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[Jonathon Tanks](https://orcid.org/0000-0002-0232-8240), Yoshihiko Arao, Masatoshi Kubouchi

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[Network-level Analysis of Damage in Amine-crosslinked Diglycidyl Ether Resins Degraded by Acid](https://mdr.nims.go.jp/datasets/86338e2e-53f3-493f-8f0f-2f7eb22b6ccd)

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Layout 11. IntroductionEpoxy represents one of the most commonly usedthermosetting polymers for a variety of applications,including adhesives, anti-corrosion coatings, andcomposites. When used in acidic environments suchas chemical storage tanks or sewer systems, the acidpenetration rate and consequent degradation reactionsmust be characterized to inform lifetime predictions[1–7]. Anhydride-cured epoxy contains hydrolyzableester groups, which means prolonged exposure toaqueous environments will lead to a decrease incrosslink density and subsequently mechanical prop-erties [1, 3, 8–10]. These systems are rather conven-ient for evaluating such degradation behavior exper-imentally or analytically, since various techniquescan be readily applied for measuring crosslink density,segmental molecular weight of leached reactionproducts, and relative concentrations of carbonyls [9,10]. Amine-cured epoxy resins offer advantages overother types, such as low curing temperatures and highalkali resistance, but several studies have shown thattheir lifetime is reduced in acidic environments [1–4, 7, 11–14]. Nitric acid [15–17] and glacial aceticacid [18] have proven effective for breaking downthe amine-epoxy network in the case of chemical re-cycling, but these acids are special cases comparedto common inorganic acids such as H2SO4 and HCl.The relationship between specific degradation mech-anisms and corresponding changes in mechanicalproperties that occur in amine-crosslinked epoxieshave not been elucidated theoretically, despite numer-ous experimental observations [1–4, 18].488Network-level analysis of damage in amine-crosslinkeddiglycidyl ether resins degraded by acidJonathon Tanks1* , Yoshihiko Arao2 , Masatoshi Kubouchi31National Institute for Materials Science, Research Center for Structural Materials, Tsukuba, Japan2Waseda University, Department of Applied Mechanics and Aerospace Engineering, Tokyo, Japan3Tokyo Institute of Technology, Department of Chemical Science and Engineering, Tokyo, JapanReceived 15 October 2021; accepted in revised form 5 January 2022Abstract.Amine-crosslinked epoxy resins represent a large fraction of polymers used in structural and coating applications,meaning the characterization and modeling of environmental durability and mechanical reliability is of utmost importance.In particular, chemical storage tanks, sewage systems, and oil/gas infrastructure involve prolonged exposure to organic andinorganic acids. However, the majority of meso-scale models for polymer network degradation are more appropriate for hy-drogels than stiff thermosets; meanwhile, other models developed for acid degradation of ester-containing networks are notapplicable to amine-cured epoxies due to the assumption that crosslink density decreases. In this paper, we report the aciduptake and subsequent degradation behavior of bisphenol-F epoxy cured with an aliphatic amine, as well as propose a simplebut physically meaningful model for an ideal 2D network that effectively relates acid uptake and polymer structure to thedecrease in elastic properties over time. Parameters include crosslink density and acid penetration rate, and the only best-fitparameter is the reaction rate constant. This analysis can be extended to more complex network structures and environmentalconditions to model neat resins and composites.Keywords: thermosetting resins, damage mechanism, mechanical properties, modelling and simulationExpress Polymer Letters Vol.16, No.5 (2022) 488–499Available online at www.expresspolymlett.comhttps://doi.org/10.3144/expresspolymlett.2022.37Research article*Corresponding author, e-mail: tanks.jonathon@nims.go.jp© BME-PT   p  pohttps://orcid.org/0000-0002-0932-180Xhttps://orcid.org/0000-0003-1534-3420https://orcid.org/0000-0002-0232-8240Tanks et al. [19] proposed a degradation model thatelucidates the relationship between network struc-ture and equilibrium acid uptake through an arylether cleavage-based chain scission scheme, namely:(1) dehydration of hydroxyls near crosslink sites toform enol-ether intermediates, followed by (2) hydra-tion of the protonated intermediate to form an alde-hyde and phenol. This pathway can be found in bio -degradation studies on compounds containing arylether linkages [20, 21]. However, in order to applysuch a degradation model to the life prediction ofepoxy resins (i.e., prediction of long-term mechani-cal properties), another model is necessary for con-necting the molecular scale and the bulk scale – aso-called meso-scale ‘black box’ model.Shaw [22] conducted a chain-level analysis of untan-gled elastomers to show how random scission due tooxidation or mechanical load decreases long-termstiffness. This is a purely physical model that relateselastic properties to the network connectivity, with-out consideration for time-dependent processes ormolecular structure of the network. Several other keystudies [23–30] have proposed network-level analyt-ical models for polymer degradation, but they arespecifically intended for hydrogels and consider mass-loss/molecular weight as the target variable for pre-dicting degradation. Furthermore, these models applyto polymers with reactive crosslinks that are typical-ly assumed to react with the environment so thatdegradation can be measured by the decrease in cross -link density, which is unrepresentative of highly-crosslinked epoxy/amine resins. Li et al. [26] pro-posed a model for hydrogels considering chain scis-sion rather than crosslink breakage, where the Flory-Rehner equation is coupled with a first-order kineticslaw to produce a statistical model for swelling causedby the degradation of poly(ethylene glycol) with dif-ferent chain lengths. Gilormini et al. [27] focused oncrosslinked polymers rather than hydrogels, and al-though they used the affine network assumption toestimate the elastic shear modulus from crosslinkdensity, they also assumed crosslinks as the primaryreaction site, which is applicable to anhydride-curedepoxy. Advances in molecular modelling technologyhave prompted an increase in numerical analysisstudies of elastic properties of soft networks [31–33],but only one model focused on the effect of water onthe elastic modulus of thin polymer films [34].There are currently no analytical models which candirectly relate the local chemical environment andpolymer structure to the bulk network elastic prop-erties – i.e., elastic properties in the damaged state,where chain scission is the degradation mechanism.This paper presents such a model for the service lifeprediction of amine-crosslinked bisphenol-basedepoxy materials used in acid environments, only re-quiring five parameters: crosslink density and chainfunctionality (measurable), penetration rate and equi-librium acid uptake (measurable), and reaction rateconstant (approximated by fitting).2. Experimental study2.1. MaterialsThe model epoxy resin used in this study was a mix-ture of bisphenol-F epoxy (DGEBF, Epiclon 830,Mw ~ 320) and poly(ether)amine (PEA, JeffamineD230, Mw ~ 230). These monomers react to form acrosslinked network with tertiary amine crosslinksand a mixture of aliphatic and aromatic ethers.DGEBF and PEA were mixed with a 100:33.4 ratioby weight, degassed and oven-cured at 80°C for 24 hrand 120°C for 24 hr, from which rectangular speci-mens with dimensions of 60×20×2 mm3 were cutusing a water-cooled silicon carbide blade.2.2. Experimental procedureIn applications where epoxy resin is exposed toaqueous acid, only chemical tanks would constitutea case of highly-concentrated acid, whereas serviceconditions for epoxy linings or other structural ap-plications would involve low concentrations. Aque-ous H2SO4 at concentrations of 0.1, 1.0, and 10 wt%and temperatures of 60 and 80 °C were selected asmodel environmental conditions, in which speci-mens were immersed for various durations, and theacid penetration was monitored by measuring thechange in mass. At several durations, the immersedspecimens were dried in vacuum until the mass sta-bilized (usually several weeks to months), thenrecorded as the ‘dry’ state. This was done to approx-imate the actual water-free acid uptake. Previousstudies have shown that the penetration depth ofH2SO4 in amine-cured epoxy as measured elementalsulfur mapping by EDS can be accurately represent-ed by the mass change since it does not exhibit aFickian concentration gradient [4, 14]. FTIR wasalso used to identify degradation, as mentioned indetail in previous work [19].Strength is dependent on various factors, includingdefects and stress concentrations, making it difficultJ. Tanks et al. – Express Polymer Letters Vol.16, No.5 (2022) 488–499489to control the scatter with a small sampling size;thus, the elastic modulus was chosen as the mechan-ical property to assess chemical damage by acid.Water can significantly affect the elastic modulus,convoluting the effects of polymer network damage,so the elastic modulus was measured on dried spec-imens by three-point bending on an electromechan-ical testing machine at a crosshead speed of1 mm/min and span length of 40 mm (ASTM D790).The tangent modulus of elasticity Eb for a rectangu-lar cross-section is given by [35] (Equation (1)):(1)where L is the span length, b is the specimen width,d is the specimen thickness, and m is the slope of thelinear portion of the force-deflection curve (i.e., atsmall strains).2.3. ResultsThe mass uptake behavior of H2SO4 is shown for thewet and dry conditions in Figure 1. There seems tobe insignificant influence of temperature on equilib-rium uptake, which is primarily determined by acidconcentration. However, the diffusion rate is expect-edly temperature-dependent. Note that the dry con-dition represents an estimation of penetrated acid,which is hereafter the assumed state when discussingmass change.Although the gravimetric profiles resemble Fickiandiffusion behavior, EDS elemental mapping of sulfurions confirms a sharp concentration gradient that in-validates the underlying assumptions of Fick’s sec-ond law [4, 14] (Equation (2)):(2)where c is the concentration of penetrant in the ma-terial thickness dimension x, D is the diffusivity, andt is time. This integrates to the approximate solution[36] (Equation (3)):(3)For curve-fitting purposes, this mathematical formof the approximate solution to Fick’s second law isconvenient for obtaining an approximate diffusivityD. However, using the relation xA = MA⁄M∞, A to de-fine the penetration depth for inorganic acids, a sim-ple approximation for xA at time t the Equation (4):(4)where DA is the penetration rate coefficient (units ofhr–1/2). Figure 2 summarizes the acid uptake behav-ior in terms of equilibrium mass change (after dry-ing) M∞, A, D and DA, all of which increase with acidconcentration in a similar way to that reported in pre-vious work [2, 4, 7]. Higher temperature increasesDA as expected, but no significant difference can beseen for M∞, A except for a small increase in the caseof 10 wt%. These two parameters are largely respon-sible for controlling the progression and maximumlevel of damage in the polymer.FTIR spectra – taken from dry specimens after reach-ing equilibrium in different acid concentrations –suggest that a small amount of aryl ether cleavage(resulting in phenol and aldehyde) takes place [19–21], providing a pathway for chain scission that doesnot involve amine crosslinks (Figure 3a). A smallpeak is visible that indicates some unreacted epoxidegroups are present, but this does not change notice-ably during immersion, and excess epoxy has a small-er influence on the proton reactions than excessamine (see Section 3). Therefore, we did not consid-er this to be a problem. The photographs in Figure 3bEbdL m433b =x D t1 2A A=ddtcDxt2222=expMMn hD n t182 1142 12 2022 2tnrr= -+- +33= QU QVV Z/J. Tanks et al. – Express Polymer Letters Vol.16, No.5 (2022) 488–499490Figure 1. Acid penetration behavior for various H2SO4 concentrations, in the wet (a) and dry (b) conditions.of specimens after 48 hr of immersion reveal visualchanges indicative of penetration and reaction by sul-furic acid; the once-colorless epoxy turns red as thereaction progresses. Furthermore, the distinct pene-trated layer is clearly visible even to the eye (inset).Figure 4a shows representative force-deflection curvesof the control (pristine) epoxy and degraded epoxyafter drying. The slope m was taken in the linear re-gion (<2 mm) and used to calculate the tangent mod-ulus in Equation (1). Evidence of reaction kinetics-dominated damage progression is shown in Figure 4b,where the residual modulus shows no decreaseeven after a significant amount of acid has penetrat-ed the epoxy (acid uptake ~7%), until a reductionoccurs – gradually in the case of low concentrations,and abruptly in the case of high concentrations.Temperature does not appear to influence the overalldamage mechanism since all data from both temper-atures follow the same trend, but the rate should beaffected by temperature according to Arrhenius ki-netics. These experimental results, in continuationwith the experimental and analytical results in Tankset al. [19], form the basis for the key analytical modeldevelopment in Section 3.3. Analysis of network damaged by acidThis section presents a theoretical investigation ofthe mechanical damage that occurs in amine-curedbisphenol-diglycidylether epoxy systems as a resultof inorganic acid penetration; i.e., we present an an-alytical model for relating acid uptake behavior toreductions in elastic properties.J. Tanks et al. – Express Polymer Letters Vol.16, No.5 (2022) 488–499491Figure 2. Equilibrium mass change (a) and diffusivity (b) for different conditions (The lines are only intended to help theeye).Figure 3. Chemical changes in acid-immersed epoxy shown by (a) FTIR analysis of carbonyl and ether groups, and (b) pho-tographs of significant color change and visible penetration layer.Figure 4. Flexural properties of pristine and degraded DGEBF/amine networks: (a) typical force-deflection curves showingthe approximation method of the tangent modulus, and (b) normalized modulus as a function of dry mass change.3.1. Key assumptionsFollowing the work in Tanks et al. [19], there areseveral simplifying assumptions that must be madefor the current model:(1) The crosslinked structure is ideal; i.e., no loopsor uncured functional groups, so that every cross -link point is a tertiary amine and there are noepoxy chain having both ends connected to thesame amine chain.(2) Reactions occur randomly in the acid penetratedlayer (no biased direction).(3) Based on the crosslink density, the threshold en-vironmental acid concentration needed to satisfyonly the amine protonation reaction is 0.009 M(corresponding to an equilibrium mass changeof ~12%), so any excess acid above this con-tributes to chain scission through ether cleavage.Thus, the selected concentrations of 0.1, 1.0 and10 wt% (0.019, 0.189 and 1.889 M, respective-ly) are assumed to cause chain scission of dif-ferent degrees.(4) Amine crosslinks are protonated by acid but donot break; aryl ether cleavage is the primarymechanism of chain scission.(5) The polymer network is affine in both the initialand damaged states, such that elastic propertiescan be estimated from network structure usingthe Gaussian phantom network theory [27].Figure 5 illustrates the network structure and degra-dation process described by this 2D analysis, wherepolyetheramine (PEA) and bisphenol-F diglycidyl -ether monomers constitute a model epoxy system.3.2. Network featuresFeatures of the network structure considered for thedegradation model by inorganic acid [19] are de-scribed in more detail here. Consider an ideal 2Dcrosslinked network, represented by a unit cell withn nodes and dimensions a×b (out-of-plane thicknessis unity), as shown in Figure 6. This unit cell is suf-ficiently large that assumption #2 is valid and thatthe calculation of mechanical properties is not affect-ed by boundary conditions or damage distribution.The crosslink density ν can be expressed as (Equa-tion (5)):(5)The density of elastically effective chain connections– meaning the number of load-bearing chains con-nected to each crosslink per unit volume – is givenby Equation (6):(6)for diamine crosslinker, which is tetra-functional(i.e., a maximum of two epoxy segments connectedto each amine). From assumption #1, it follows thatabn3co =abno =J. Tanks et al. – Express Polymer Letters Vol.16, No.5 (2022) 488–499492Figure 5. Model system comprised of an ideal 2D network undergoing layer-wise penetration and degradation by acid. Eachcrosslink consists of a tertiary amine (NR3), two secondary alcohols (ROH) and two aryl ethers (ROR).Figure 6. Schematic of ideal 2D network damage process,consisting of alcohol elimination followed by arylether cleavage in a H2SO4 environment.the number of elastically effective chain density –the number of load-bearing chains per unit volume–can be approximated by Equation (7):(7)for the initial state only. The relationship between ncand νc no longer holds once damage is present; Equa-tions (6) and (7) are used for relating the elastic mod-ulus in the initial state to the chain density rather thancrosslink density since n is assumed to be constant.Using the affine network assumption [27, 32, 37],the elastic modulus in the initial state E0 can be re-lated to crosslink density by Equation (8):(8)where R is the universal gas constant and T is thetemperature. To calculate the elastic modulus in thedamaged state, we used chain density instead ofcrosslink density (Equation (9)):(9)3.3. Degree of degradation at localequilibriumBecause the degree of degradation is represented asa statistical average per crosslink site, the expressionfor local degradation in the penetrated layer comesfrom Equation (6) (Equation (10 )):(10)where fROR is the average number of chain scissionreactions that can occur given a certain amount ofavailable acid (H+), which follows from Equa- tion (11), (12) [19]:(11)(12)where νexp is the experimentally determined cross -link density, M∞, A is the equilibrium acid uptake bymass (excluding water), aH is the number of disso-ciated protons per acid molecule, mA is the molecu-lar weight of the acid, and ρ0 is the initial polymerdensity. The total functionality of the unit cell f–=fR3N) + fOH + fROR is the summation of reactive sites:amine, alcohol, and ether (respectively). Chain scis-sion is proposed to occur at the aryl ethers in thebisphenol epoxy segment, according to previouswork [19–21], where fROR accounts for the presenceof aliphatic ethers in the amine segment.The approximate degraded elastic modulus in thepenetrated layer can be written by combining Equa-tions (9) and (10) (Equation (13)):(13)which can be normalized by the initial modulus toyield (Equation (14)):(14)The bulk material properties – as measured experi-mentally – in the damaged state can be approximatedas the average of Ê and E0 proportional to their re-spective volumes. Thus, the global normalized elas-tic modulus can be written as (Equation (15)):(15)which finally reduces to (Equation (16)):(16)where xA = MA/M∞, A represents the relative pene-tration depth of the acid.3.4. Reaction kinetics and penetration rateThe previous section describes the model for pre-dicting the elastic modulus in the damaged state atequilibrium (i.e., all reactions have been completed).However, at short times the penetration rate of theacid may be much higher than the chain scissionrate, such that the penetrated layer is not immediate-ly degraded. This is clearly observed in the experi-mental data (Figure 4b), since the residual modulusdoes not decrease linearly with acid uptake startingat t = 0. Therefore, a kinetic model is required to ad-equately capture the complex degradation process.In this work, a pseudo-first order reaction kineticslaw is assumed, using the parameter fROR to signifythe reactive species (Equation (17)):(17)tfk f A k fdd0RORROR ROR=- =- r" " ! "% % $ %EEnnx x10 ccA A= + -t R WEE fx1 30RORA= -E n RT2 c=t tEEnn0 cc=t tmMfa,expA 0AHot=3rfmM a211,expA 0RORAHto= -3U ZE nL RT30 c=abn f3 RORco =-tR Wn 21c c. oE RT30 o=J. Tanks et al. – Express Polymer Letters Vol.16, No.5 (2022) 488–499493where k is the second order reaction rate constant;the concentration of acid [A] is assumed to be essen-tially constant due to the external source, leading tothe use of a new pseudo-first order rate constant k–.The probability that a reactive chain remains intactis (Equation (18)):(18)This leads to the expression for the extent of reactionξi within penetrated layer i (Equation (19)):(19)where τ is the reaction time – also referred to as thedwell time. It starts counting once layer i has beenfully penetrated at time ti (meaning τj = ti + 1), withincrement size equal to the global time increment(i.e., penetration time) so that ∆τ = ∆t. Equation (16)can then be rewritten to express the extent of reac-tion in layer i (penetrated at time ti) at dwell time in-crement i as Equation (20):(20)The acid continues to penetrate the polymer duringthis time period. Thus, the elastic modulus of layer iin the damaged state at time ti + 1 is Equation (21):(21)which is the specific expression for thermodynami-cally-bounded local value at any point in time in aone-dimensional system.Now we must select an appropriate model to de-scribe the acid penetration behavior, since this ac-counts for the reactant supply in Equation (17). Asdiscussed in Sections 2.2 and 3.2, there is sufficientexperimental evidence confirming that inorganicacids do not follow Fick’s laws [4, 14, 18]. While itis trivial to fit the mass uptake data with an arbitraryfunction that serves as the input to Equation (20),there are several phenomenological models availablein the literature for diffusion-reaction systems. Bor-rowing from the concept of bound and mobile watermolecules in the so-called Langmuir-type diffusionmodel [36, 38], we speculate that acid molecules maybe mobile or bound (‘trapped’) in polar sites withinthe global penetration scheme. Consider that the totalacid uptake is comprised of bound and mobile acidmolecules whose concentration follows a power-law(Equations (22), (23)):(22)(23)where β ≥ 1 is a concentration-dependent shape pa-rameter. There is currently no experimental evidencein the literature that confirms or denies this behavior– we simply propose a possible explanation for theobserved concentration dependency.3.5. Elastic modulus of a layer-wise degradedbeamIn the case of uniaxial tensile loading, the local elas-tic modulus in Equation (21) can be homogenizedacross the bulk volume of the polymer (having mlayers) at time ti to give Equation (24):(24)However, our study employs a simply supportedthree-point bending test to evaluate the elastic mod-ulus, so a flexural analysis of layered beams afterBîrsan et al. [39] is more accurate. The equivalentflexural stiffness for the structure shown in Figure 7acan be expressed as Equation (25):(25)where yi is the distance from the neutral axis of thebeam to the penetrated layer i. The elastic moduluscan then be estimated by the layer-wise summationof stiffnesses at each time increment and normaliz-ing by the moment of inertia I. Since the degradationbegins on the outer surface and progresses inward,the pristine epoxy stiffness is continually reducedwhile being symmetrical around the neutral axis.This means that the global modulus calculated byEquation (25) will be slightly lower than Equa-tion (24) at intermediate durations (i.e., they are thesame at t = 0 and equilibrium). As a result, the pre-dictions are conservative.M M MA, m A A, b= -M f M MA, b A A= =bR RW Wexp k1ijjp x= - - rR WexpEE ffk t t1 31 310ijRORijRORi k ip= - == - - - -+trU TT RZ YY W" " % %expffktROR0ROR= - rQ V""%%exp k1ip x= - - rQ VEEEE0 01ijijim==tU UZ Z/EI y yy y Ey y31i i iini iin3 311121 21212eq i i 1in= - ---+=+=+=Q RURRV WWWZ///J. Tanks et al. – Express Polymer Letters Vol.16, No.5 (2022) 488–499494The model was solved using a 1D central differencescheme written in python, as illustrated in Figure 7b.Assuming penetration occurs only through the thick-ness of the epoxy, the body was divided into layersin which the reaction only begins once the acid hasfully penetrated (ti), at which point the reaction pro-gresses through each time step (τj) until complete.4. Analysis results4.1. Instantaneous scission vs kinetic scissionThe kinetic behavior described by Equation (19) isdemonstrated in Figure 8a for various values of k–,such that a higher value equates to a shorter time untilthe layer is fully reacted (i.e., maximum damage). Itis extremely difficult to isolate the reactions thatoccur in a polymer system to measure the reactionrates of that functional group alone accurately; werefer to other studies [20, 21] on acid cleavage ofaryl ethers to approximate the general range of k– ̅val-ues, which appears to be around 0.008–0.1 hr–1. Forfaster reaction rates, the reaction in a given layer willreach completion faster, until the reacted region isnearly equivalent to the penetrated region at a giventime. This can be considered instantaneous scission,wherein the change in mechanical properties of thenetwork is directly proportional to acid uptake; thiswas not experimentally observed for the H2SO4 con-centrations in this study. Numerical examples ofMA = MA, b + MA, m are shown in Figure 8b, 8c. AsJ. Tanks et al. – Express Polymer Letters Vol.16, No.5 (2022) 488–499495Figure 7. Scheme to solve for the reaction variable and resid-ual elastic modulus through the thickness in thecase of layer-wise acid penetration: (a) schematicof the layered beam model, and (b) schematic of1D reaction model.Figure 8. (a) Relationship between rate constant k– and extent of reaction ξ, (b) influence of shape parameter β on the fractionof bound acid, and (c) example of decomposed acid uptake curve into bound and mobile concentrations.β increases, the relative portion of bound acid mol-ecules – available for reaction – becomes delayed atshort timescales and accelerated at longer ones. Thecombination of concentration-dependent diffusionand reaction leads to a flexible model for capturingvarious complex mechanisms.4.2. Degraded network propertiesWe found that for high acid concentrations (10 wt%),the modulus was maintained until a higher degree ofacid penetration before suddenly falling, comparedto the more gradual decline seen with low concen-trations (Figure 4b). However, as higher concentra-tion would not exhibit lower reaction rates thanlower concentrations, we must assume that the localacid concentration and proton activity is different forhigher concentrations due to significantly higherpenetration rates and stoichiometric changes [18].Thus we consider different values of β to explain thedisparity between experiment and analysis for agiven rate constant.Model predictions are plotted against experimentaldata in Figure 9, showing good agreement using thebest-fit parameters of k– in Table 1 (crosslink densityfor the model polymer is taken [19] as νexp =1.621 mol/cm3). The model appears particularly validfor low concentrations, using rate constants similarto those in the literature. Higher concentrations pres-ent a challenge for the model when calculating acidconcentration vs penetration depth via Equation (4),since the comparatively high penetration rate quicklyinitiates the reaction at short durations (i.e., <10 hr);however, based on experimental data, we propose thatthe reaction takes more time to initiate so that the re-action time becomes relatively smaller as the systemapproaches equilibrium (i.e., long durations). Thus,the selected function in Equation (22) shows betteragreement with experiments by dividing the localconcentration of protons in early-stage penetratedlayers into bound and mobile species. A more sophis-ticated physiochemical model is needed to describethis complex diffusion/reaction behavior accurately.Using the information on the network structure (νexp,J. Tanks et al. – Express Polymer Letters Vol.16, No.5 (2022) 488–499496Table 1. Model parameters for DGEBF/PEA system degrad-ed by H2SO4.Parameter0.1 wt% 1.0 wt% 10 wt%60°C 80°C 60°C 80°C 60°C 80°CM∞, A  [%] 15.600 15.900 17.600 18.000 26.100 26.700fROR    [–] 0.457 0.476 0.642 0.662 0.865 0.934k–         [hr–1] 0.019 0.028 0.037 0.058 0.046 0.092DA      [hr–1/2] 0.033 0.048 0.044 0.080 0.091 0.133Figure 9. Model predictions of residual elastic modulus during exposure to H2SO4 at various concentrations and temperatures,comparing instantaneous vs. kinetic for different values of β.fROR), this model directly predicts the change in elasticproperties based on a specific reaction mechanism.As mentioned above, the reaction rate constants k– andactivation energies Ea – 16.3, 22.0, and 33.9 kJ/molfor 0.1, 1.0, and 10 wt%, respectively – correspon-ding to the model were obtained by fitting the exper-imental data. However, this study demonstrates theutility of our network analysis approach for relatingchemical processes to mechanical properties; al-though we did not find comparable data in the liter-ature, they follow pseudo-first order kinetics and ap-pear valid (Figure 10). Future work will include awider range of temperatures, acid types, and networkstructures. For example, weak organic acids presenta difficult challenge due to their typically higher pKavalues and solvent swelling properties, so the effectsof swelling stresses would need to be incorporatedin the stiffness formulation. As for different epoxynetworks, a mixed-hardener system involving poly -amides or esters would constitute a different reactionscheme, complicating the determining of rate con-stants. However, by systematically varying the chem-ical structure and employing the basic approach ofour model, it would be possible to formulate the spe-cific expression relating reaction schemes to mechan-ical properties. In the future, a random 3D modelwill replace the ideal 2D model to reflect the con-nectivity (elastic chain density) more accurately. Thiswill require automated node and edge generation,which was outside the scope of the current paper.5. ConclusionsExperimental investigations revealed a layer-wisepenetration process of inorganic acid (H2SO4) inamine-cured epoxy resin, accompanied by physicaldamage manifested in reduced elastic properties. Amechanistic model was proposed for relating theenvironmental conditions and polymer structure tothe long-term elastic properties of an amine-crosslinked phenolic epoxy resin, based on a simple2D network analysis. The only fitting parameter inthis model is the reaction rate constant k– ̅, which isdifficult to determine directly through isolated ex-periments, but the best-fit values were within theranges for aryl ether cleavage in the literature. Theanalytical results show good agreement with exper-imental data. We show that features of the networkstructure such as functionality and crosslink densitydetermine the equilibrium damaged state, while thedegradation kinetics are primarily influenced by thepenetration and reaction rates. This framework canbe used for other material-environment systems thatexhibit coupled diffusion-reaction damage process-es, given that the degradation mechanism is under-stood. Furthermore, it serves as a ‘black box’ for ap-proximating reaction rate constants in complexnetworks if the inputs (i.e., polymer structure, envi-ronmental conditions) and outputs (i.e., mechanicalproperties, spectroscopic analyses) are known sincedirect measurement is unfeasible. Extending thismodel to 3D random networks with partial curing ef-fects would be particularly useful for large-scale nu-merical simulations of thermosetting resins and theircomposites in applications such as automotive, pipe -line, and aerospace structures.AcknowledgementsJ. T. acknowledges the financial support of the Ministry ofEducation, Culture, Sports, Science and Technology (MEXT)Japan.References[1] Møller V. B., Dam-Johansen K., Frankær S. M., Kiil S.:Acid-resistant organic coatings for the chemical indus-try: A review. Journal of Coatings Technology and Re-search, 14, 279–306 (2017).https://doi.org/10.1007/s11998-016-9905-2[2] Sembokuya H., Negishi Y., Kubouchi M., Tsuda K.:Corrosion behavior of epoxy resin cured with differentamount of hardener in corrosive solutions. Journal ofthe Society of Materials Science Japan, 9, 230–234(2003).https://doi.org/10.2472/jsms.52.9Appendix_230[3] Tsuda K.: Behavior and mechanisms of degradation ofthermosetting plastics in liquid environments. Journalof the Japan Petroleum Institute, 50, 240–248 (2007).https://doi.org/10.1627/jpi.50.240J. Tanks et al. – Express Polymer Letters Vol.16, No.5 (2022) 488–499497Figure 10. Approximated activation energy Ea for increasingH2SO4 concentrations.[4] Abacha N., Kubouchi M., Tsuda K., Sakai T.: Perform-ance of epoxy-nanocomposite under corrosive environ-ment. Express Polymer Letters, 1, 364–369 (2007).https://doi.org/10.3144/expresspolymlett.2007.51[5] Frank K., Childers C., Dutta D., Gidley D., Jackson M.,Ward S., Maskell R., Wiggins J.: Fluid uptake behaviorof multifunctional epoxy blends. Polymer, 54, 403–410(2013).https://doi.org/10.1016/j.polymer.2012.11.065[6] Pitarresi G., Toscano A., Alessi S.: Fracture toughnessof synthesized high-performance epoxies subject to ac-celerated water aging. Polymer Testing, 68, 248–260(2018).https://doi.org/10.1016/j.polymertesting.2018.04.010[7] Padarthi Y., Mohanta S., Gupta J., Neogi S.: Quantifi-cation of swelling stress induced mechanical propertyreduction of glass fiber/epoxy composites immersed inaqueous 10% sulphuric acid by instrumenting with dis-tributed optical fiber sensors. Fibers and Polymers, inpress (2021).https://doi.org/10.1007/s12221-021-0317-2[8] El Yagoubi J., Lubineau G., Traidia A., Verdu J.: Mon-itoring and simulations of hydrolysis in epoxy matrixcomposites during hygrothermal aging. CompositesPart A: Applied Science and Manufacturing, 68, 184–192 (2015).https://doi.org/10.1016/j.compositesa.2014.10.002[9] Capiel G., Miccio L. A., Montemartini P. E., SchwartzG. A.: Water diffusion and hydrolysis effect on thestructure and dynamics of epoxy-anhydride networks.Polymer Degradation and Stability, 143, 57–63 (2017).https://doi.org/10.1016/j.polymdegradstab.2017.06.010[10] Capiel G., Uicich J., Fasce D., Montemartini P. E.: Dif-fusion and hydrolysis effects during water aging on anepoxy-anhydride system. Polymer Degradation andStability, 153, 165–171 (2018).https://doi.org/10.1016/j.polymdegradstab.2018.04.030[11] Barkoula N. M., Karabela M., Zafeiropoulos N. E.,Tsotra P.: Fast curing versus conventional resins –Degradation due to hygrothermal and UV exposure.Express Polymer Letters, 14, 401–415 (2020).https://doi.org/10.3144/expresspolymlett.2020.34[12] Zhang A. Y., Li D. H., Zhang D. X., Lu H. B., Xiao H.Y., Jia J.: Qualitative separation of the effect of voidson the static mechanical properties of hygrothermallyconditioned carbon/epoxy composites. Express Poly-mer Letters, 5, 708–716 (2011).https://doi.org/10.3144/expresspolymlett.2011.69[13] Chow W. S.: Water absorption of epoxy/glass fiber/organo-montmorillonite nanocomposites. Express Poly-mer Letters, 1, 104–108 (2007).https://doi.org/10.3144/expresspolymlett.2007.18[14] Kusano M., Kubouchi M., Bulgarevich D. S., ShiwaM.: Non-destructive evaluation by terahertz spectro -scopy for penetration of acid solutions into epoxy resin.Express Polymer Letters, 10, 941–949 (2016). https://doi.org/10.3144/expresspolymlett.2016.87[15] Dang W., Kubouchi M., Sembokuya H., Tsuda K.:Chemical recycling of glass fiber reinforced epoxy resincured with amine using nitric acid. Polymer, 46, 1905–1912 (2005).https://doi.org/10.1016/j.polymer.2004.12.035[16] Dang W., Kubouchi M., Yamamoto S., Sembokuya H.,Tsuda K.: An approach to chemical recycling of epoxyresin cured with amine using nitric acid. Polymer, 43,2953–2958 (2002).https://doi.org/10.1016/S0032-3861(02)00100-3[17] Hanaoka T., Arao Y., Kayaki Y., Kuwata S., KubouchiM.: Analysis of nitric acid decomposition of epoxyresin network structures for chemical recycling. Poly-mer Degradation and Stability, 186, 109537 (2021).https://doi.org/10.1016/j.polymdegradstab.2021.109537[18] Lim J. S. K., Gan C. L., Hu X. M.: Unraveling themechanistic origins of epoxy degradation in acids. ACSOmega, 4, 10799–10808 (2019).https://doi.org/10.1021/acsomega.9b00859[19] Tanks J., Kubouchi M., Arao Y.: Influence of networkstructure on the degradation of poly(ether)amine-curedepoxy resins by inorganic acid. Polymer Degradationand Stability, 157, 153–159 (2018).https://doi.org/10.1016/j.polymdegradstab.2018.10.011[20] Sturgeon M. R., Kim S., Lawrence K., Paton R. S.,Chmely S. C., Nimlos M., Foust T. D., Beckham G. T.:A mechanistic investigation of acid-catalyzed cleavageof aryl-ether linkages: Implications for lignin depoly-merization in acidic environments. ACS SustainableChemistry and Engineering, 2, 472–485 (2014).https://doi.org/10.1021/sc400384w[21] Jasiukaitytė-Grojzdek E., Huš M., Likozar B.: Acid-cat-alyzed α-O-4 aryl-ether cleavage mechanisms in (aque-ous) γ-valerolactone: Catalytic depolymerization reac-tions of lignin model compound during organosolv pre-treatment. ACS Sustainable Chemistry and Engineer-ing, 8, 17475–17486 (2020).https://doi.org/10.1021/acssuschemeng.0c06099[22] Shaw M. T.: Models of a degrading polymer network.Journal of Chemical Physics, 54, 2172–2177 (1971).https://doi.org/10.1063/1.1675149[23] Metters A. T., Anseth K. S., Bowman C. N.: A statisticalkinetic model for the bulk degradation of PLA-b-PEG-b-PLA hydrogel networks:  Incorporating network non-idealities. Journal of Physical Chemistry B, 105, 8069–8076 (2001).https://doi.org/10.1021/jp004083h[24] Metters A., Hubbell J.: Network formation and degra-dation behavior of hydrogels formed by Michael-typeaddition reactions. Biomacromolecules, 6, 290–301(2005).https://doi.org/10.1021/bm049607o[25] Reddy S. K., Anseth K. S., Bowman C. N.: Modelingof network degradation in mixed step-chain growthpolymerizations. Polymer, 46, 4212–4222 (2005).https://doi.org/10.1016/j.polymer.2005.02.050J. Tanks et al. – Express Polymer Letters Vol.16, No.5 (2022) 488–499498https://doi.org/10.1016/S0032-3861(02)00100-3[26] Li X., Kondo S., Chung U-I., Sakai T.: Degradation be-havior of polymer gels caused by nonspecific cleavagesof network strands. Chemistry of Materials, 26, 5352–5357 (2014).https://doi.org/10.1021/cm502480f[27] Gilormini P., Richaud E., Verdu J.: A statistical theoryof polymer network degradation. Polymer, 55, 3811–3817 (2014).https://doi.org/10.1016/j.polymer.2014.05.008[28] Escobar F., Anseth K. S., Schultz K. M.: Dynamicchanges in material properties and degradation of poly(ethylene glycol)-hydrazone gels as a function of pH.Macromolecules, 50, 7351–7360 (2017).https://doi.org/10.1021/acs.macromol.7b01246[29] Jahanmir G., Abdekhodaie M. J., Chau Y.: Stochasticmodeling of degradation behavior of hydrogels. Macro-molecules, 51, 3941–3952 (2018).https://doi.org/10.1021/acs.macromol.8b00165[30] Machatschek R., Saretia S., Lendlein A.: The interplaybetween network morphology and degradation kineticsof polymers: Theoretical and experimental analysis bymeans of a 2D model system. MRS Advances, 5, 679–691 (2020).https://doi.org/10.1557/adv.2019.457[31] Liu S., Wang P., Huang G., Wang L., Zhou J., Lu T. J.,Xu F., Lin M.: Reaction-induced swelling of ionic gels.Soft Matter, 5, 449–455 (2015).https://doi.org/10.1039/C4SM02252A[32] Heidemann K. M., Sharma B., Rehfeldt F., Schmidt C.F., Wardetzky M.: Elasticity of 3D networks with rigidfilaments and compliant crosslinks. Soft Matter, 11,343–354 (2015).https://doi.org/10.1039/C4SM01789G[33] Kawai R., Tanaka H., Matsubara S., Ida S., Uchida M.,Okumura D.: Implicit rule on the elastic function of aswollen polyacrylamide hydrogel. Soft Matter, 17, 4979–4988 (2021).https://doi.org/10.1039/D1SM00346A[34] Chen R., Wang Z., Li S., Du H.: A novel degradationmechanism of the elastic modulus of wet polymer sub-strates under nanoindentation. Soft Matter, 16, 5009–5019 (2020).https://doi.org/10.1039/D0SM00645A[35] ASTM D790-03: Standard test methods for flexuralproperties of unreinforced and reinforced plastics andelectrical insulating materials (2010).[36] Ameli A., Datla N. V., Papini M., Spelt J. K.: Hygro -thermal properties of highly toughened epoxy adhe-sives. Journal of Adhesion, 86, 698–725 (2010).https://doi.org/10.1080/00218464.2010.482405[37] Khonakdar H. A., Morshedian J., Wagenknecht U., JafariS. H.: An investigation of chemical crosslinking effecton properties of high-density polyethylene. Polymer,15, 4301–4309 (2003).https://doi.org/10.1016/S0032-3861(03)00363-X[38] El Yagoubi J., Lubineau G., Roger F., Verdu J.: A fullycoupled diffusion-reaction scheme for moisture sorp-tion-desorption in an anhydride-cured epoxy resin. Poly-mer, 53, 5582–5595 (2012).https://doi.org/10.1016/j.polymer.2012.09.037[39] Bîrsan M., Pietras D., Sadowski T.: Determination ofeffective stiffness properties of multilayered compositebeams. Continuum Mechanics and Thermodynamics,33, 1781–1803 (2021).https://doi.org/10.1007/s00161-021-01006-2J. Tanks et al. – Express Polymer Letters Vol.16, No.5 (2022) 488–499499https://doi.org/10.1016/S0032-3861(03)00363-X