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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 61 (2013) 2625–2637

0022-50http://d

n CorrE-m

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

Coarse-grained simulation of molecular mechanisms ofrecovery in thermally activated shape-memory polymers

Brendan C. Abberton a, Wing Kam Liu b, Sinan Keten c,n

a Theoretical & Applied Mechanics, Northwestern University, Evanston, IL 60208, United Statesb Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, United Statesc Departments of Mechanical Engineering and Civil & Environmental Engineering, Northwestern University, Evanston, IL 60208,United States

a r t i c l e i n f o

Article history:Received 19 February 2013Received in revised form13 June 2013Accepted 7 August 2013Available online 20 August 2013

Keywords:Shape-memory polymerMolecular dynamicsCoarse-grained molecular dynamicsMechanicsRelaxationRecovery

96/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.jmps.2013.08.003

esponding author. Tel.: þ1 847 491 5282.ail address: s-keten@northwestern.edu (S. K

a b s t r a c t

Thermally actuated shape-memory polymers (SMPs) are capable of being programmedinto a temporary shape and then recovering their permanent reference shape uponexposure to heat, which facilitates a phase transition that allows dramatic increase inmolecular mobility. Experimental, analytical, and computational studies have establishedempirical relations of the thermomechanical behavior of SMPs that have been instru-mental in device design. However, the underlying mechanisms of the recovery behaviorand dependence on polymer microstructure remain to be fully understood for copolymersystems. This presents an opportunity for bottom-up studies through molecular model-ing; however, the limited time-scales of atomistic simulations prohibit the study of keyperformance metrics pertaining to recovery. In order to elucidate the effects of phasefraction, recovery temperature, and deformation temperature on shape recovery, here weinvestigate the shape-memory behavior in a copolymer model with coarse-grainedpotentials using a two-phase molecular model that reproduces physical crosslinking.Our simulation protocol allows observation of upwards of 90% strain recovery in somecases, at time-scales that are on the order of the timescale of the relevant relaxationmechanism (stress relaxation in the unentangled soft-phase). Partial disintegration of theglassy phase during mechanical deformation is found to contribute to irrecoverable strain.Temperature dependence of the recovery indicates nearly full elastic recovery above thetrigger temperature, which is near the glass-transition temperature of the rubberyswitching matrix. We find that the trigger temperature is also directly correlated withthe deformation temperature, indicating that deformation temperature influences therecovery temperatures required to obtain a given amount of shape recovery, untilthe plateau regions overlap above the transition region. Increasing the fraction of glassyphase results in higher strain recovery at low to intermediate temperatures, a widening ofthe transition region, and an eventual crossover at high temperatures. Our resultscorroborate experimental findings on shape-memory behavior and provide new insightinto factors governing deformation recovery that can be leveraged in biomaterials design.The established computational methodology can be extended in straightforward ways toinvestigate the effects of monomer chemistry, low-molecular-weight solvents, physicaland chemical crosslinking, different phase-separation morphologies, and more compli-cated mechanical deformation toward predictive modeling capabilities for stimuli-responsive polymers.

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eten).

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1. Introduction

Shape-memory polymers (SMPs) are a promising class of responsive polymeric materials with the capability of beingprogrammed into a temporary shape and then recovering their permanent shape upon exposure to post-deformation externalstimuli, including moisture, light, and temperature (Gall et al., 2002; Lendlein et al., 2005; Sokolowski et al., 2007; Gunes and Jana,2008; Huang et al., 2010; Behl et al., 2010). A wide variety of applications have been conceived for these materials, including self-healing or shape-shifting structural materials, variable-stiffness substrates for cell culturing, and smart biomaterials such as stentsand sutures (Wischke and Lendlein, 2010; Yakacki et al., 2007, 2008; Yoshida et al., 2006; Lendlein and Langer, 2002; Bellin et al.,2006; Vaia and Baur, 2008; Davis et al., 2011). The shape-memory behavior in polymers emerges as a result of the material'smicrostructure, which simplistically can be thought of as a combination of static netpoints that facilitate structural memory anddynamic switching domains, which undergo phase transformation upon triggering. In copolymer SMPs, such as polyurethane-poly(ϵ�caprolactone) (PU-PCL), a miscibility gap between molecular blocks leads to a phase-separated microstructure consisting offrozen hard and active soft phases (D'hollander et al., 2010). The rapid change in molecular mobility upon phase transition in theactive phase is the driving force that enables the transition from kinetic arrest in the deformed shape to shape recovery uponrelaxation (Lendlein et al., 2005; Behl and Lendlein, 2007; Mather et al., 2009; Behl et al., 2010).

Earlier experimental, analytical, and computational studies have focused on developing a description of the thermomechanicalbehavior of thermally actuated SMPs. Thus far, constitutive formulations have focused primarily on thermosets with irreversiblecovalent cross-links. Nonlinear thermoviscoelastic models with elastic coefficients that depend exponentially on temperature havebeen proposed to reproduce the large change in material properties above and below Tg (Tobushi et al., 2001; Diani et al., 2006).This was combined with a phenomenological viscosity–temperature relationship. Follow-up studies have incorporated a differentapproach where the distribution of the glassy or rubbery phases has been defined by simple evolution laws of frozen (glassy) andactive (rubbery) microdomains (Liu et al., 2006; Qi et al., 2008). A homogenization approach was used to predict the behavior of theSMP from the stress formulation of the individual phases. In an attempt to escape from phenomenological descriptions andcircumvent the need to assume volume fractions and their evolution, Nguyen et al. (2008) incorporated the underlying molecularmechanisms responsible for the shape effect to the constitutive model by using a nonlinear Adam-Gibbs model for structuralrelaxation. More recently, Srivastava et al. proposed a generally applicable theory for SMPs that also takes into account theviscoplastic behavior (Srivastava et al., 2010a, 2010b; Chester and Anand, 2010).

Advancements in phenomenological models have facilitated improved description of the thermomechanical behavior of SMPs,which has been instrumental in device design. However, the underlying physical principles of the mechanical behavior of thesematerials are very complex, involving intricate relations between temperature, moisture, polymer chemical structure, as well asmorphological dynamics of the hierarchical polymer network. For example, it is currently not possible to accurately predict theshape transition temperature and shape-recovery kinetics in SMPs simply from the molecular design of the polymer. As such, thereis a need for molecular modeling and a bottom-up approach to the study of SMPs and the effects of molecular design parameters onshape recovery. In pursuit of this fundamental understanding, Diani and Gall (2007) have performed the pioneering work usingfully atomistic molecular dynamics (MD) to study the shape-memory properties of cis-polyisoprene (cis-PI), a homopolymer thatcan be used to form the switching domains in an SMP. However, the need to take into account the microstructural organization andlonger term relaxation behavior in SMP copolymers requires length and time scales that exceed all-atom molecular dynamicssimulations. Coarse-grained molecular dynamics (CGMD) offers a mesoscopic modeling paradigm that can potentially bridge thegap between continuummodels and all-atom simulations systematically, thereby closing the loop of a bottom-up strategy. A CGMDmethodology allows one to reach greater length and time scales, making it possible to impose greater programmed deformationsand to observe a larger fraction of the recovery process (perhaps, for some systems, its entirety). Additionally, it allows the study ofsystems composed of both switching domains and netpoints through straightforward assignment of soft- and hard-phaseproperties within designated geometric/block regions through generic effective potentials. However, there are also significantchallenges in using CGMD. For instance, in order to study the effect of polymer chemical structure, a coarse-grained potential mustbe created for each structure of interest using mathematically rigorous methodologies such as the Inverse BoltzmannMethod (IBM)(Müller-Plathe, 2002; Faller, 2004; Sun and Faller, 2006).

In a first attempt to assess the potential of CGMD approach for studying shape-memory behavior in copolymer systems,here we present an analysis of the shape recovery behavior of a two-phase CG polymer model with generic potential termsfor hard and soft phases. We reproduce the shape-memory cycle of a thermally activated, binary, shape-memory copolymerthrough coarse-grained simulation, and we investigate and explain the effects of phase fraction, deformation temperature,and recovery temperature on shape recovery. We discuss how molecular design parameters as well as shape programmingconditions can be tailored to alter recovery characteristics, such as the recovery temperature and its breadth. Finally, weconclude with an outlook of the modeling approach on elucidating the effects of moisture, polymer chemistry as well asphase morphology on SMP performance.

2. Materials and methods

2.1. Generic coarse-grained potentials

The key components of a thermally activated copolymer SMP are a glassy phase that remains glassy, and a rubbery phasethat undergoes a glass transition, over the range of operating temperatures. It is this combination of glassy and rubbery

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phases that produces the shape-memory effect. It follows that if we wish to examine the molecular mechanisms involved, asa first application representative of a broad class of materials, it is sufficient to use “generic” (i.e. not intended to representany polymer in particular) coarse-grained potentials, as long as they reproduce this combination of a glassy phase and arubbery phase in transition. To model the rubbery phase (soft beads), we use a potential based on one for cis-PI developedby Li et al. (2011). The relationship between the Li potential and the generic rubbery potential used here is illustrated inFig. 1. The inter-monomer bonding potential is unaltered, while the bending potential is simplified, and the non-bondedpotential is adjusted to include both attractive and repulsive terms. Quadratic forms were used for the bond and anglepotentials, while a 12-6 Lennard–Jones (LJ) potential was used for the non-bonded interactions, as in Eqs. (1)–(3), wherekb¼105.37 ϵ=s2, b0 ¼ 6 ffiffiffi

2p

s, kθ ¼ 2:35ϵ, and θ0 ¼ π. In implementation, the LJ potential was truncated at a cutoff distance ofrc ¼ 5s. The conversions between LJ and real units are shown in Table 1.

Ub ¼ kbðb�b0Þ2; ð1Þ

Uθ ¼ kθðθ�θ0Þ2; ð2Þ

ULJ ¼ 4ϵsr

� �12� s

r

� �6� �

ð3Þ

To model the glassy phase (hard beads), we construct a new potential by scaling the parameters of the rubbery potential,as shown in Fig. 2, which serves to stiffen polymer segments and to increase the glass-transition temperature Tg. The scalingfactors used are also shown in Fig. 2, where a superscript of HH or SS indicates the quantity pertains to hard–hard (glassy) orsoft–soft (rubbery) interactions, respectively. For the sake of simplicity, as commonly done in CG models, the masses of thehard and soft beads are assumed to be equal.

Using these potentials, we construct polymeric system models containing soft and hard beads, wherein the soft beadsbehave similarly to cis-PI, and the hard beads are significantly stiffer (i.e. have an increased Tg). Moreover, using mixing rulesto calculate the potentials of cross-interaction, we construct a system in which these soft and hard beads are intermingled. Ageometric mixing rule was used for the LJ potential, while an arithmetic mixing rule was used for the bond and anglepotentials. The relevant scenario in this work, wherein separation of soft and hard phases occurs, is when the beads areconnected to resemble block copolymers.

2.2. Model generation

Three unique simulation boxes with periodic boundary conditions were created containing 50 chains, treated as self-avoiding randomwalks, with N¼ 100 monomers per chain. These simulation boxes were then equilibrated in two steps, firstusing a soft (cosine) non-bonded potential (Kremer and Grest, 1990) for 10k timesteps (92.4 ps with dt ¼ 0:005τLJ), and thenusing the LJ potential in Eq. (3) in combination with the bond-swap algorithm (Sides et al., 2004) for 2M timesteps (18.5 ns).In both cases, the same quadratic potentials for bond and angle interactions described in Eqs. (1) and (2) were used. Hard-phase regions were then inserted as spheres of radius rw, as given by Eq. (4), where w is the hard-phase fraction and l0 is the

Fig. 1. The generic rubbery potential used in this work (solid lines) and the Li potential for cis-PI (dashed lines).

Table 1Conversions between Lennard–Jones and real units.

Unit Symbol Expression Value

Energy ϵ — 0.6050 kcal/molDistance s — 3.564 ÅMass m — 68.11 g/molTime τLJ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffims2=ϵ

p1847 fs

Temperature Θ ϵ=kB 304.4 KPressure Π ϵ=s3 950.0 atm

Fig. 2. Potentials for the soft–soft (solid lines) and hard–hard (dashed lines) bead interactions. Mixed bond and angle potentials are calculated usingarithmetic rules, while mixed non-bonded potentials are calculated using a geometric rule.

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initial edge length of the simulation box (� 18:35s).

rw ¼ 3w4π

� �1=3

l0 ð4Þ

The result is a copolymer with aspects of both block and random nature. The systems with nonzero hard-phase fraction werethen further equilibrated using the combined LJ/bond-swap method for 4M timesteps (36.9 ns) to help remove initial stresses inthe hard-phase region. This was done for w¼0, 0.05, and 0.1, making a total of nine simulation boxes. The results presented for agiven value of w in this paper are the composite results of the three simulation boxes with that value of w. A post-equilibrationimage of one such simulation box with w¼0.1 is shown in Fig. 5(a). Note that this is one cell in a periodic structure; that is, weare actually simulating a simple cubic lattice of spherical hard-phase regions within a soft-phase matrix. All simulations in thiswork were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator1 (LAMMPS) (Plimpton, 1995).

2.3. Glass transition simulations

The systems with 0% and 10% hard phase were then equilibrated for 2M timesteps (18.5 ns) using a Langevin thermostatand Berendsen barostat (P¼0) at each of ten temperatures between 0.5 and 1 ϵ=kB, and the volume was recorded at eachtemperature, as in Fig. 3. The glass transition is observed at Tg ¼ 0:79ϵ=kB in both cases, but the transition is less apparent forw¼0.1. This is due to a reduction in the composite coefficient of thermal expansion in the binary system caused by thepresence of the hard (glassy) phase. In the thermomechanical simulations described in the following section, thistemperature of T ¼ 0:79ϵ=kB (240 K) will be used as the transition temperature for shape-memory behavior. Due to thepotential modifications, we note that this material no longer behaves identically to cis-PI, so it is not surprising that the Tg isoutside the 200–233 K Tg range for cis-PI (Van Krevelen and Te Nijenhuis, 2009).

2.4. Thermomechanical simulation cycles

Each system was subjected to the thermomechanical simulation cycle illustrated in Fig. 4, where the deformationtemperature is Th ¼ 1ϵ=kB (1:27Tg), the cooling temperature is Tl¼0.1 ϵ=kB (0.127 Tg), and the recovery temperature is Tr

varies between 0.1 and 1.5 ϵ=kB (0.127 and 1.90 Tg). For w¼0.1, simulations with Th ¼ 1:3ϵ=kB (1.65 Tg) were also performed.First, the systems were again equilibrated at Th for 1M timesteps (9.24 ns) using a Langevin thermostat and Berendsenbarostat (P¼0). The systems were then subjected to the isochoric mechanical deformation described by the deformationgradient Fm in Eq. (5), over 2M timesteps (18.5 ns) using a Langevin thermostat. Typical strain amplitudes in theexperimental literature are on the order of the 100% strain used in the current work, but are often geometrically complex.After the mechanical deformation, the temperature was decreased linearly at fixed volume from Th to Tl over 2M timesteps(18.5 ns). This led to a decrease in volume and an excess in hydrostatic pressure, which was relieved in the following stepusing a fully coupled barostat over 2M timesteps. This was followed by re-heating to Tr at constant volume over anadditional 2M timesteps, and subsequent removal of hydrostatic pressure over yet another 2M timesteps. Note that forTr ¼ Tl, the previous two steps are omitted. At this point in the cycle, all that remains is the recovery step. This wasperformed by de-coupling the spatial dimensions of the barostat (i.e. removing the remaining deviatoric stress), whichallows the system to “snap back” in the direction of the initial state. This recovery was allowed to progress over 10Mtimesteps (92.4 ns). Note that the ideal case is illustrated in Fig. 4, wherein the initial and final configurations are equivalent.In actuality, irrecoverable strains prevent the cycle from being closed.

Fm ¼ 2 e1 � e1ð Þþ 1ffiffiffi2

p e2 � e2þe3 � e3ð Þ ð5Þ

Fu ¼ Fr � F2T � F1T � Fm ð6Þ

Fig. 3. Volume vs. temperature plots for (a) 0% and (b) 10% hard phase. The glass transition at T¼0.79 ϵ=kB is observable in each case, but it is lessprominent for w¼0.1.

Fig. 4. Generalized schematic of thermomechanical simulation cycles (in the ideal case of full recovery).

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χ ¼ 1�ðFu�1Þ � ðFm�1Þ�1 ð7Þ

χ ¼ xm�xuxm�x0

ð8Þ

The ultimate state of the simulation box is described by the deformation gradient Fu, as in Eq. (6), which is thecomposition of the deformation gradients Fm, F1T , F

2T , and Fr, whose individual roles are indicated in Fig. 4. Specifically, Fm

imposes the isochoric mechanical deformation described by Eq. (5); F1T and F2T allow the volumetric contraction andexpansion associated with the temperature changes �ΔT1 ¼�ðTh�TlÞ and ΔT2 ¼ Tr�Tl, respectively, whose amplitudes varydepending on the simulation, and are indicated by the values of Th, Tl, and Tr given previously; and Fr finally brings thesystem to its ultimate state through removal of the remaining deviatoric stress after reheating. Now, in the ideal case, theentirety of the applied mechanical strain should be undone at the end of the cycle, so it is useful to measure the shape-memory behavior of these systems in terms of the fraction of strain recovered. Thus, we define a strain-recovery tensor χ asin Eq. (7). Due to the simple nature of the imposed deformation, this tensor is symmetric and its components are triviallydiagonalized (principal stretches are along coordinate axes), but it may also be applied in cases with more complicatedloading. We are primarily concerned with recovery in the direction of extension e1, so in the following sections we focus onthe scalar χ ¼ e1 � χ � e1, which can be calculated directly using Eq. (8), where x0, xm, and xu are shown in Fig. 4.

3. Results and discussion

In this section, we analyze the results of our shape-memory cycle simulations. We begin by examining an initial,individual shape memory cycle, wherein the recovery temperature and the deformation temperature are the same(Tr ¼ Th ¼ 1:27Tg), and the material has 10% hard phase (w¼0.1). This leads to a discussion of the relevant timescales ofrelaxation and the mapping between CGMD-simulation and real time. We follow this with an analysis of the effect ofrecovery temperature on strain recovery, whereby we observe a well-defined shape-memory transition, and we examinethe effect of deformation temperature on this transition. We use recovery temperatures ranging between 0.127 and 1.90 Tg,

Fig. 5. (a) Simulation path through stress–strain–temperature space. The paths are as follows: 1–2, NVT isochoric mechanical deformation; 2–3, NVTcooling; 3–4, NPT hydrostatic relaxation; 4–5, NVT heating; 5–6, NPT hydrostatic relaxation; and 6–7, NPT deviatoric relaxation (recovery). (b–d) Snapshotsof the thermomechanical cycle for w¼0.1 and T r ¼ Th ¼ 1:27Tg: (b) initial configuration (point 1 in (a)), (c) after programming (point 4 in (a)), and (d) afterrecovery (point 7 in (a)). Soft-phase beads are shown as diffuse particles, while hard-phase beads are shown in green. Partial disintegration of the hard-phase core is evidenced by the isolation of hard-phase beads from the central cluster in (c) and (d). (For interpretation of the references to color in thisfigure caption, the reader is referred to the web version of this paper.)

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and we use deformation temperatures of 1.27 Tg and 1.65 Tg. Finally, we investigate the effect of hard-phase fraction onstrain recovery over the full range of recovery temperatures but with constant deformation temperature of Th ¼ 1:27Tg,using additional phase fractions of w¼ 0 and 0.05.

3.1. Recovery and timescales

The standard shape-memory cycle used in experiments is one in which the deformation temperature and the recoverytemperature are the same. For this reason, we begin by presenting a shape-memory cycle defined by w¼0.1 andTr ¼ Th ¼ 1:27Tg, whose progress is illustrated in Fig. 5, and addressing the issues of irrecoverable strain and recoverytimescale. The full path of the simulation in stress–strain–temperature space is shown in Fig. 5(a), while snapshots of thesystem at key points in this path are shown in Fig. 5(b–d), with the soft phase shown as diffuse particles and the hard phaseshown in green. Note that in comparing Figs. 4 and 5(a), the mappings associated with each deformation gradient can berepresented as Fm : 1-2, F1T : 3-4, F2T : 5-6, and Fr : 6-7. The deformation gradients associated with the mappings 2-3and 4-5 are both the unit tensor 1, as these legs of the simulation are performed in the NVT ensemble. In all three systems,as the initial mechanical strain is imposed, the hard-phase core experiences partial disintegration, which is not resolved atthe end of the cycle. Approximately one third of the imposed mechanical strain remains unrecovered at the end of theshape-memory cycle. Thus, we propose that this partial disintegration (i.e. plastic deformation of the glassy phase) is asignificant contributor to this unrecoverable strain. However, as shown in the following section, roughly half of this excessstrain can be removed by allowing the system to recover at higher temperatures (Tr4Th). That is, either the timescale ofrecovery at Tr ¼ 1:27Tg is greater than the simulation time allotted, or the system is still partially arrested and cannotrecover even for Tr ¼ Th.

The amount of shape recovery observed here necessarily depends on the amount of stress relaxation within the softphase. Thus, additional equilibrium simulations were performed, starting from the initial, undeformed configurations, inorder to observe the decay of the shear-stress and end-to-end-vector correlation functions at T¼1.27 Tg for both w¼0 andw¼0.10 over a time period equal to the recovery time tr (results not shown). Neither of the end-to-end correlations nor the

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stress correlation for w¼0.10 were observed to decay below 70%. However, the stress correlation for w¼0 was observed todecay to approximately 4%. Again continuing our analogy to cis-PI, the polymer chain length used in this study (N¼100) ison the order of the entanglement length (Ne) of cis-PI predicted by Li et al. (2011) (Ne ¼ 76) and measured by Fetters et al.(1994) (Ne ¼ 46) and Riedel et al. (2010) (Ne ¼ 74). Thus, it is unsurprising that the stress would relax much more quicklythan the end-to-end vector in our model. The result is that our recovery time tr is large enough to account for the vastmajority of the recovery possible at this temperature. However, if the polymer network were highly entangled (NbNe), therecovery time would need to be extended significantly. Specifically, it would need to be on the order of the disentanglementtime of the tube model developed by Doi and Edwards (1988) and De Gennes (1979).

The recovery time tr used in our simulations was 10M timesteps or 9:234� 10�8 s. However, it has been shown that anon-trivial dynamic mapping exists between all-atom and coarse-grained models, due to the loss of friction that occurs as aresult of the coarse-graining process; that is, CGMD simulations sample through configurations faster than all-atomsimulations due to the loss of friction and reduced degrees of freedom. As such, a time-scaling factor is required to properlyframe the results of dynamic CGMD simulations (Padding and Briels, 2002; Harmandaris and Kremer, 2009; Padding andBriels, 2011). Li et al. (2012) calculated this scaling factor as the ratio of decorrelation times and found that their CGMDmodel was 11.47 times faster than the MD model fromwhich it was derived. The time-scaling factor is a measure of the lossof dissipative degrees of freedom sustained during the coarse-graining process; that is, it relates the complexity of thecoarse-grained model to that of the original. Since our CG model is generic and does not pertain to a particular all-atompolymer structure, there is no singularly relevant time-scaling factor that can be identified. However, if we also assume thatthe chemical structure of the hypothetical material we are modeling shows similar behavior (relative to our CG model) tothat of cis-PI (relative to the Li model), then we can claim the relevant time-scaling factor is on the order of the Li result (Liet al., 2012). Time-scaling factors on the order of 2–10 are common for low degrees of coarse graining, such as monomer tosingle bead mappings of linear polymers (Depa and Maranas, 2005; Padding and Briels, 2011; Li et al., 2012). Thus, we claimthat the actual time of recovery t′r in our simulations is approximately 1� 10�6 s.

Deformation- and heating/cooling-rate sensitivity were examined by also performing shape-memory cycles for w¼0.10and Tr ¼ Th ¼ 1:27Tg with deformation and heating/cooling periods two and three times as long as mentioned in Section 2.4.The effects on recovery were found to be minor, and are therefore not shown. However, as the material is viscoelastic, rateeffects can play a significant role in shape-memory cycles performed on experimental timescales, and should therefore betaken into consideration in that context. Castro et al. (2010) examined the effect of thermal rates on the thermomechanicalbehavior of amorphous SMPs; the interested reader is referred to their work.

3.2. The effect of recovery and deformation temperatures

First we examine the effect of recovery temperature at fixed hard-phase fraction and deformation temperature. The fullshape-memory cycle was performed and the strain recovery ratio calculated at multiple recovery temperatures for w¼0.1 andTh ¼ 1:27Tg, as shown in Fig. 6(a), where error bars indicate variation in behavior among the three initial configurations. BelowTg, increasing the recovery temperature gradually increases the mobility of polymer chains, leading to the observed lineartrend. In the vicinity of the glass transition, a change from glassy to rubbery kinetics occurs within the soft phase, dramaticallyincreasing the number of accessible conformations, causing a sharp upturn. However, recovery does not plateau to a finitemaximum value until the recovery temperature exceeds the deformation temperature (Tr4Th). This is emphasized in Fig. 6(b), where we see a discontinuity in recovery times near Tr ¼ Th for large recovery ratios. The overall sigmoidal trend inrecovery agrees qualitatively with experimental results in unconstrained recovery; however, in experiments, the recoveryplateau is reached before Tr ¼ Th (Qi et al., 2008; Nguyen et al., 2008; Srivastava et al., 2010b). The dynamics of the recoveryprocess are illustrated in Fig. 6(c). This figure illustrates that increasing the recovery temperature dramatically increases theamount of recovery at short to intermediate times. This is also observed experimentally, but with an initial delay before theonset of recovery (Westbrook et al., 2011), a difference we attribute to scale effects.

Next, we examine the effect of deformation temperature at fixed hard-phase fraction. The full series of shape-memorycycles with different recovery temperatures was performed again using an increased deformation temperature ofTh ¼ 1:65Tg with w¼0.1. The results are shown in Fig. 6(a). Increasing the deformation temperature causes the χ curve tobe shifted toward higher temperatures. That is, at low and intermediate temperatures, an increase in recovery temperatureis required to obtain the same amount of recovery. A similar phenomenon was observed by Yakacki et al. (2012) whencomparing experiments with deformation temperatures above and below Tg. Hu et al. (2005) also observed that for constantrecovery and fixing temperatures, increasing the deformation temperature above Tg leads to a decrease in recovery ratio.A simple explanation for this behavior is that when the system is deformed at higher temperatures, its components respondwithin a greater volume of phase space, which is not available during recovery at lower temperatures, even within thetransition region observed previously. That is, deformation proceeds relatively unimpeded through a free-energy landscapeover the multidimensional phase space, due to high deformation temperature and imposed force, and the system isprogrammed into a different state than it would have been at a lower deformation temperature. On the return path over thismultidimensional phase space, at a given temperature below Th, impediments cannot be easily overcome or circumventedwithout the additional volume of accessible phase space afforded by the higher deformation temperature. When themultidimensional free-energy landscape is mapped to a single reaction coordinate (in this case, the edge length of thesimulation box), the result is the appearance of an energy barrier between the partially and fully recovered states, which can

Fig. 6. (a) Strain recovery ratio vs. recovery temperature for Th ¼ 1:27Tg (solid line) and Th ¼ 1:65Tg (dashed line), (b) recovery time vs. recoverytemperature with Th ¼ 1:27Tg for several percentages of strain recovery, (c) strain vs. time during recovery for multiple recovery temperatures with w¼0.1and Tr ¼ 1:27Tg, and (d) a graphical illustration of the recovery effect of deformation temperature in a simplified, one-dimensional system. Error bars in (a)indicate variation in behavior among the three initial configurations. The recovery temperature ranges from 0.127 to 1.90 Tg. The shape-transitiontemperature depends on the glass transition temperature of the amorphous switching domains, as well as the original deformation temperature duringprogramming. As Th increases, the polymers become intermingled in such a way during programming that they cannot deconvolute during recoverywithout high temperature.

B.C. Abberton et al. / J. Mech. Phys. Solids 61 (2013) 2625–26372632

only be overcome (on the timescales observed here) at or above the deformation temperature during recovery. This effect isillustrated in the schematic shown in Fig. 6(d), where the energy barrier along a reaction coordinate between the partiallyand fully recovered states at a given recovery temperature increases as a function of the original deformation temperature.Note that this is an idealized schematic, and the landscape will be much rougher in actuality.

3.3. The effect of hard-phase fraction

Here we examine the effect of hard-phase fraction at various recovery temperatures. The full shape-memory cycle wasperformed at each recovery temperature for two additional hard-phase fractions (w¼0 and w¼0.05), with a deformationtemperature of Th ¼ 1:27Tg. Increasing the hard-phase fraction increases resistance to flow during mechanical deformation,leading to greater overall stresses, as shown in Fig. 7(a). The effect of the hard-phase fraction on strain recovery behavior isillustrated in Fig. 7(b). A larger hard-phase fraction leads to higher strain recovery at low to intermediate temperatures,widening of the transition region, and an eventual crossover at high temperatures. It also increases stress heterogeneity inboth phases, as defined by the standard deviation of the per-monomer stress (not shown). As shown in Fig. 8(a–c), thedistribution of per-atom stress changes with increasing w such that the hard phase carries a greater fraction of the load. Thisis a result of the connectivity of the hard- and soft-phase regions across their boundary, and the magnitude of the imposedstrain. This load transfer occurs throughout the deformation process, and is inhomogeneous over the surface of the phaseboundary, leading to stress concentrations and partial disintegration of the hard phase. The normalized mean-squareddeviations (MSD) of the individual beads at the end of the shape-memory cycle with Tr ¼ Th ¼ 1:27Tg are shown in

Fig. 7. (a) Stress vs. strain during mechanical deformation, and (b) recovery ratio χ as a function of recovery temperature Tr. Note that the deformationtemperature is Th ¼ 1:27Tg. Increasing the fraction of glassy phase results in resistance to flow during deformation, higher strain recovery at low tointermediate temperatures, a widening of the transition region, and an eventual crossover at high temperatures. Error bars in (b) indicate variation inbehavior among the three initial configurations. This variation is largest at T r ¼ Th, which reinforces the importance of the deformation temperature as acritical transition point, as discussed in Section 3.2.

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Fig. 8(d–f) to give a measure of mobility. The MSD is distributed heterogeneously throughout the bodies, and it is larger inthe soft phase than in the hard phase. However, we do see concentrations of large motion within the hard phase,corresponding to those beads that were pulled out of the core during mechanical deformation.

The deformation and recovery of the hard phase specifically are illustrated in Fig. 9. The asphericity of the hard-phasecore was calculated from the eigenvalues of the normalized gyration tensor S, as in Eqs. (9) and (10), where the positionvectors rðkÞ are with respect to the center of mass of the hard-phase core, and plotted alongside the rigid and affine limitingcases during mechanical deformation (Fig. 9(a)) and recovery (Fig. 9(b) and (c)):

S ¼ S � S�1t ¼ 0; Sij ¼

1N

∑N

k ¼ 1rðkÞi rðkÞj ð9Þ

β¼ λ1�12 λ2þλ3ð Þ; λ14λ24λ3 ð10Þ

A transition between rigid and compliant response occurs during mechanical deformation, which is the cooperativeeffect of elastic and plastic deformation. In Fig. 9(b), we see that a significant portion of this asphericity remains for bothw¼0.05 and w¼0.10 after recovery at Tr ¼ Th ¼ 1:27Tg. However, after recovery at Tr ¼ 1:90Tg, the asphericity appears tovanish for w¼0.05, yet remain for w¼0.10. Thus, a larger fraction of the hard-phase deformation is plastic for w¼0.10 thanfor w¼0.05. This is not to say that the hard-phase core is fully intact at the end of the cycle for w¼0.05; in fact, about 5% ofthe hard-phase beads remain dislocated from the core even after the higher-temperature recovery. This kind of plasticdeformation, and the irrecoverable strain associated with it, can be minimized by promoting chemical crosslinking withinthe hard-phase core and by using highly immiscible hard- and soft-phase monomers. Alternatively, irrecoverable strains canbe controlled by limiting the maximum strain imposed (and thus the amount of plastic deformation) during SMP cycling;this is observed experimentally to be the case (Hu et al., 2005).

Increasing the hard-phase fraction also leads to an increase in strain recovery at low and intermediate temperatures,causing the shape-memory transition to become less well-defined, but a crossover in recovery ratio occurs for recoverytemperatures above Th, as shown in Fig. 7(b), where error bars indicate variation in behavior among the three initialconfigurations. Variation in recovery behavior among the three initial configurations is largest at the deformationtemperature Th, which further highlights the importance of this temperature discussed in Section 3.2. The increase inrecovery at temperatures below Th occurs because a greater portion of the initial mechanical deformation is elastic, due tothe increased capability of elastic energy storage within the hard-phase core. This leads to an apparent increase of theshape-memory-transition breadth, as defined by the reciprocal slope of the recovery vs. temperature curve at transition. Inthe extreme case when there is no soft phase present in the material, and the temperatures are well below Tg of the hardphase, we expect to observe no shape memory behavior. In this case, the strain recovery curve would be linear with a smallslope due to thermal expansion, but near zero because the majority of the deformation would be plastic. The crossover athigh temperatures between w¼0.05 and w¼0.10 is due to the relative amounts of hard-phase disintegration. That is,increasing w increases stress heterogeneity and the amount of irrecoverable strain developed via this mechanism of plasticdeformation.

In summary, we find that partial disintegration of the glassy phase contributes to irrecoverable strain; that nearly fullrecovery is observed above the trigger temperature, which is near the glass-transition temperature of the soft-phase Tg, but

Fig. 8. Normalized axial per-atom stress after mechanical deformation (a–c) and normalized mean-squared deviation (MSD) at the end of the shape-memory cycle (d–f) for T r ¼ Th ¼ 1:27Tg with w¼0 (a, d), w¼0.05 (b, e), and w¼0.10 (c, f). Beads are shown in their initial configurations with radii scaledaccording to the relative magnitude of stress or MSD in the appropriate configuration. The mean drift has been removed from the MSD calculations. Thesigns of the per-atom stresses are color-coded as follows: in the soft phase, pink indicates tension and blue indicates compression; while in the hard phase,green indicates tension and red indicates compression. Normalizations are not absolute (i.e. bead sizes should not be compared across different values ofw). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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is correlated with the initial deformation temperature Th; and that increasing the amount of hard phase leads to increasedelastic recovery at temperatures below Th, causing an apparent widening of the shape-memory transition, but leads to acrossover at high temperatures due to increased disintegration of the hard-phase core.

4. Conclusion

Toward advancing the field of bottom-up study of shape memory polymers through molecular simulation, and with theimmediate goal of circumventing the limitation of timescale associated with all-atom molecular dynamics, we havereproduced shape-memory behavior in a copolymer model defined with coarse-grained potentials, by inserting sphericalregions of hard phase within soft-phase matrices, and subjecting the systems to multiple shape-memory cycles. The timeallowed for recovery at the end of these cycles is on the order of the relevant relaxation mechanisms. Cycles were performed

Fig. 9. Asphericity of the hard-phase core (a) during mechanical deformation, and during recovery with (b) T r ¼ Th ¼ 1:27Tg and (c) Tr ¼ 1:90Tg. The casesof affine deformation (solid or dashed red) and perfect rigidity (dashed black) are also shown for comparison. There occurs during mechanical deformationa rigid-to-compliant transition, which is the result of partial disintegration and elastic stretching of the hard phase. Recovery at Tr ¼ Th ¼ 1:27Tg does notfully remove the asphericity for either hard-phase fraction, but at Tr ¼ 1:90Tg, the asphericity appears to vanish for w¼0.05. Thus, a greater amount of thehard-phase deformation is plastic for w¼0.10. (For interpretation of the references to color in this figure caption, the reader is referred to the web versionof this paper.)

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using various hard-phase fractions, recovery temperatures, and deformation temperatures, in order to elucidate howmolecular design parameters and shape-programming conditions can be tailored to alter recovery characteristics.

Increasing the recovery temperature increases the strain recovery ratio, linearly in the glassy region, with a noticeableupturn near the glass transition temperature of the soft phase, leading into an apparent plateau at high temperatures.Increasing the deformation temperature shifts the shape-memory transition (trigger) temperature, such that a higherrecovery temperature is required to recover the same amount of imposed strain, until the plateau regions overlap. That is,the shape-memory transition temperature is near the glass transition temperature of the soft phase, but it is also affected bythe deformation temperature used during programming. As such, one can fine-tune the trigger temperature by altering thedeformation temperature.

Increasing the hard-phase fraction reduces softening during mechanical deformation (with more elastic energy beingstored in the hard-phase core), which will lead to increased recovery stresses in the case of constrained recovery, which isdesirable in many biomaterial applications. In addition, increasing the hard-phase fraction increases the strain recovery ratioat recovery temperatures below the deformation temperature, due to the increase in elastic energy storage within the hard-phase core, causing the shape-memory transition to become less well-defined, but it also results in an eventual crossover athigh recovery temperatures, due to an increase in localized plastic deformation (partial disintegration of the hard-phasecore). The result is that recovery stress and the shape of the strain-recovery curve can be adjusted through phase fraction.

Future investigations of shape-memory behavior will address the strain-recovery effects of monomer chemistry, low-molecular-weight solvents, physical and chemical crosslinking, different phase-separation morphologies, and morecomplicated mechanical deformation. The methodology used in this work can be extended in straightforward ways tostudy each of these effects. Chemistry-specific coarse-grained potentials can be incorporated to study the effect of monomerchemistry, solvent molecules can be included in the initial construction of the simulation boxes, crosslinks may be simulatedby inserting intermolecular bonds between monomers, dissipative particle dynamics or other “soft” potentials may be used

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to create more realistic phase morphology, and more complicated mechanical deformations can be imposed by using atriclinic box structure. The ability to model these chemical, microstructural, and environmental aspects within molecularsimulation will propel the state of the art toward predictive modeling capabilities for stimuli-responsive polymers.Specifically, we look forward to developing in our future work a predictive model based on molecular modeling that willestimate performances such as strain recovery, optimal recovery temperature, and recovery stress, using material andprocessing parameters as input; and to using this molecular-level model to inform existing or develop new continuum-levelmodels.

Acknowledgments

B.C.A is grateful to Ying Li, Luis Ruiz, and Miguel Bessa for their participation in multiple helpful discussions. W.K.L.would like to acknowledge support from NSF CMMI Grants 0823327 and 0928320, and NSF IDR CMM Grant I 1130948, aswell as the World Class University Program through the National Research Foundation of Korea (NRF) funded by theMinistry of Education, Science and Technology (R33-10079). S.K. would like to acknowledge support from NSF CBET Grant1234305. Resources of the Quest and Hercules computing clusters at NU were utilized in this research.

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