics, which we subsequently use to model the rubbery shear modulusdependence on the extent of reaction as well as the rate and form of the network topology evolution (permanent shapechange). Finally, we present the equilibrium and non-equilibrium Helmholtz free energy densities and calculate associatedthermodynamic sensitivities and the equation of motion for the temperature field.

3.1. Kinematics

We begin a discussion of kinematics and disclose the notation used throughout the rest of the text. Consider ahomogenous body composed of a network polymer with reversible linkages which initially occupies the volume Ω0 and hasa boundary, ∂Ω0. This initial configuration is taken to be the time-independent reference configuration. The position of amaterial point within Ω0 is denoted by Xj, where we use Einstein's index notation. We restrict constitutive modeldevelopment to be within Cartesian frames of reference both to avoid the distracting complexities of spatially varyingmetric tensors and for ease of implementation in standard finite element programs. The motion of a material point from itsposition, Xj, in the reference configuration to its position, xi, in the current configuration, wherein the body occupies thevolume Ω with an associated boundary ∂Ω is assumed to be a smooth, bijective mapping, so that the inverse mappingalways exists. The motion, displacement field ui, material velocity, deformation gradient, and the volume ratio between thetwo configurations are

xi ¼ χ Xið Þ ¼ Xiþui; vi ¼ _xi ; Fij ¼∂xi∂Xj

; J ¼ det Fij� �

40; ð1Þ

here the overhead dot denotes the time derivative of a material point quantity. The notation used in this work distinguishesbetween reference and current configuration quantities via upper case and lower case letters respectively. For somequantities, however, the reference configuration is distinguished with an underscore 0. For example, the body in thereference and current configuration occupies the volume Ω0 and ðΩÞ, respectively.

We take the polar decomposition of the deformation gradient to obtain the rotation, Rik, and material stretch tensor, Ukj,and take the logarithm of the latter to generate the logarithmic (Hencky) strain, which is defined in the referenceconfiguration:

Fij ¼ RikUkj; Υ ij ¼ logðUijÞ: ð2Þ

As is standard, we may additively split the Hencky strain into its spherical and deviatoric parts:

Υ spij ¼ Υkk

3δij; Υdev

ij ¼ Υ ij�Υ spij ; ð3Þ

here we have employed the Kroenecker delta, δij, to represent the identity tensor.We use two invariants of the Hencky strain in the Helmholtz free energy function that are distinct from the standard

Cayley–Hamilton invariants. Advantages of this choice have been discussed in other work relevant to polymer constitutivemodeling (Caruthers et al., 2004). We define the following two invariants of the Hencky strain tensor (using the propertiesof logarithms):

I1½Υ ij� ¼ Υ ii ¼ logðJÞ ð4Þ

I2 Υ ij� �¼ Υ ijΥ ij ¼ Υdev

ij Υdevij þ I1½Υ ij�2

3: ð5Þ

To separate invariant volumetric and deviatoric kinematics, we will use I1½Υ ij� and I2 Υdevij

h i¼ I2 Υ ij

� ��13I1½Υ ij�2.

3.2. Mass, species, and momentum balances

We briefly state the balance laws to provide a setting for constitutive model development. The chemical species mustsatisfy individual conservation statements. Consider a particular species, labeled α, with number density defined in thereference configuration, Nα. The rate of change of the total number of species α within a subregion of the body, ω0 �Ω0 with


the boundary ∂ω0, is

_Zω0

Nα dV ¼ �Z∂ω0

Hαi Ni dAþ

Zω0

Hα dV ð6Þ

here Hαi , H

α, and Ni denote respectively the reference configuration species flux (number per area time), species source(number per volume), and unit normal vector associated with the differential reference area, dA. The overhead dot denotes amaterial time derivative, which commutes with the integral over the time-independent reference volume. Thus, with thedivergence theorem, the local continuity equation for species density α in the reference configuration is

_Nα ¼ �∂Hα

i

∂XiþHα: ð7Þ

Since the species concentrations change in time and space, the referential mass density at a material point may alsochange in time unless diffusion of all species is negligible. Using Eq. (7), the balance of mass at a material point in thereference configuration is

Dρ0Dt

¼∑αMα _Nα ¼∑

αMα �∂Hα

i

∂XiþHα

� �; ð8Þ

here ρ0 denotes the mass density in the reference configuration, while Mα represents the mass per particle of species α. Thesum in Eq. (8) is over all species at the material point and includes chemically inert portions of the polymer chains, which donot change in time.

Finally, we require the balance of linear momentum, which we write in the spatial configuration and ignore inertia terms:

∂sij∂xi

þbj ¼ 0j; ð9Þ

here sij and bj represent the Cauchy stress and body force vector defined per unit spatial volume respectively. We assume anabsence of micro-polar moments, so that the angular momentum balance restricts the Cauchy stress to be symmetric;a detail is employed in Eq. (9). Later, we will need stress and strain rate measures defined in the reference configuration, and sowe introduce the Second Piola–Kirchoff stress and its work conjugate Green–Lagrange strain as well as the First Piola–Kirchoffstress, which is work conjugate to the material time derivative of the deformation gradient:

Sij ¼ JF �1ij sjkF

�1lk ; Eik ¼ 1

2 FjiFjk�δik� �

; Pik ¼ JsijF�1kj : ð10Þ

3.3. Energy balance

We treat the body as a homogenous, single phase material that contains a total number of molecules of each chemicalspecies. Neglecting kinetic energy, the time rate of change of the total energy in the body is composed of three quantities:the rate of mechanical work the body does against its surroundings ð _WoutÞ, the rate of chemical work that occurs as specieschange ð _EspeciesÞ, and the rate of thermal energy flowing into or generated within the body ð _Q inÞ. The rate of total internalenergy change in the body can then be written as

_Etotal ¼ _Q total in� _Wby systemþ _Especies ð11Þ

here we are considering a closed system and neglect species transport across the boundary of the body although thermaland mechanical energies can be exchanged between the body and its surroundings. From Eq. (11) and following Gurtin et al.(2010), we state the local form of the energy balance, with an internal energy density defined per unit reference volume, ε0.Again, consider a subregion of the body in the reference configuration, denoted as ω0 �Ω0, with its boundary ∂ω0. The rateof change of the total internal energy in this region is

_Zω0

ε0dV ¼ �Z∂ω0

QiNi dAþZω0

Q dVþZ∂ω0

PijNjvi dAþZω0

Jbivi dVþ∑α

�Z∂ω0

μαHαi Ni dAþ

Zω0

μαHα dV� �

ð12Þ

here Qi and Q represent the referential thermal flux vector (energy per area time) and thermal source (energy per volumetime) respectively. The energy associated with changing the referential number density of species α is characterized by thechemical potential, μα (energy per number of species α). Employing the divergence theorem, the time-invariance of thereference configuration, the referential species conservation statements (Eq. (7)), and the fact that the size of the subregioncan be made arbitrarily small, the local form of the energy balance is

_ε0 ¼ �∂Qk

∂XkþQþSEij _Eijþ∑

αμα _N

α�Hαi∂μα

∂Xi

� �; ð13Þ

wherein we have taken advantage of the fact that the internal stress power with respect to the First Piola–Kirchoff stress canalso be written in terms of the Second Piola–Kirchoff stress and the Green–Lagrange strain (Pij

_F ij ¼ SEij _Eij).


3.4. Entropy production inequality

We use the Clausius–Duhem form of the second law of thermodynamics to enforce entropy production between any twostates of the material. Ultimately, this requirement restricts the evolution of thermodynamic potentials and state variablesthat describe material at hand. In the reference configuration at a material point, the Clausius–Duhem entropy productioninequality is

_η0 �QΘþ1Θ

∂Qi

∂Xi�Qk

Θ2

∂Θ∂Xk

Z0; ð14Þ

where Θ is the absolute temperature. By combining the local forms of the energy balance (Eq. (13)) and the entropyproduction inequality (Eq. (14)), we arrive at a constraint on both the entropy and energy density evolutions:

Θ _η0 � _ε0 þSEij _Eij�Qk

Θ

∂Θ∂Xk

þ∑α

μα _Nα�Hα

i∂μα

∂Xi

� �Z0: ð15Þ

The natural thermodynamic state variables for the internal energy density are the entropy, deformation gradient, andspecies number densities. It is more convenient in the context of numerical implementation to choose a thermodynamicpotential that uses temperature, rather than entropy, as a natural variable. Thus, the Helmholtz free energy density, definedper unit reference volume, is preferred and follows from the standard Legendre transform:

Ψ ¼ ε0�Θη0; ð16Þand upon taking the material time derivative of Eq. (16) and substituting the result into Eq. (15), we arrive at the principalinequality of rational mechanics (PIRM) (Coleman and Gurtin, 1967):

_Ψ þη0 _Θ0�SEij _EijþQk

Θ

∂Θ∂Xk

�∑α

μα _Nα�Hα

i∂μα

∂Xi

� �r0; ð17Þ

which shows that the time evolution of the thermodynamic state occurs in the direction that minimizes the free energydensity (Gurtin et al., 2010).

If we assume in general for reacting solid materials that the Helmholtz free energy is a function of Eij, Θ, Nα, and

fZ1; Z2;…Zng, where Zβ is the β indexed internal state variable that may not be directly measurable, then the material timederivative of the Helmholtz free energy density (per unit reference volume) is

_Ψ ¼ ∂Ψ∂Eij

_Eijþ∂Ψ∂Θ

_Θþ∑α

∂Ψ∂Nα

_Nα

� �þ∑

β

∂Ψ∂Zβ

_Zβ

� �: ð18Þ

Following the Coleman and Noll (1963) procedure, we collect terms associated with the rate of change of eachthermodynamic variable ( _Eij, and _Θ, and _N

α). Then, to guarantee satisfaction of the PIRM (Eq. (17)) under arbitrary

variations in these variables, we are restricted to define the stress, entropy, and chemical potential of each species as

Sij ¼∂Ψ∂Eij

; η0 ¼ �∂Ψ∂Θ

; μα ¼ ∂Ψ∂Nα: ð19Þ

It should be noted that the free energy descriptions in subsequent sections will depend on the first two invariants of thelogarithmic strain tensor as defined by Eqs. (4) and (5), and thus, to recover the Second Piola–Kirchoff stress, the followingtransformation is needed (Caruthers et al., 2004):

SEij ¼∂Ψ∂Υab

∂Υab

∂Eij¼ SΥab

∂Υab

∂Eij: ð20Þ

The fourth rank transformation between the logarithmic and the Green–Lagrange strains can be derived by consideringdifferential changes to both principal stretches and directions but is not presented here for brevity. In Eq. (20), SΥab is thestress work conjugate to differential changes to the logarithmic strain, and so we call this quantity the Hencky stress. Theremaining terms in the PIRM collectively represent a constraint on the rate of free energy dissipation that occurs while statevariables are held constant:

∑β

∂Ψ∂Zβ

_Z β

� �þ∑

αHα

i∂μα

∂Xi

� �þQk

Θ

∂Θ∂Xk

r0: ð21Þ

In general, the internal state variables ðZβÞ, species fluxes ðHαi Þ, and thermal energy flux (Qk) may vary independently. So, to

guarantee that the PIRM is satisfied for arbitrary changes of these variables, we further restrict their associated timeevolutions by requiring that each term in Eq. (21) satisfies the inequality separately:

∑β

∂Ψ∂Zβ

_Z β

� �r0; ∑

αHα

i∂μα

∂Xi

� �r0;

Qk

Θ

∂Θ∂Xk

r0: ð22Þ

The choice of evolution rules for each internal variable, Zβ , and constitutive relationships for the species flux, Hαi , and heat

flux, Qi, must be chosen to satisfy the constraints imposed in Eq. (22) to guarantee consistency with the PIRM. Theseconstraints are discussed in accordance with the constitutive model developed in subsequent sections.


3.5. Equation of motion for the temperature field

We define the heat capacity per unit mass of the material at a constant state of deformation and relate it to the secondorder sensitivity of the Helmholtz free energy per unit mass through the Legendre transform, Eq. (16), and the entropydensity relationship in Eq. (19):

ρ0CF ¼∂ε0∂Θ

� �¼ �Θ

∂2Ψ∂Θ∂Θ

� �ð23Þ

here the overbar indicates a per unit mass quantity. We derive the equation of motion for the temperature field by using therelationship between the entropy and Helmholtz free energy densities, Eq. (19), the material time derivative of theHelmholtz free energy, Eq. (18), the Legendre transform, Eq. (16), the energy balance, Eq. (13), and the specific heat capacity,Eq. (23), to arrive at (the heat equation):

ρ0CF _Θ ¼ Q�∂Qk

∂Xkþ∑

αΘ

∂2Ψ∂Θ ∂Nα

_Nα�Hα

i∂μα

∂Xi

� �þΘ

∂2Ψ∂Θ ∂Eij

_Eijþ∑β

Θ∂2Ψ

∂Θ ∂Zβ�∂Ψ∂Zβ

� �_Zβ: ð24Þ

The heat equation is specified in subsequent sections to the constitutive model developed in this work.

3.6. Species and thermal energy transport

Typically species and heat fluxes are constitutively specified to scale with the spatial gradients of the chemical potentialand temperature respectively. We assume here that isotropic conditions occur so that in the current and referenceconfigurations, the species and heat fluxes are constitutively specified as

hαi ¼ �Dα∂μα

∂xi; qi ¼ �κ

∂Θ∂xi

; ð25Þ

Hαi ¼ �DαJC�1

ij∂μα

∂Xi; Qi ¼ �κJC�1

ij∂Θ∂Xi

ð26Þ

here Dα and κ are the isotropic α-species and heat diffusion coefficients, defined in the current configuration and arematerial constants. With these constitutive rules, it is simple to show that the term-by-term requirements on the material'sfree energy dissipation rate are satisfied for the thermal and species diffusion terms, Eq. (22)(2 and 3).

For the system considered in this work, we simplify our treatment by neglecting diffusion of the Diels–Alder species, A, F,and M so that Dα ¼ 0 for all α. This simplification is reasonable because chains that have been broken at F and Mfunctionalities are still covalently tethered to the network. Thus, unless whole segments of chains are broken from thenetwork, the individual species cannot diffuse long distances. Hence, we neglect diffusion of all of the chemical species.Under this simplification, the species number density conservation statements, Eq. (7), reduce to

_Nα ¼Hα: ð27Þ

Therefore, the time evolution of species densities is related only to source terms, and so these conservation statementsreduce a system of ordinary differential equations at each material point rather than partial differential equations across thebody (when diffusion occurs). As a consequence of Eq. (27), the mass continuity equation, Eq. (8), reduces to a constantreferential mass density and hence to the familiar form that ρ0 ¼ Jρ.

3.7. Diels–Alder chemistry

We now describe the chemical reaction involved in this system as the species densities, and their associated chemicalpotentials, will directly enter into the free energy density that we develop. As discussed in Section 1, we are considering athermally reversible chemical reaction of the form:

FþM2A ð28ÞThis reaction is responsible for connecting chains (forward reaction) or breaking them apart (reverse reaction), and so, thenumber density of chains and shear modulus will scale with the extent to which the reaction is in the bonded (forward)state. The total species referential number density, which should not be confused with the reference configuration surfacenormal vector, is

N¼NAþNFþNM ð29ÞThe total number density, N, in Eq. (29) is not a conserved quantity, but assuming that Eq. (28) is the only chemical reactionoccurring in the system and that there is no species diffusion (see Section 3.6), there is a conserved species density, whichrepresents the maximum referential density that any species can obtain, which is defined as

ϕ¼NAþ12 NFþNM�

: ð30Þ


With ϕ, we can define the extent of the chemical reaction:

NA ¼ ϕx; xA ½0;1�: ð31ÞIn the experimental system from the literature that will be modeled in this work, the polymer network is formulated withan equal mole fraction of furan and maleimide species, and since they bond in a 1:1 ratio (from Eq. (28)), the numberdensities of furan and maleimide species are the same in this work. Therefore, we may write the number densities in termsof the extent of reaction and the total conserved species density, ϕ:

NF ¼NM ¼ ϕð1�xÞ; N¼ ϕð2�xÞ: ð32ÞFor the experimental system considered in this work, Bowman and co-workers measured the equilibrium extent of

reaction as a function of temperature in a state of constant stress. Specifically, the specimens in their studies were notsubjected to any applied mechanical stresses other than ambient pressure. Under such conditions, the Gibbs free energy ofthe system is the natural thermodynamic potential with which to consider the equilibrium conditions for the reaction. Wedefine the reaction equilibrium constant following the stoichiometric relationship of the chemical reaction (Eq. (28)):

K1 Θ½ � ¼ NA1=N

ðNF1=NÞðNM

1=NÞ¼ xð2�xÞð1�xÞ2

; ð33Þ

wherein ”1” indicates that the quantities have equilibrated at the temperature, Θ, and do not change in time. Note that thisequilibrium constant definition is different compared with the one used by Bowman and co-workers, who use a definitionthat is not dimensionless (Adzima et al., 2008). By empirically examining the temperature dependence of K1½Θ� in Eq. (33),Bowman and co-workers calculated the standard enthalpy and entropy of the reaction via the Van't Hoff equation:

log K1 ¼ �Δg○rxnRΘ

¼ ��ΔH○rxn

RΘþΔη○rxn

R; ð34Þ

wherein R denotes the universal gas constant, and Δg○rxn, ΔH○rxn, and Δη○rxn denote the standard changes in Gibbs free energy,

enthalpy, and entropy, respectively, due to the reaction. These quantities determine the associated quantity change that isaccompanied when one mole of the products (species A) is formed from the reactants (species F and M) under standardconditions. To enforce the same chemical equilibrium temperature dependence while using Eq. (33), we modify thestandard reaction enthalpy and entropy.

Under a constant state of stress and at a constant temperature, the equilibrium condition for the material is that theGibbs free energy is at a minimum so that its total differential is zero

0¼ dgjΘ;SΥij ¼ Υ ij dSΥij �η0 dΘþ ∑

α ¼ A;F;Mðμα dNαþdμαNαÞ ¼ ∑

α ¼ A;F;MðμαdNαÞ; ð35Þ

where we have used the fact that under the equilibrium conditions considered here, dSΥij ¼ 0ij, dΘ¼ 0, and the Gibbs–Duhemequality:

∑α ¼ A;F;M

ðdμαNαÞ ¼ �Υ ijdSΥij þη0dΘ¼ 0: ð36Þ

Eq. (35) gives the expected statement of chemical equilibrium at a fixed temperature and stress. To produce thetemperature-dependent equilibrium constant, Eqs. (33) and (34), we model the chemical potentials for each species as

μA ¼ μA○þRΘ log aAð Þ; aA ¼NA

N; ð37Þ

μF ¼ μF○þRΘ log aFð Þ; aF ¼NF

N; ð38Þ

μM ¼ μM○þRΘ log aMð Þ; aM ¼NM

N; ð39Þ

here aα represents the ideal activity of species α. Non-ideal behavior is captured by the temperature dependent standardchemical potentials of each species, μα○. From the condition of equilibrium at a constant stress and a temperature, Eq. (35),the equilibrium constant relationships (Eqs. (33) and (34)), and the chemical potentials, the Gibbs–Duhem equality issatisfied (Eq. (36)) and that

μA○�μF○�μM○ ¼Δg○rxn: ð40ÞUsing the extent of reaction variable, x, a single chemical potential may be used to characterize the thermodynamic forceassociated with changes to the chemical state of the system:

μðxÞ ¼ ∂g∂x: ð41Þ

The condition of chemical equilibriumwith respect to the extent of reaction is μðxÞ ¼ 0, which reproduces the previous statementof chemical equilibrium, Eq. (40). However, via Eq. (41), we have a convenient way to include the effects on chemical equilibrium


of coupling terms in the free energy. That is, we may include couplings between the extent of reaction, x, the absolutetemperature, Θ, and the first and second invariants of the logarithmic strain, I1Υ and I2Υ respectively. If such terms are included,then the condition of chemical equilibrium may differ from Eqs. (40) and (41).

3.8. Diels–Alder reaction kinetics

Our objective here is to determine the rates of the forward and reverse Diels–Alder reactions (Eq. (28)), which is relatedto the rates at which the number density of network chains is increasing and decreasing. The forward and reverse reactionsdetermine the source terms in the species balance laws, Eq. (7), and so without diffusion, the reaction kinetics determine therates of change of the species densities directly; see Eq. (27). Recall that an equal stoichiometry of furan and malemide wasused in the system under examination (see Eq. (32)). Following Bowman and co-workers, we model that the Diels–Alderreaction kinetics via a second-order thermally activated model so that the referential species densities along with theconservation requirement, Eq. (30), obey in the following kinetics:

_NF ¼ _N

M ¼ �kf NFNMþkrNA; _NA ¼ kf NFNM�krNA: ð42Þ

These kinetics are a manifestation of the law of mass action, which is reasonable here since we expect that the Diels–Alderreaction occurs as a single step reaction. Here, kf and kr are the temperature dependent forward and reverse reaction rateconstants respectively. The first term in Eq. (42)(1) represents the rate at which the number density of chains is increasingdue to the forward reaction, while the second term gives the rate of decrease of the number density of chains associatedwith the reverse reaction. We may relate the reverse and forward reaction rates to the rate of change of the extent ofreaction from Eqs. (31), (32) and (42)

_x� ¼ krϕx; _xþ ¼ kfϕð1�xÞ2; _x ¼ _xþ � _x� : ð43ÞAt equilibrium, the species evolutions are zero, so that the forward and reverse rate constants are related to the

equilibrium constant via

kf

kr¼ K1½Θ�

ϕ: ð44Þ

Bowman and co-workers assumed that these rate constants are thermally activated, so that, along with Eq. (44), they arespecified as

kr ¼ k0 exp�EactRΘ

�; ð45Þ

where the prefactor, k0, and activation energy, Eact, are determined experimentally. However, these reaction kinetics do notaccount for the arresting effects of vitrification. Following the previous work (Adolf and Chambers, 2007; Martin and Adolf,1990; Adolf and Martin, 1990), we assume that the thermally activated rate acts in parallel with the material (viscous) timescale of the network provided that this viscous time scale is dilated compared with the laboratory time scale. That is, theglassy behavior can slow the Diels–Alder reaction kinetics

kr ¼ k0

1þaexp

�EactRΘ

�; ð46Þ

here 1=ð1þaÞ represents the material time scale dilation (shift) factor from viscoelasticity. In the rubbery state, a51, so thatthe chain relaxation kinetics do not inhibit the Diels–Alder kinetics. Below the glass transition, ab1 so that effectively, theshift factor arrests the Diels–Alder kinetics. The viscoelastic shift factor and material time are discussed further insubsequent sections.

3.9. Rubbery shear modulus dependence on the chemical species

In the flexible chain limit of rubber elasticity, the shear modulus of the network depends linearly on the number densityof chains beyond the gel-point extent of reaction as well as linearly on the absolute temperature (Rubenstein and Colby,2003). Near the gel point transition, the number density of chains is low relative to network topologies at larger extent ofreactions so that the connection between the rubbery shear modulus, extent of reaction, and temperature should obey theflexible chain limit behavior (Gaussian chain statistics). To keep model parameterization simple, we assume that this linearscaling is valid further away from the gel point transition. Hence, we model the equilibrium shear modulus via

G1 Θ; x½ � ¼ G1refΘðx�xgelÞ

Θref ðxref �xgelÞ¼ GΘ x�xgel

� �; ð47Þ

which allows us to calibrate the shear modulus the experimentally measured value at a reference temperature, G1ref . Note

that G lumps the reference property material constants into one variable for convenience. The gel point extent of reaction,xgel, is a fundamental geometric/topological property of the network and does not depend on temperature, state ofstress, etc.


Under conditions of chemical equilibrium, the rubbery shear modulus in Eq. (47) depends non-linearly on temperaturesince the equilibrium the extent of reaction depends exponentially on temperature. Hence, this shear modulus representa-tion may vary substantially more over a given temperature range than a conventional thermoset. Mechanistically, thismodel feature is sensible since in thermosets with thermally reversible functionalities, the number density of chainschanges is not constant.

However, in the short chain limit, the shear modulus of the network scales non-linearly with the number density ofchains. For example, for epoxy networks cured via a step-growth reaction with di-functional epoxy monomers, Adolf andChambers (2007) report that the shear modulus scales with the curing extent of reaction via

G x½ � ¼ G x¼ 1½ �ðx2�x2gelÞ2:7

1�x2gel

!; ð48Þ

here xgel is the curing extent of reaction when the network has reached the percolation limit (gel point). The temperaturedependence is not included in Eq. (48), and it need not be linear in temperature. For simplicity, we adopt the ideal shearmodulus and temperature dependence scaling of Eq. (47) and recognize that this choice may be a substantialoversimplification.

Using the reaction kinetics written in terms of the reaction extent, Eq. (43), and the shear modulus dependence on theadduct species density, Eq. (47), the material time derivate of the shear modulus takes the following form:

_G1 ¼ G _Θðx�xgelÞþGΘ _x ¼ G _Θðx�xgelÞþ _G1þ � _G

1� ; ð49Þ

_G1� ¼ GΘϕkrx; _G

1þ ¼ GΘϕkf ð1�xÞ2; ð50Þ

here _G1þ and _G

1� denote the rates of increase and decrease of the shear modulus due to the addition and destruction of the

number density of chains respectively. As the material vitrifies, these rates of increase and decrease of the shear modulusdue to changes in the number density of change are arrested since they depend on the forward and reverse rate constantsfrom the thermal chemistry and the viscoelastic shift factor from Eq. (46).

3.10. Evolution of the stress-free configuration

Next, we discuss the effects of network topology evolution on the stress-free shape of the body, which provides the network amechanism to take on a new permanent shape. Phenomenologically, we capture this behavior through the time evolution of thestress-free strain tensor internal state variable following the polymer curing work of Adolf and Chambers (Adolf and Chambers,2007). We note that considerable theoretical and computational efforts have explored the concept of stress-free configurations asa means of explaining compression set in elastomers. Tobolsky proposed the Two-Network hypothesis (Andrews et al., 1946),examined theoretically by Flory (1960), and computationally by Rottach et al. (2007). This approach has been applied withsuccess to a variety of problems associated with the continuum thermomechanical behavior of temperature sensitive elastomers(Shaw et al., 2005; Wineman and Shaw, 2002) and photo-mechanically coupled polymers (Long et al., 2009). The Two-Networkapproach is based on a multiplicative decomposition of the deformation gradient:

Ftotalij ¼ F2ikF1kj; ð51Þ

such that the elastic free energy of the first and second network volume fractions depend on Ftotalij and F2ik respectively. Forexample, often in the literature, the total elastic free energy of a two-network material is

Ψ elasticF2F1 ¼ v1Ψ ðF2ikF1kjÞþv2Ψ ðF2ijÞ ð52Þ

There are two issues with this multiplicative split. First, the stress-free configuration of the material depends on the specificconstitutive functions involved ðΨ ð�ÞÞ, and second, it is theoretically and computationally challenging to evolve the intermediateconfiguration F1kj during continuous deformation. Some work has looked into evolving the intermediate configuration forpolymers experiencing microstructural evolution (Ge et al., 2012; Long et al., 2010b; Rajagopal and Srinivasa, 1998), but wechoose a different and simpler approach.

As mentioned above, we model the effect of an evolving network topology by directly evolving a stress-free strain tensor,ξdevij , subject to the following assumptions:

�
Volumetric deformation, which is dominated by Van-der-Waals interactions in polymer networks, does not induce achange in the stress free configuration.
�
Deviatoric deformation moves chains past each other, so that as chains break and reform, they may do so in differentconfigurations. An aggregate number of such events reduces the elastic free energy.
�
The stress-free configuration may only evolve if it is different from the current state of deformation, Υdevij �ξdevij a0ij � The rate of change of the stress-free configuration scales with the rate of increase of the equilibrium shear modulus due
to the formation of chains, _Gþ, in Eq. (50).

�
The rate of decrease of the equilibrium shear modulus due to chain scission does not significantly affect the stress-freeconfiguration.


tensor is
Following Adolf et al. (2009) and these hypothesis, we model the material time derivative of the stress free strain
_ξdevij ¼

_Gþ

G1 Υdevij �ξdevij

� : ð53Þ

Although simple in form, Eq. (53) is affected both by the Diels–Alder chemistry and by the viscoelsticity through _Gþ.

Recall that the rate of shear modulus growth in Eq. (50) is driven both by the rate of formation of Diels–Alder linkages(through kf) and by the material time shift factor, 1=ð1þaÞ, from Eq. (46). Thus, there is a competing relationship betweenthe rate of Diels–Alder linkage formation and chain vitrification. As the material is cooled below the glass transition, thestress-free strain tensor ceases to evolve as the material time scale becomes much longer than chemical kinetics time scale.

One important inadequacy of this rule is that it does not account for the change of the stress-free configuration due tochain scission (assumption 4), which, as pointed out in many theoretical investigations, involves a load transfer from chainsas they are scissioned to the surrounding network (Rottach et al., 2007; Flory, 1960). However, these theoretical treatmentsconsider the special case of just 2 states of strain. Here, we are concerned with an arbitrary number of strainedconfigurations, and so in this work, we neglect the influence of scissioned chains on the evolution of the stress freeconfiguration.

Regarding objectivity, by construction of the evolution rule (Eq. (53)), the stress-free strain tensor is a deviatoric, secondrank material tensor and hence is indifferent to observer transformations. The stress-free strain tensor is taken to be thezero tensor at the start of an analysis. The fact that ξdevij is a material tensor is evident as follows. Initially, when zero, itevolves proportional to the deviatoric Hencky strain and hence is a material tensor, which maps a material vector to anothermaterial vector. Since addition is a linear operator, the difference of two material tensors is also a material tensor. Therefore,subsequent evolution of the stress-free strain tensor is proportional to a material tensor as well, and so the stress-free straintensor is a material tensor by construction. Material tensors are frame indifferent as is shown in standard continuummechanics texts. Hence, the model satisfies objectivity.

3.11. Additive split of the Helmholtz free energy density

From the experimental observations in Section 2, we develop a Helmholtz free energy density, consistent with the PIRM(Eq. (17)), to describe the thermal–chemical–mechanical behavior of network polymers with reversible linkages. We assumethat the Helmholtz free energy, per unit reference volume, is a function of the following state variables:

Υ ij
the logarithmic strain tensor Θ the absolute temperature Nα the referential number density of each chemical species, α¼ A; F;M. These number densities are related to the extent of
reaction definition, Eq. (31), and they obey their own conservation statements, Eq. (7)
ξdevij the stress-free strain tensor, which represents the evolving permanent shape of the material point due to the
topological rearrangement of the network

We model the material's referential Helmholtz free energy in two parts:

Ψ ðΥ ij;Θ;Nα; ξdevij Þ ¼Ψ1þΨ visco: ð54Þ

The first component, Ψ1, represents the equilibrium network response with respect to changes to the thermodynamic state.For example, from it, the rate independent heat capacity, stress, or chemical state can be derived. However, if the networktopology is changing, the equilibrium free energy will decay in time as stress is relaxed in the network. The second term,Ψ visco, represents a non-equilibrium free energy penalty that the material suffers when the thermodynamic state is changedtoo quickly relatively to its own internal time scale. Classically, this free energy penalty gives rise to viscous stresses, heatcapacities, and thermal expansion behaviors that distinguish the polymer's non-equilibrium glassy state compared with itsequilibrium rubbery state.

The additive split of the free energy density into equilibrium and non-equilibrium parts (Eq. (54)) results in an additivesplit of each of the generalized thermodynamic forces into equilibrium and non-equilibrium parts. This result is possiblesince the differential operators with respect to material coordinates, temperature, and species concentration are linearoperators and therefore are distributive with respect to addition. Thus, from Eqs. (54) and (22) the Second Piola–Kirchoffstress, the entropy, and the chemical potential of each species also split into equilibrium and non-equilibrium contributions

Sij ¼ SE1ij þSEviscoij ; η0 ¼ η10 þηvisco0 ; μðxÞ ¼ μðxÞ1þμðxÞvisco: ð55Þ

Analogously, the thermodynamic work conjugate fluxes associated with the internal state variables ðZβÞ also may be splitinto equilibrium and non-equilibrium parts.


3.12. Equilibrium free energy contributions

We first examine the equilibrium referential Helmholtz free energy density, which we further divide into four partsarising from elastic, thermal, chemical, and mixed free energy contributions:

Ψ1ðΥ ij;Θ;Nα; ξdevij Þ ¼Ψ1

elasticþΨ1thermalþΨ1

chemicalþΨ1mixedþΨ ref ; ð56Þ

Ψ1elastic ¼ Pref I1Υ þ

K1

2ðI1Υ Þ2þG x;Θ½ �I2 Υdev

ij �ξdevij

h i; ð57Þ

Ψ1thermal ¼ ρ0CF0 Θ�Θref �Θ log

Θ

Θref

� �� ρ0CF1

2ΘrefðΘ�Θref Þ2; ð58Þ

Ψ1chemical ¼∑

αðμαNαÞ; ð59Þ

Ψ1mixed ¼ �K1β1ðx�xref ÞI1Υ �Kα1ðΘ�Θref ÞI1Υ : ð60Þ

Throughout these equations, a reference state is referred to which characterizes the initial free energy density. The extent ofthe chemical reaction, x, is also used for convenience and is related to the referential species densities through Eq. (31). Thechemical potential, associated with each species, is modeled via Eq. (37)–(39). We summarize the material properties inTable 2.

It is important to note that a simple form of the elastic free energy density is used. For other systems, anotherconstitutive equation for the elastic free energy may be more appropriate. Also, note that the shear modulus, G½x;Θ�, isassumed to be a function of the extent of reaction as well as the absolute temperature as discussed in Section 3.9. There areseveral additional assumptions built into the equilibrium free energy density that will be discussed in subsequent sections.

Following Eqs. (19) and (20), we derive the equilibrium contribution to

SΥ1ij ¼ ∂Ψ1

∂Υ ij¼ 2G1 Υdev

ij �ξdevij

� þ Pref þK1I1Υ �K1β1 x�xref

� ��K1α1 Θ�Θref� ��

δij: ð61Þ

We also calculate the referential entropy density and chemical potential with respect to a change in the extent of reactionvariable, x, from Eqs. (19) and (41)

η10 ¼ ρ0CF0 logΘ

Θref

� �þρ0CF1

ΘrefΘ�Θref� ��G x�xgel

� �I2 Υdev

ij �ξdevij

h iþK1α1I1Υ ; ð62Þ

μðxÞ1 ¼ ϕ RΘ logxð2�xÞð1�xÞ2

!þΔH○

rxn�ΘΔη○rxn

!�K1β1I1Υ þGΘI2 Υdev

ij �ξdevij

h i; ð63Þ

where the chemical potentials (Eqs. (37)–(39)), standard reaction enthalpy and entropy densities (Eq. (40)), and equilibriumshear modulus behavior (Eq. (47)) have been used. Two interesting results emerge from the choice of the equilibriumHelmholtz free energy. First, the referential entropy density depends on the volumetric state of deformation and shearmodulus sensitivity with respect to temperature. However, these dependencies do not influence the specific heat capacity atconstant deformation from Eq. (23)(2) given the linear sensitivity of the shear modulus with respect to temperature fromEq. (47). Second, the condition of chemical equilibrium with respect to the extent of reaction, that μðxÞ ¼ 0, now includesdependencies on the volumetric deformation (I1), shear deformation (I2), and the shear modulus sensitivity with respect to x.Even under isothermal conditions, chemical equilibrium can be changed if the material is subjected to substantialdeformation.

Finally, we check that the evolution of the permanent shape of the body (stress-free strain tensor Eq. (53)) satisfies thefree energy dissipation requirements from the second law, Eq. (22),

∂Ψ1

∂ξdevij

_ξdevij ¼ �2 _G

þΥdevij �ξdevij

� Υdevij �ξdevij

� r0; ð64Þ

which is always satisfied since _Gþ

is always a positive quantity. Thus the evolution of ξdevij is a thermodynamicallyadmissible, irreversible process associated with the change in the equilibrium free energy contribution.

3.13. Non-equilibrium Helmholtz free energy contributions

Next, we develop the non-equilibrium free energy penalties that the system experiences when its thermodynamic stateis changed faster than its characteristic time scale. Here, we adopt the simplified potential energy clock model (SPEC)developed by Adolf et al. (2009) to model glassy thermosets. We summarize the pertinent results needed to represent theviscous behavior of polymers with removable linkages. We assume that the non-equilibrium contributions are sufficientlysmall that we may approximate them with functional Taylor expansions about the equilibrium state. We considerviscoelastic dependencies related to the time histories of the logarithmic strain different from the stress-free strain tensor


ðξdevij Þ, the absolute temperature, and the extent of reaction each to second-order along with certain combinations of cross-terms. Distinctively missing are cross-terms that couple the shear deformation history with the extent of reaction as well aswith the temperature history. As discussed in Section 3.10, we have neglected the coupling between the network topologyand volumetric responses. The non-equilibrium free energy contribution is

Ψ visco ¼ 12

KG�K1� Z t

0dsZ t

0du f 1 t′�s′; t′�u′ð ÞdI1Υ

dsdI1Υdu

þ GG�G1� Z t

0dsZ t

0du f 2 t′�s′; t′�u′ð Þ

dðΥdevij �ξdevij Þds

dðΥdevij �ξdevij Þdu

�ρ0ðCFG�CF11Þ2Θref

Z t

0dsZ t

0du f 5 t′�s′; t′�u′ð ÞdΘ

dsdΘdu

� KGαG�K1α1� Z t

0dsZ t


dsdΘdu

� KGβG�K1β1� Z t

0dsZ t


dsdxdu

þ ΔηrxnG�Δη○rxn� � Z t

0dsZ t

0du f 6 t′�s′; t′�u′ð ÞdΘ

dsdxdu

ð65Þ

The functions, ff kg, are different relaxation functions, expanded here as Proney series, which satisfy the dissipationinequality requirements (see Adolf, 2010)

f k t′�s′; t′�u′ð Þ ¼ ∑m

j ¼ 1Aj exp

�ðt′�s′Þτj

� �exp

�ðt′�u′Þτj

� �ð66Þ

The Proney series satisfy a normalization condition that f kð0;0Þ ¼∑mj ¼ 1A

j ¼ 1, and the arguments, t′�s′ and t′�u′, representthe difference in material time between t′ and s′. The material time is related to the laboratory time scale, t, through theviscoelastic shift factor, a, such that t′¼ t=a. The viscoelastic shift factor is an internal state variable which is itself a functionof the deformation, temperature, and reaction histories. The component pairs (τj, A

j) represent individual relaxation timesand weights associated with the relaxation spectrum fk. Typically, these experimentally determined relaxation spectra arefirst represented through stretched exponentials which are later fit with a least-squared projection onto the Proney basis inEq. (66). For example, the volumetric and shear relaxation spectra at the viscoelastic reference temperature, which may bechosen to be the glass transition temperature, are characterized by two parameters each ðτv; γvÞ and ðτs; γsÞ through

f v tð Þ ¼ exp�tτv

� �γv� �

; f s tð Þ ¼ exp�tτs

� �γs� �

; ð67Þ

here ðτk; γkÞ and ðτm; γmÞ are material constants and no sum is applied on these subscripts.To simplify matters, we assume that all relaxation spectra obey a common time–temperature superposition, which is the

statement rheological simplicity. This assumption may be invalid if the material is transitioned across its gel-point andbecomes a liquid. As an example, for a rheologically simple material, the complex shear modulus as measured from dynamicmechanical analysis obeys the following relationship:

Gnðω;ΘÞ ¼ GnðaΘω;Θref Þ ð68ÞFollowing the SPEC model, we use the phenomenological representation of the viscoelastic shift factor's dependence on thetemperature, shear, and volumetric deformation histories:

log a¼ �C1NC2þN

; ð69Þ

N¼ Θ�Θglass�Z t

0ds f v t′�s′;0ð ÞdΘ

ds

� �þC3 I1Υ �

Z t

0ds f v t′�s′;0ð ÞdI1Υ

ds

� �

þC4

Z t

0dsZ t

0du f s t′�s′; t′�u′ð Þ


dðΥdevij �ξdevij Þdu

; ð70Þ

here C1�4 are material constants that determine the sensitivity of the viscoelastic time scale with respect to temperaturechange ðC1;C2Þ, volumetric deformation (C3), and shear deformation ðC4). The material time scale can then be calculated as afunction of the thermal, volumetric and shear deformation histories through

t′�s′¼Z t

s

dzaðzÞ dz: ð71Þ

The viscoelastic shift factor also constitutes an internal state variable, and therefore its evolution must satisfy the second lawrequirement from Eq. (22). A general treatment and proof of the thermodynamic consistency of materials with memorybased on a material time scale was performed by Lustig et al. (1996).

From the equilibrium and non-equilibrium Helmholtz Free energy densities in Eqs. (56) and (65), the Hencky stress isderived, with the equilibrium term denoted by SΥ1ij , given by Eq. (61)

SΥij ¼∂Ψ∂Υ ij

¼ SΥ1ij þ KG�K1� Z t


dsδijþ2 GG�G1

� Z t

0ds f s t′�s′;0ð Þ



� KGαG�K1α1� Z t


dsδij� KGβG�K1β1

� Z t

0ds f v t′�s′;0ð Þdx

dsδij ð72Þ

Note that the stress free strain tensor ðξdevij Þ influences only mechanical quantities in both the equilibrium and non-equilibrium free energies. We again check if its evolution is consistent with the satisfaction of the dissipation inequality,Eq. (22). Proceeding as with Eq. (64), we find

∂Ψ visco

∂ξdevij

_ξdevij ¼ �2

_Gþ ðGG�G1Þ


� Z t



r0: ð73Þ

The prefactor associated with the shear moduli is always positive, but for arbitrary histories of ξdevij and Υdevij , it is not obvious

that the kinematic portion is always positive. We expect that as the material is deformed in the glassy state, the integratedhistory of Υdev

ij �ξdevij and the difference between these quantities will have the same sign for each component so that thedissipation requirement would be satisfied. However, once external inputs to the system are ceased, so that the free energyof the system evolves on its own accord (associated with the viscoelastic relaxation), it is possible that Υdev

ij �ξdevij and therate of Υdev

ij �ξdevij could have opposite signs (for all components). Specifically, we have not proven that Υdevij �ξdevij and the

integrated rate of Υdevij �ξdevij weighted by the relaxation function fs will have the same sign so as to satisfy the dissipation

requirement. We accept this issue as a potential shortcoming of the model.The referential entropy density and chemical potential associated with the extent of reaction are now given by

η0 ¼ � ∂Ψ∂Θ

� �¼ η1þ KGαG�K1α1

� Z t


dsþρ0

ðCFG�CF11ÞΘref

Z t


ds

� ΔηrxnG�Δη○rxn� � Z t

0du f v t′�s′;0ð Þdx

du: ð74Þ

μðxÞ ¼ μðxÞ1� KGβG�K1β1� Z t


dsþ ΔηrxnG�Δη○rxn� � Z t

0du f v t′�s′;0ð ÞdΘ

du: ð75Þ

3.14. The heat equation revisited

With the full model and thermodynamic machinery, we populate terms in the equation of motion for the temperaturefield, Eq. (24)

ρ0 CF0þCFGΘ

Θref

!_Θ ¼ Q�∂Qk

∂Xkþϕ RΘ log

xð2�xÞð1�xÞ2

!�ΘΔη○rxn

!_xþΘ GI2 Υdev

ij �ξdevij

h iþ ΔηrxnG�Δη○rxn� ��

_x

þ2G1 Υdevij �ξdevij �

Z t

0ds f s t′�s′;0ð ÞdðΥ

devij �ξdevij Þds

!∂Υ ij

∂Ekl_Ekl�δijΘK

GαG∂Υ ij

∂Ekl_Ekl

þ2_GþGG


� Z t



ð76Þ

There are five sets of terms that deserve discussion. On the left hand side, the heat capacity at constant deformation per unitvolume is composed of both an equilibrium term and one associated glassy behavior, the latter of which appears becausethe glassy thermal Helmholtz free energy density defined in Eq. (65) includes a difference term (CFG�CF1). On the righthand side of Eq. (76), there are four categories of energy generation. The first collection of terms are the classic thermal heatsource and flux terms.

The second group of terms, modified by _x, represent the energy sources due to advancing the Diels–Alder chemicalreaction. At equilibrium, the second line of Eq. (76) is �ϕΔH○

rxn _x, according to Eqs. (34) and (33), which shows that anequilibrated temperature involves a latent energy source associated with the change in the extent of reaction. The third lineof Eq. (76) provides the energy source associated with the equilibrium shear modulus dependence on the extent of reaction(provided the material is under a state of strain with Υdev

ij aξdevij ) as well as a contribution due to the difference in reactionentropies in the glassy vs. equilibrium states. Typically this last term should be negligible as the glassy state changes thekinetics not the energetics of the Diels–Alder reaction.

The third grouping of terms by the material time derivative of the Green–Lagrange strain, _Ekl, are the traditional thermal-elastic heating contributions. Here, we have assumed that the glassy mechanical properties KG and GG are not functions oftemperature. Under equilibrium conditions where the glassy history dependent integrals are zero, one recovers amechanical–thermal coupling associated with the temperature dependence of the shear modulus (first term) and thermalexpansion. Note again that, given the form of the equilibrium and glassy thermal expansion terms in the free energy, thethermal expansion coupling is associated with glassy response.

The final grouping of terms are associated with the evolution of ξdevij , the stress-free log strain tensor. Remarkably, theevolution of this internal state variable only produces a temperature change if viscoelastic behavior is active. Althoughcounterintuitive, this result emerges from the linear dependence of the equilibrium shear modulus on temperature, Eq. (47),

Table 1Properties of the Diels–Alder chemistry and transport.

Parameter Units Description

Θref K Initial temperaturexref None Initial extent of reactionxgel Gel-point extent of reactionDα m2 s�1 Isotropic spatial diffusivity of species α

κ J m�1 K�1 s�1 Isotropic, spatial thermal conductivityβ1 none Equilibrium volumetric deformation associated with the chemical reaction extentϕ mol m�3 The maximum concentration (per unit reference volume) that any species can obtainΔH○

rxn J Standard enthalpy change associated with the Diels–Alder reactionΔη○rxn J K�1 Standard entropy change associated with the Diels–Alder reactionk0 s�1 Prefactor to the Arrhenius chemical kineticsEact J Activation energy associated with the Arrhenius reaction kinetics

Table 2Equilibrium Helmholtz free energy density parameters.


Pref Pa First Piola–Kirchoff pressure datum (ambient pressure)K1 Pa Equilibrium bulk modulusG1ref Pa Equilibrium shear modulus at the reference temperature, Θref

CF0 J kg�1 K�1 Specific heat capacity at a fixed state of deformation. This constant weights a term thatproduces a constant response with respect to temperature

CF1 J kg�1 K�1 Specific heat capacity at a fixed state of deformation. This constant weights a term thatproduces a linear response with respect to temperature

α1 K�1 Equilibrium volumetric thermal expansion coefficient

Table 3Non-equilibrium Helmholtz free energy density parameters.


Θglass K Nominal glass transition temperatureΨ ref Pa Helmholtz free energy density datum in the initial state of the material pointKG Pa Glassy bulk modulusGG Pa Non-equilibrium shear modulusCFG J kg�1 K�1 Specific heat capacity at a fixed state of deformation in the glassy state. This constant

weights a term that produces a linear response with respect to temperatureαG K�1 Non-equilibrium volumetric thermal expansion coefficientβG None Non-equilibrium volumetric deformation associated with the chemical reaction extentηrxnG J m�3 Non-equilibrium entropy density (per unit reference volume) associated with the thermal chemistryfi None Relaxation process associated with the ith reactionτi s Stretched exponential time constant associated with the ith spectrumγi None Stretched exponent associated with the ith spectrumC1 None Viscoelastic clock parameter associated with temperatureC2 K Viscoelastic clock parameter associated with temperatureC3 K Viscoelastic clock parameter associated with volumetric deformationC4 K Viscoelastic clock parameter associated with shear deformation


and the quadratic dependence of the equilibrium free energy on Υdevij �ξdevij . For any other temperature dependence, an

evolution of the stress-free strain tensor would produce a thermal source from both the equilibrium and non-equilibriumfree energies.

3.15. Model parameters and experiments necessary to populate them

The removable polymer model involves three categories of parameters. The first set contains the thermal–chemicalequilibrium and kinetics constants associated with the Diels–Alder chemistry, which is summarized in Table 1. The secondand third sets are respectively associated with the rubbery and glassy thermal–mechanical properties and are summarizedin Tables 2 and 3.


Bowman and co-workers discuss using FTIR to measure the temperature dependence of the equilibrium constant as wellas to determine the thermally activated parameters associated with the forward and reverse reactions (Adzima et al., 2008),and supplementary material for this reference describes the process of fitting the Arrhenius kinetics parameters. However,they did not measure the volume change as a function of the extent of reaction, and likely assumed that it is negligibleparticularly in the gelled state of the material. Still, other systems could involve a volume change with the extent of reaction,which would be difficult to deconvolve from thermal expansion. Another property related to the structure of the polymernetwork and the reversible chemistry is the gel point extent of reaction, which can be determined as the extent of reactioncorresponding to the temperature at which the storage and loss-moduli have similar frequency scalings, which is known asthe Winter-Cambrion criterion (Winter, 1987). This property can therefore be determined from isothermal frequency sweepDMA data performed under oscillatory shear.

Mechanical properties associated with the equilibrium Helmholtz free energy density (Eq. (56)) are summarized in Table 2.For the RPM model, two isotropic (and isothermal) elastic constants are needed, which could be derived from storage shearand compression (or tensile) moduli in DMA. Thermal properties involve measurements of the thermal conductivity and theenthalpy, from which the specific heat capacity at constant stress can be derived.

The final set of properties are associated with the glassy behavior. A detailed discussion of this topic can be found in Adolfet al. (2004), and the interested reader is directed there. However, there is an additional subtlety in calibrating the glassybehavior of RPM materials compared with conventional thermosets. For RPM materials, viscoelastic relaxation attemperatures near the glass transition may be difficult to separate from relaxation behavior due to network evolution.For the specific material investigated here, the glass and gel point transitions are 40 and 90 1C respectively (Adzima et al.,2008), and so these phenomena occur in different time and temperature regimes and can be independently examinedexperimentally. If the gel point temperature falls within the glassy region, then glassy behavior could still be distinguishedby examining volumetric relaxation and thermal expansion. Other tests, such as creep or compression testing or digitalscanning calorimetry, would show effects from both phenomena near the glass transition. Far below the glass transition,these data could provide useful guidance in calibrating such materials. A summary of system parameters is supplied below.

4. Results

In addition to model development, an objective of this paper is to examine the difference in thermal–mechanicalbehavior between removable and conventional thermosetting encapsulations, and so we focus on scenarios in which thetemperature field is controlled. Furthermore, since we neglect species diffusion (see Section 3.6), we may focus only onbalancing momentum as the material is subjected to different thermal–mechanical boundary value problems. Weimplement the constitutive model into Sandia's SIERRA Code Framework and run simulations using its implicit, quasi-static finite deformation momentum balance application, SIERRA/SM. Simulations employ 3D linear hexahedral finiteelements with selective deviatoric integration. The model may be linked with SIERRA's energy and species balanceapplication codes as needed to fully represent transient thermal–chemical–mechanical analysis. We present three sets ofresults with associated objectives:

�
We develop (semi-)analytic solutions to problems involving applied uniaxial extension and thermal histories of theremovable polymer model under rubbery conditions. These results showcase the network relaxation behavior due to theDiels–Alder chemistry without the added complexity the non-equilibrium viscoelasticity.
�
We calibrate and validate the constitutive model against data in the literature to illustrate its predictive capabilities. � We contrast the behaviors of a removable vs. non-removable thermosetting polymers in encapsulation scenarios. These
results have important ramifications on the use of removable polymers in electronics packaging.

4.1. Analytic solutions of rubbery removable encapsulation

Our goal is to determine the thermal–chemical–mechanical behavior of the removable polymer model when we subjecta material point to an applied uniaxial stress and temperature history. Immediately, we neglect the non-equilibriumviscoelastic behavior discussed in Section 3.13 as well as any volume change associated with evolving the extent of reaction(β¼ 0 in Eq. (61)). Consider the scenario in which we apply a strain history in the 11 direction (Υ11 ¼ Υ11½t�; tZ0) while theother two principal directions remain traction free (SΥ22 ¼ SΥ33 ¼ 0; tZ0). We focus on the stress work conjugate to thelogarithmic strain, Eq. (61), which can be transformed to the Second Piola–Kirchoff stress via Eq. (20) as desired. We seek tocompute the axial stress (SΥ11½t�), the transverse strain ðΥ22½t� ¼ Υ33½t�Þ, and the permanent axial and transverse strains thatarises from the Diels–Alder chemistry and network rearrangement (ξ11½t�; ξ22½t� ¼ ξ33½t�). Requiring that the transversestresses are zero, one may solve for the transverse strain as a function of time from Eq. (61)

Υ22 ¼Kα1ðΘ�Θref Þ

2þ G1

3�K2

� �Υ11þG1ξ22

� �G1

3þK

� ��1

: ð77Þ


The permanent deformation evolves from Eq. (53), such that the transverse component obeys

_ξ22 ¼_G1þ

3G1Kðα1ðΘ�Θref Þ�3Υ11Þ

2G1

3þK

� � þξ22G1

G1

3þK

�3

0BB@

1CCA

0BB@

1CCA: ð78Þ

Since the permanent strain tensor, ξdevij , is deviatoric by construction (see Eq. (53)), the permanent axial deformation isξ11 ¼ �2ξ22. If Eq. (78) can be integrated, then the permanent deformation tensor and total logarithmic strain tensors areknown (the latter via Eq. (77)), and consequently, the axial stress can be obtained from Eq. (61). However, Eq. (78) is difficultto integrate under arbitrary applied axial strain and temperature histories because, as they change, chemical equilibriumchanges in accordance with Eqs. (33) and (34), and so the equilibrium shear modulus (G1) changes via Eq. (47) along withthe rate of increase of the shear modulus ( _G

1þ ) due to the forward Diels–Alder reaction (Eqs. (49) and (50)).

If we further restrict ourselves to isothermal conditions, then the system is in chemical equilibrium, and G1, G1þ , and

α1ðΘ�Θref Þ are all constants. Then, Eq. (78) can be integrated for certain applied axial strain histories. Consider anisothermal, a uniaxial stress relaxation scenario in which the applied axial strain history is Υ11 ¼ Υ ; tZ0. The correspondingtransverse permanent strain from Eq. (78) is

ξ22 ¼ �ab1�exp �bt

� �� ; a¼

_G1þ

3G13Υ �α1ðΘ�Θref Þ

2G1

3þK

� �0BB@

1CCAK; b¼

_G1þ

3G1 3� G1

G1

3þK

0BB@

1CCA: ð79Þ

The total transverse strain can be computed from Eqs. (77) and (79), and finally, the axial stress may be computedstraightforwardly from Eq. (61).

The main result from the isothermal stress relaxation boundary value problem is the exponential evolution of thetransverse strain, permanent strain, and axial stress. Since there is only one time scale, which is set by G1= _G

1þ , a single set of

results characterizes isothermal, rubbery stress relaxation due to network topology evolution in Fig. 4. As thermal expansiononly serves to offset the initial applied deformation, it is neglected in these analytic results. From Fig. 4b, the axial stress inthe material approaches zero after a sufficient amount of dimensionless time. For the stress tensor to be zero, both thedeviatoric and volumetric contributions must vanish, which is exactly what the evolution of the stress free strain tensoraccomplishes in Fig. 4a. The stress decays because the permanent shape of the material evolves to become the deformedshape. An interesting feature of this process is that the stress free strain tensor is by construction deviatoric, and so we seethe relaxation of elastic volumetric deformation due to the removable of deviatoric stresses. Another interesting feature ofthis result is that regardless of the ratio of bulk to shear moduli, the zero stress state is the same, and so all solutions willarrive at the permanent deformations in Fig. 4a although their initial trajectories may differ.

Fig. 4. Analytic solution of an RNP subjected to an isothermal, stress relaxation protocol in its rubbery state. Thermal expansion is neglected without loss ofsolution generality. A sudden logarithmic strain of 1 is applied at t¼0, and the ratio of the bulk to shear moduli is 106 corresponding to the nearlyincompressible limit of elastomers. (a) Stress free ðξdevij Þ and transverse log strains ðΥ transÞ normalized to the applied log strain ðΥaxÞ. (b) Axial stressnormalized by Young's modulus.


Isothermal uniaxial extension at a constant logarithmic strain rate is also modeled analytically. Here, the applied axialstrain is given by Υ11 ¼ _Υ t; tZ0, where _Υ is a constant. The associated permanent transverse logarithmic strain can beintegrated from Eq. (77)

ξ22 ¼ða�bcÞð1�exp½�bt�Þ�abt

b2; a¼

_G1þ

3G13 _ΥK

2G1

3þK

� �0BB@

1CCA; b¼

_G1þ

3G1 3� G1

G1

3þK

0BB@

1CCA; c¼ �

_G1þ

3G1α1ðΘ�Θref Þ

2G1

3þK

� �0BB@

1CCA

ð80ÞAgain, the total transverse logarithmic strain and axial stress can be computed from Eqs. (77) and (61) respectively usingEq. (80). In contrast to the sudden isothermal stress relaxation problem, there are two time scales in the isothermal, uniaxialextension scenario, one associated with the strain rate, _Υ , while the other is associated with the rate of permanent shapechange due to network topology evolution, _G

1þ =G

1. The ratio of these time scales determines the model's response,_ΥG1= _G

1þ , which is shown for different ratios in Fig. 5.

There are two clear limiting regimes. If _ΥG1= _G1þ b1, then there is no significant evolution to the network topology, and

so the model exhibits a linear elastic response as expected. On the other hand, if _ΥG1= _G1þ 51, then the evolution of the

network topology and associated stress relaxation rate is much faster than the strain rate. Hence, the material “flows” withapplied deformation and no stress is produced. When these two rates are comparable, as is the case in Fig. 5, then a steadystate behavior is set by the ratio itself. Specifically, the maximum axial stress (normalized by Young's modulus) approachesthis ratio, which can be clearly seen in Fig. 5b for the values of _ΥG1= _G

1þ ¼ 0:1;1.

4.2. Model calibration and validation

We now endeavor to validate the model's capability to predict the polymer dynamics associated with the Diels–Alderchemistry and concomitant stress relaxation and (coupled) viscoelasticity. Our objective is to predict the dynamicmechanical analysis (DMA) behavior of a specific RNP from Adzima et al. (2008). This task is challenging task because itinvolves three time scales: viscoelastic glassy behavior, Diels–Alder chemistry and associated network relaxation, and thetime scale of mechanical perturbation. Furthermore, it sweeps a range of temperatures and perturbation frequencies. TheDMA protocol involves isothermal frequencies sweeps in oscillatory shear. The model fit (75 1C), predictions (all othertemperatures), and experimental data are shown in Fig. 6.

Although the equilibrium and Diels–Alder chemistry/relaxation portions of the model can be fully calibrated from data inAdzima et al. (2008) without any adjustable parameters, there is no supporting calibration data for the glassy behavior.Since the shear DMA data is exclusively above Tg, which is stated to be approximately 40 1C, one needs only to calibrate theWLF coefficients (C1 and C2 from Eq. (69) provided C3 ¼ C4 ¼ 0 from Eq. (70)) and a characteristic shear stress relaxation

Fig. 5. Analytic solution of an RNP subjected to an isothermal, constant logarithmic strain rate uniaxial extension protocol in its rubbery state. The axial andtransverse strains are provided and represent the limit when the strain rate is much faster than the relaxation rate due to network topology evolution.Thermal expansion is neglected without loss of solution generality. the ratio of the bulk to shear moduli is 106 corresponding to the nearly incompressiblelimit of elastomers. (a) Stress free strains ðξdevij Þ. (b) Axial stress.

Fig. 6. Comparison between model predictions and experiments in dynamic mechanical analysis under oscillatory shear. Storage shear moduli arepresented for three isothermal frequency sweeps. Experimental data is reproduced from Adzima et al. (2008). At lower frequencies, the storage modulusdeclines at each temperature as the relaxation effects of the Diels–Alder chemistry become more prominent.


curve at the reference temperature, which is the process of constructing a master curve. The reference temperature can beany temperature above the glass transition but sufficiently below the gel point temperature such that chemical relaxation isnegligible relative to the glassy response. Construction of a master curve is a standard task for viscoelastic polymers aboveTg (Rubenstein and Colby, 2003). However, for this system with chemical relaxation near the gel point temperature, theprocess is a bit different. Ideally, one would still employ horizontal frequency shifting to obtain the WLF parameterssufficiently below the gel point temperature. However, the characteristic relaxation curve (characterized by ðτ; βÞ is better fitdirectly in the time domain at the reference temperature. Thus, stress relaxation or creep compliance curves are needed tofit these parameters.

Without adequate frequency or time domain data, we use a single characteristic relaxation curve ðτ; βÞ from a typicalepoxy from ref (Adolf et al., 2009) and calibrate the WLF coefficients ðC1;C2Þ to the lowest temperature storage shearmodulus curve (75 1C) using standard function optimization from Matlab. This approach is approximate because of theuncertainty in the characteristic relaxation curve and because the isothermal frequency sweep data only spans 0.02–20 rad s�1. A larger frequency range, especially at the higher frequency bound, would improve the fit. We calibrate the bulkmechanical behavior by selecting the lowest frequency rubbery storage shear modulus at 75 1C, and choose the bulkmodulus to be four orders of magnitude larger to replicate nearly incompressible elastomeric behavior. The chemical,rubbery, and viscoelastic model parameters are summarized in Table 4. All other temperature curves are then predicted andplotted against experiments for validation.

The experimental data can be divided into three regimes dominated by chemical relaxation, equilibrium elastomericbehavior, and viscoelasticity. The first regime occurs at low frequencies at lower temperatures and increases over thefrequency range as the temperature approaches the gel point. The calibrated RPM model reasonably captures the qualitativebehavior of the storage modulus trends. At slower frequencies and higher temperatures, the rate of network topologyevolution (permanent shape change) is so fast that the material essentially “flows” with the applied deformation and cannotstore elastic energy. The model likely has the wrong temperature scaling for the equilibrium shear modulus as it does notdiminish sufficiently fast with increasing temperature although it is reasonable very close to the gel point.

In the middle, especially for the 75 and 80 1C curves, both the experiment and the model demonstrate similar rubberyplateau's in which the storage modulus is less sensitive to frequency on a given isotherm. This result make sense in thecontext of the rubbery RPM model. Without viscoelasticity, only two time scales are active, and if the applied mechanicaltime scale is much faster than the chemistry/network topology evolution, then the network behaves in an elastic manner.

Finally, at the fastest frequencies, both the model and the experimental data show viscoelastic stiffening. The calibrationof viscoelasticity to the 75 1C curve appears to adequately describe the high frequency experimental behavior at each of theother temperatures, which suggests that viscoelasticity continues to dominate at high enough frequencies even as the gelpoint temperature is approached. This result is expected.

Table 4Calibrated material properties associated with the dynamic mechanical analysis validation predictions.Quantities are taken directly from Adzima et al. (2008) with the exception of K1 , β1 , C1–C4, τ1, τ2, γ1, and γ2,which were chosen as discussed in the text.

Θref 75 1C xgel 1ffiffiffi2

p

K1 3.8e8 Pa G1ref 3.8e4 Pa

Dα 0 m2 s�1 β1 0ΔH○

rxn �400 J mol�1 Δη○rxn �106 J K�1

C1 20 C2 180 KC3 0 K C4 0 Kτ1 6 s γ1 0.24τ2 0.12 s γ2 0.22k0 5.6e9 s�1 Eact 88e4 J

0 2 4 6 8 10

−20

0

20

40

60

80

Time (hours)

Tem

pera

ture

(C)

Fig. 7. Temperature history applied in the homogenous motion, uniaxial stress encapsulation scenario.


4.3. Removable thermosets in underfill scenarios

The objective of this section is to distinguish the behavior of RNP and conventional thermoset encapsulations underthermal–mechanical cycling. We are particularly interested in underfill applications wherein an encapsulation's thermal–mechanical behavior may strongly influence the conditions for solder-joint fatigue (see Fig. 3) due to the large thermalexpansion mismatch of the polymer underfill and neighboring ceramic and metallic components. Clearly, the RNP isbeneficial/detrimental compared with conventional thermoset encapsulation if it experiences lower/higher stresses duringthermal cycling, which ultimately are applied to surrounding components. To make this comparison, we examine a simple,homogenous motion boundary value problem in which the displacement field is constrained along one axis while the othertwo axes unconstrained (and their outward surfaces are traction free), which may primitively represent underfill scenarios(neglecting adhesion constraints at the interfaces). The body is subjected to a simple thermal cycle shown in Fig. 7, and asingle element is used to compute the body's response.

The behavior of the removable network polymer model is determined by three competing time scales:

�
The experimental time scale associated with the rates of applied deformation and temperature change. � Chemical kinetics, such as the half-life of a Diels–Alder linkage. � The non-linear viscoelastic time scale ð1=ð1þaÞÞ.
The analytic solutions in Section 4.1 dealt only with the first two time scales under special conditions in which they arefixed. However, in encapsulation scenarios, materials are subjected to thermal cycles in which all three time scales vary.If the material is in the glassy state throughout the entire thermal cycle, then the model will predict no significant network

Table 5Properties used in the two thermal–mechanical simulations that compare removable vs. conventional thermosets. The Proney terms are logarithmicallyspaced evenly between 10�8 and 108 s. Two Proney series are fit in a least squares sense to the stretched exponential relaxation curves defined by (τ1, γ1),(τ2, γ2) in Eq. (67). Viscoelastic parameters were taken from Adolf et al. (2009). Thermal-chemical constants were taken from Adzima et al. (2008).

Θref 25 1C xgel 1ffiffiffi2

p

K1 3.2e9 Pa Gref 4.5e6 PaKglass 4.9e9 Pa Gglass 0.9e9 PaDα 0 m2 s�1 β1 0ΔH○

rxn �40 kJ mol�1 Δη○rxn �106 J mol�1 K�1

k0 5.6e9 s�1 Eact 88e4 Jα1 600 ppm K�1 αglass 170 ppm K�1

C1 16.5 C2 54.5 KC3 1000 K C4 11,800 Kτ1 6 s γ1 0.24τ2 0.12 s γ2 0.22Number of Proney terms 17 Θglass �25 1C, 25 1C


topology evolution based on our assumption that chemical and viscoelastic kinetics act in parallel which influences the rateof shape change (see Eq. (46)), and so we do not observe differences between removable and conventional encapsulationunder that condition. However, if the glass transition occurs within or below the thermal cycle temperature range, thendifferences will arise.

We consider two scenarios, one in which Θglass is equal to the minimum temperature in the thermal cycle (�25 1C) andthe other in which Θglass is in the middle of the thermal cycle (25 1C). For these analyses, we choose a different encapsulationmaterial, namely an epoxy thermoset (828 DEA) that has been extensively studied and characterized for encapsulationapplications (Adolf et al., 2011). To compare it to the model in this paper, we “turn-on” the additional thermal-chemistry andconsider that the RPM's rubbery properties are calibrated to 25 1C instead of 75 1C. To examine the role of vitrification, wechange the reference temperatures of the glass transition between the two scenarios. A table of the system parameters foreach scenario is reported in Table 5. In the first scenario, Θglass ¼Θmin from Fig. 7, so that the materials are in the rubberystate the whole time. The body is assumed to be initially stress free, which is reasonable since the glass transition occurs inthe vicinity or well below the starting temperature in Fig. 7.

The removable and conventional thermosets differ in their axial stress responses in Fig. 8 due to both the changing stressfree configuration of the removable thermoset as seen in Fig. 8(a) and (b) and the non-linear temperature dependence ofRNP's rubbery shear modulus. The conventional thermoset behavior is straightforward; the constrained thermal expansiongives rise first to an axial compressive stress during heating and later to a tensile stress as the material is cooled below theinitial (stress-free) temperature. Some transience in the stress state is observed at the lowest temperature 25 1C, whichcorresponds to the glass transition temperature. This location is the only one in which glassy behavior plays a (minor) role,which is clear from shift factor time history in Fig. 8(b) for both materials.

The behavior of the RNP is more complicated. During the initial temperature rise, the kinetics of the DA chemistryare slow as evidenced by the half-life of a DA linkage near 25 1C from Fig. 3b. However, by 50 1C, the kinetics becomesufficiently fast so that the permanent shape evolves quickly in Fig. 8d, which correspondingly relaxes the axial stressresponse to zero. Subsequent temperature rises involves no stress accumulation since the rate of change of the permanentshape is so fast. During this heating process, the shear modulus also substantially diminishes owing to a large change inthe extent of reaction in Fig. 8c and Eq. (47), which serves to further accelerate the rate of permanent shape change throughEq. (53).

On cooling, initially the RNP accumulates little axial stress since the permanent shape continues to evolve rapidly relativeto the temperature rate. Around 50 1C, the permanent shape change evolution becomes slows with the chemical kinetics asevidenced by the plateau in Fig. 8c, and the a tensile axial stress builds. Note the difference in slope between the RNP andthe conventional thermoset behavior on further cooling. Although they have the same rubbery shear modulus at 25 1C inequilibrium, the RNP chemical kinetics are insufficient to increase the extent of reaction on further cooling, and so the RNPretains a low, non-equilibrium shear modulus as the material is cooled below 25 1C. During the final reheat, the chemicalkinetics again begin to increase along with the extent of reaction, shear modulus, and stress response on furthertemperature change, though an equilibrium extent of reaction is not reached by the end of the simulation.

In the second scenario, the glass transition is assumed to occur in both materials at 25 1C, which is in the middle of thethermal cycle. Vitrification plays a dominant role, and the material responses are only slightly distinguishable with respectto the axial stress response. Here, the viscoelastic shift factor transitions from rubbery to glassy regimes and is almostidentical between the two materials, the small differences arising from the changes to the permanent shape of the RNP(Fig. 9). The initial heating, high temperature hold, and first half of the cooling response proceed as in the previous case. Oncooling below the glass transition, permanent shape change and chemical kinetics are arrested, and the glassy responsetrumps the differences in the rubbery behaviors of the two materials. These results demonstrate that under thermal cyclingconditions that cross the glass transition, the intrinsic relaxation behavior of RNPs is insignificant.

Fig. 8. Single element encapsulation scenario in which Θglass ¼Θmin from Fig. 7. Along one axis, the displacement field is fixed while traction free conditionsare applied to the outer surfaces in the orthogonal directions. In this scenario, both materials remain rubbery throughout the thermal cycle, and asubstantial difference in the axial stress response is observed between the two materials. (a) Axial stress: Θglass ¼ �25 1C. (b) Viscoelastic shift factor:Θglass ¼ �25 1C. (c) Extent of reaction: Θglass ¼ �25 1C. (d) Permanent logarithmic strains: Θglass ¼ �25 1C.


5. Conclusions

We developed a thermal–chemical–mechanical constitutive model for network polymers with thermallyreversible linkages based on the Diels–Alder cycloaddition reaction. The reversible chemistry continuously addsand subtracts chains from the network, which causes these materials to exhibit three distinct behaviors whencompared with conventional thermosetting polymers: permanent shape evolution (stress relaxation), heightenedsensitivity of mechanical properties to temperature change, and a clear gel-point transition above which they behave asa viscous liquid. This latter property allows these network polymers to be removed, and hence we have labeled themthermally removable network polymers (RNP). The constitutive model includes internal variables to represent thepermanent shape change that results from adding and removing chains as well as a material clock variable that drivesglassy constitutive behavior.

Motivated by the need to use RNPs as non-destructively removable encapsulation materials, we focused ondistinguishing RNP from conventional network polymer behavior when the materials are subjected to thermal–mechanical

Fig. 9. Single element encapsulation scenario in which Θglass ¼Θroom (25 1C) from Fig. 7. Along one axis, the displacement field is fixed while traction freeconditions are applied to the outer surfaces in the orthogonal directions. In this scenario, each material transitions from a rubber to a glass as evidenced bythe viscoelastic shift factor, and only a very small difference is observed between their stress responses. (a) Axial stress: Θglass ¼ �25 1C. (b) Viscoelasticshift factor: Θglass ¼ �25 1C. (c) Extent of reaction: Θglass ¼ �25 1C. (d) Permanent logarithmic strains: Θglass ¼ �25 1C.


loading in the solid polymer (gelled) state. Three time scales determine the thermal, chemical, and mechanical responses ofsuch materials:

�
The rate of applied temperature change and/or deformation. � The kinetics of the reversible Diels–Alder reaction. � The viscoelastic time scale associated with the material's glassy behavior.
In the rubbery state, analytic solutions were developed for uniaxial stress relaxation and extension for RNPs thatcharacterize the interplay between the applied loading and the natural relaxation time scale of the material. Using limiteddynamic mechanical analysis data from the literature, the rubbery model was validated against experiments over a range oftemperatures and applied perturbation frequencies. Note that the model involved no tunable parameters, and the effects ofvitrification were ignored due to the lack of supporting data.

In the last series of results, simple thermal–mechanical cycling scenarios were studied to compare the RNP andconventional thermoset behaviors. Results demonstrated that if viscoelasticity dominates the applied thermal–mechanicalcycles, the RNPs and conventional thermosets respond identically. Also, if the thermal–mechanical conditions are such that


vitrification is avoided and the rates of applied deformation and temperature change are considerably faster than the Diels–Alder chemical kinetics, then removable and conventional thermosets behave similarly as elastomers. However, since thenumber density of chains, and therefore the rubbery shear modulus, evolves with temperature for RNPs, their thermal–mechanical behavior is distinct.

When vitrification is either fully or partially avoided in the applied thermal–mechanical cycles, and the rates ofmechanical and thermal stimuli are slow or comparable to the Diels–Alder chemical kinetics (and associated networktopology evolution), then removable network polymers behave in a significantly different manner compared withconventional thermosets. Given enough time, they can change their shape to become stress free in any configuration thatdoes not involve volumetric deformation.

Acknowledgments

The author is grateful for the many important discussions with William Scherzinger, David Wheeler, James Aubert, andRobert Chambers that helped shape this work. This work was supported under and Early Career Laboratory DirectedResearch and Development program at Sandia National Laboratories. Sandia National Laboratories is a multi-programlaboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S.Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

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The mechanics of network polymers with thermally reversible linkages