Solid -state nanofoaming experiments are conducted on two PMMA grades of markedly different molecular weight using CO 2 as the blowing agent. The dependence of p orosity of the nanofoams upon foaming time and foaming temperature is measured . Also, the microstructure of the PMMA nanofoams is charac terized in terms of cell size and cell nucleation density. A one dimensional numerical model is developed to predict the growth of spherical, gas -filled voids during the solid -state foaming process. Diffusion of CO 2 within the PMMA matrix is sufficiently r apid for the concentration of CO 2 to remain almost uniform spatially . The foaming model makes use o f experimentally calibrated constitutive laws for the PMMA grades , and t he effect of dissolved CO 2 is accounted for by a shift in the glass transition temper ature of the PMMA . The observed limit of achievable porosity is interpreted in terms of cell wall tearing; it is deduced that the failure criterion is sensitive to cell wall thickness. Keywords: solid -state foaming, PMMA nanofoams, molecular weight, void growth model , porosity limit , deformation maps 1. Introduction Polymeric nanofoams are polymer foams with a n average cell size of below 1 micrometer [1]. This relatively new class of porous solids has the potential to offer unique and attractive combinatio ns of thermal, mechanica l, and optical properties [2–4]. For example, t he thermal Preprint of http://doi.org/10.1098/rspa.2019.0339 2 conductivity of polymeric nanofoam s can be lower than that of air (= 0.025 W m-1K-1). When the average cell size is in the order of the mean free path of the gas molecules in the cells (close to 70 nm for air at standard conditions ), the thermal conductivity of the gas in the foam is significantly reduced due to the Knudsen effect [5,6] . Wang et al. [7] calculated that a polymeric nanofoam has a thermal conductivity close to or below 0.025 W m-1K-1 when the average cell size l is below 200 nm and the p orosity f exceeds 0.85. To achieve this morphology , the cell nucleation density dN must exceed 1021 m-3 [1]. A large number of experimental studies focus on the effect of processin g conditions and the choice of polymer precursor upon the cell nucleation density dN , the void size l and the porosity f of polymeric nanofoam s, as reviewed by Costeux [1]. Many of these studies make use of the solid -state foaming method in which a physical blowing agent (e.g. CO 2) is used to nucleate and grow cells in a polymer matrix such as polymethyl methacrylate (PMMA ) [8,9] . The review by Costeux [1] on experimental studies o f solid -state nanofoaming discusses the trade -off between porosity and cell siz e of a polymeric nanofoam . Polymeric nanofoam s of l < 200 nm are reported for a nucleation density above 1021 m-3, but their porosity is limited to close to 0.85, see , for example, Aher et al. [10] and Costeux and Zhu [11].The observed porosity limit for n anofoams with a nucleation density above 1021 m-3 may be due to the fact that the walls between the nano -sized cells are limited by the end -to-end distance of the individual polymer chains [1,12] . Polymeric nanofoam s of porosity on the order of 0.8 to 0.9 have been produced, but their cell size is well above 20 0 nm (and dN << 1021 m-3 ) [1,9,13] . The microstructural requirement for polymeric nanofoam s with a thermal conductivity lower than the thermal conductivity of air is currently beyond the practical limit of the state-of-the- art solid -state nano foaming proces s [7]. The final porosity and final cell size in solid -state nanofoaming can be predicted by simulating void growth . In contrast to the substantial body of experimental work on polymeric nanofoam s produced by solid -state foaming , as reviewed by Costeux [1], and the development of cell growth model s for liquid state foaming process es [14–16], theoretical studies on cell growth during solid -state nanofoaming are limited. Costeux and co -workers [13,1 7] simulated void nucleation and void growth during the solid -state nanofoaming of acrylate co -polymers by making use of the model of Shafi et al. [18]. They conducted a series of nanofoaming Preprint of http://doi.org/10.1098/rspa.2019.0339 3 experiments but their model overestimate s the measured final porosity of the nanofoams. Th e mismatch between the simulated and the measured porosity of acrylic nanofoams may be due to (i) the assumption that cell growth continues until the foaming temperature attains the glass transition temperature of the polymer -gas solid and /or (ii) the assumption that the polymer -gas solid surrounding the cell is in a liquid (viscous ) state throughout the solid -state foaming process . In reality, void growth occurs at a temperature above and below the glass transition temperature of the solid surrounding the void. This is addressed in detail in the present study. Scope of study PMMA nanofoams are produced from two PMMA grades of widely different molecular weight ; a solid -state foaming process is used with CO 2 as the blowing agent . We characterize the microstructure of the nanofoams in terms of porosity f, cell size l, and cell nucleation density dN . In addition, we develop a void growth model , based on the constitutive law of PMMA grades close to the glass transition temperature , by b uilding on the recent study of Van Loock and Fleck [19]. Both predicted and measured final porosit ies are obtained as a function of foaming time and foaming temperature ; also, cell wall tearing mechanisms are considered in order to account for the observed limit in final porosity . 2. Nanofoaming experiments 2.1 Materials Foaming experiments were conducted on two PMMA grades : pelletized PMMA (Altuglas V825T) of average molecular weight1 wM = 92 500 g mol-1 and c ast PMMA sheets ( Altuglas CN with sheet thickne ss close to 3 mm) with wM = 3 580 000 g mol-1. We shall refer to the Altuglas V825T and Altuglas CN grades as ‘low wM PMMA’ and ‘high wM PMMA’ , respectively. Both grades have a density p equal to 1 190 kg m−3 (as measured at 23 °C and at 50% relative humidity). The glass transition temperature ( gT = 114.5 ̊C) of the low wM 1 The average molecular weight was measured by gas permeation chromatography ( GPC ) with an Agilent Technologies PL GPC220 ( USA) instrument with a nominal flow rate equal to -5 11.67 10 l s− at a test temperature equal to 30 ̊C. Preprint of http://doi.org/10.1098/rspa.2019.0339 4 PMMA is close to the glass transition temperature ( gT = 116.5 ̊C) of the high wM PMMA as measured by differential scanning calorimetry (DSC) using a heating rate of 10 ̊C min-1. 2.2 Solid -state nano foaming experiments Foaming precursors of the low wM and high wM PMMA grades were made as follows. The low wM PMMA pellets were heated to 250 ̊C for 450 s, then compressed for 60 s between two heated plates at a pressure equal to 17 MPa . The resulting sh eet was cooled to room temperature with the pressure of 17 MPa maintained. Cuboid precursors of dimension 20x10x3 mm3 were machined from the low wM PMMA sheet and from the as -received high wM PMMA sheet. Foaming experiments were performed using a pressure ve ssel2 with a pressure controller3 and temperature controller4. Medical grade CO 2 (> 99.9% purity) was used as the blowing agent for the foaming experiments. The two step solid -state foaming process was employed, as detailed in the study of Martin -de León et al. [9]. The precursor samples were held in the pressure vessel at a constant CO 2 saturation pressure equal to 31 MPa , and at a constant temperature equal to 25 ̊C for 24 hours in order to ensure saturation of the CO 2 into the PMMA. The mass concentratio n5 C, at equilibrium , is close to 24 wt% for both the low and high wM PMMA , according to the meas urement procedure detailed by Martin -de León et al. [9]. Next, the pressure was progressively released to atmospheric pressure with an instantaneous pressure drop rate close to 100 MPa s-1. The samples were then foamed in a foaming bath6 at selected foaming temperatures (25 ̊C, 40 ̊C, 60 ̊C, 80 ̊C, 100 ̊C) and selected foaming times7 (60 s, 180 s, 300 s, and 600 s). It is assumed in the remainder of the study that the foaming times are sufficiently long for the temperature to be spatially uniform8 within the sample. 2 Pressure vessel model PARR 4681 of Parr Instrument Company (USA). 3 Pressure controller pump SFT -10 of Supercritical Fluid Technologies In c (USA). 4 Temperature controller CAL 3300 of CAL Controls Ltd (UK). 5 We define the mass concentration C of CO 2 in PMMA with respect to the total mass of the PMMA -CO 2 mixture. Note that the definition of CO 2 solubility (with respect to the mass of the PMM A absent CO 2) is used in the work of Martin -de León et al. [9]. 6 Thermal bath J. P. Selecta Model 6000685 of Grupo Selecta (Spain). The time between the pressure release and the sta rt of foaming was close to 120 s. 7 Samples were immersed in a water bath at a temperature close to 1 0 ̊C at the end of the foaming time . 8 The justification for this assumption is as follows . Immersi on of the sample in water or oil provides excellent heat transfer at the surface of the sample. The time constant 2/x= ≈ Preprint of http://doi.org/10.1098/rspa.2019.0339 5 2.3 Characterization of the PMMA nanofoams Porosity The density f of the foamed samples was determined by the water -displacement method with a weight balance9. A surface layer of depth 200 μm was removed by polish ing10 to ensure that the solid skin (of thickness b elow 100 μm ) was absent before the density measurements were made . The porosity f of the samples is obtained by: f p1f =− (1) where p ( = 1 190 -3kg m ) is the density of solid PMMA . Microstructure Foamed samples were cooled in liquid nitrogen and then fractured. The fracture surfaces were coated with a layer of gold by sputtering11, and m icrographs of the coated fracture surfaces were taken by a scanning electron microscope12 (SEM). The cellular structure of each material was characterised by analysing the micrographs w ith dedicated in -house software based on ImageJ/FIJ I [20]. Microstructural parameters such as the average cell size l, standard deviation s of the observed cell size s, and cell nucleation density dN , using the method as suggested by Kumar and Suh [21], were obtained13. Open cell content The open cell content of the foamed samples was measured by gas pycnomet ry14 with nitrogen in accordance with the ASTM D6226 -15 standard [23]. The open cell content ratio vO is 20 s where x = 1.5 mm is the half -thickness of the PMMA sample and 71.1 10−= m2 s-1 is the thermal diffusivity of PMMA at ro om temperature [22]. 9 Analytical balance AT261 of Mettler -Toledo (USA). 10 Grinding and polishing system LaboPOl2 -LaboForce3 of Struers (USA). 11 Sputter coater SDC 005 of Balzers Union ( Liechtenstein). 12 Scanning electron microscope QUANTA 200 FEG of Thermo Fisher Scientific (USA). 13 At least 200 cells were analysed from multiple micrographs per foamed sample. 14 Gas pycnometer (USA) AccuPyc II 1340 of Micromeritics (USA). Preprint of http://doi.org/10.1098/rspa.2019.0339 6 defined as the ratio of the volume of open pores to the total pore volume of a foam , and is obtained by: g p s v gVVVOfV−−= (2) where gV is the geometric volume of the foam, pV is the pycnometer volume and sV is a penalty volume to account for the volume of the cells at the surface of the foam . The penalty volume sV is assumed to be close to zero in the case of nanofoams. The geometric volume gV is measured by the water -displacement method as detailed above. Foamed samples were subjected to a pressure scan f rom 0.02 MPa to 0.13 MPa in the gas pycnometer. The pycnometer volume initially decreases as the gas pressure increases until the interconnected open cells are completely filled with gas and the pycnometer volume remains constant at increased pressures. We take this constant value of pycnometer volume pV in order to calculat e vO via Eq. (2). 3. Results nanofoaming experiments The measured porosity f , average observed cell size l, standard deviation s of observed cell sizes, and cell nucleation density dN of the nanofoams are reported in Tables 1 and 2 for the low wM and high wM grades of PMMA , respectively. In addition, a representative se ries of SEM micrographs of the nanofoams are shown in Figure 1. The low wM and the high wM nanofoams have contrasting microstructures and the cell nucleation density of the low wM nanofoams ( 20 3 d2 10 m N− ) is a n order of magnitude l ess than th at of the high wM nanofoams ( 21 3 d2 10 m N− ). The average cell size l of the high wM nanofoams ranges from 20 nm to 50 nm, and is an order of magnitude smaller than the average cell size of the low wM nanofoams (of size 200 nm to 350 nm ). These values of l and dN for the low wM nanofoam s are consistent with the results of Martin -de León et al. [9] who conducted solid - state foaming experiments with an identical low wM PMMA grade . The measured average cell size l of the low wM and the high wM nanof oams , as a function of foaming time ft for fT = 60 ̊C, is plotte d in Figure 2a. Void growth typically occurs over a foaming time period of Preprint of http://doi.org/10.1098/rspa.2019.0339 7 60 s to 180 s, followed by arrest . There is a mild dependenc e of the foaming temperature fT upon the final value for l , see Tables 1 and 2 . The measured porosity f of the nanofoams is plotted as a function of ft in Figure 2b for fT = 60 ̊C and for fT = 100 ̊C. Consistent with the l versus ft curves for fT = 60 ̊C, as presented in Figure 2a, the porosity increases over a foaming period of 60 s to 180 s until a stable ( ft - independent) value of final porosity is achieved. The highest observed porosity of the low wM PMMA nanofoams ( max 0.75 f= ) is approximately 25% higher than that of the high wM PMMA nanofoams ( max 0.60 f= ). At a foaming temperature of fT = 100 ̊C, the porosity decreases with increasing foaming time beyond ft = 60 s, and this is due to collapse of the foam ed structure . This behavior is also illustrated in plots of f versus fT , over the explored range of foaming times, see Figures 2c and 2d for th e low wM and high wM PMMA nanofoams, respectively . The measured open cell content vO is plotted as a function of the measured porosity f in Figure 3a (low wM ) and in Figure 3b (high wM ) for 20 ̊C f T 80 ̊C. Nanofoams with porosities well below the highest observe d porosity maxf are closed -cell in nature . An abrupt transition to an open -celled structure occurs close to maxf . The observed c ollapse of the foam at fT = 100 ̊C is preceded by cell wall failure for the low wM nanofoams (see Figure 1b) and by the formation of cra cks interconnecting the nano -sized pores in for the high wM nanofoams (see Figure 1d). 4. Void growth model A void growth model is now developed to predict porosity as a function of foaming time and foaming temperature for the PMMA nanofoams . The expansion of a pre-existing as -nucleated cavity during solid -state nanofoaming is simulated by means of a one dimensional single cell growth model [15,24] . A finite shell surrounds the void in order to account for void -void interaction in an approximate manner. Mor e sophisticated models of an array of voids (such as periodic cell models) could be adopted but the intent here is to emphasize the strong role of the evolving constitutive response. Consider a polymer -gas solid with equisized spherical voids . A cross -section of the undeformed (reference) configuration of the spherical void , with initial Preprint of http://doi.org/10.1098/rspa.2019.0339 8 radius 0a and initial outer radius 0b , along with the adopted spherical coordinate system (,,) r , is shown in Figure 4. Assume that the initial gas pressure 0p in the as-nucleated void equals the saturation pressure during the saturation phase prior to nucleation of the voids . The deformed configuration for the void of inner radius a and outer radius b at time t is shown in Figure 4. Kinematics Assume that the void remain s spheric al during growth and that the solid surrounding the void is incompressible . Then a material point, initially at radius R, is displaced to a radius r such that: 3 3 3 3 0 r a R a−−= (3) by incompressibility. For later use, this relation is re -arranged to the form : 3 3 3 0 01 1a ra R R a =+ −   (4) Note that /rR is a function of the time-like variable ( 0/aa ) and of the Lagra ngian position variable 0/Ra . The von Mises effective strain e is defined in the usual manner as 2 e2 3ij ij  = , giving : e| 2 | 2lnr R==  (5) where  is the hoop strain. Now insert Eq. (4) into Eq. (5) to obtain: 3 3 0 e 02l1n13a a Ra+  =− (6) and take the time deriv ative of r in Eq. (3) to give : 2 rrr v aa== (7) Preprint of http://doi.org/10.1098/rspa.2019.0339 9 where rv is the radial velocity of a material element at r. Consequently, the effective strain rate e reads: 3 2 e 32r ar RRvar− ==  (8) Equilibrium Write ( rr ,  ,  ) as the active stress components in the sph erical coordinate system. Radial e quilibrium dictates that [25]: ( )120rr rrr r   +−−= (9) Due to symmetry ,  = and Eq. (9) simplifies to : () 2 2rr e rr r rr  =− = (10) where e rr  =− is the von Mises effective stress [26]. Integration of Eq. (10) leads to : e a d2ra rbr ppr= =−= (11) where p is the gas pressure inside th e cavity for a given radius a, and ap is the ambient pressure. Upon making use of Eq. (3), the above integral can be re -phrased as: 0 03 ae2dRa RbRp R pRr= =−=  (12) The effective stress e is a function of the effective strain e , the effective strain rate e , and the normalized tem perature g/TT via the constitutive law for the PMMA -CO 2 solid , of general functional form F where : () e e e gF , , / TT   = (13) Preprint of http://doi.org/10.1098/rspa.2019.0339 10 The choice of F is given below . We emphasize that the glass transition temperature gT of the PMMA is a function of CO 2 concentration and that the effect ive strate rate e scales with a as prescribed in Eq. (8). Upon substituting Eq. (13) into Eq. (12) we obtain an expression for a , and we integrate over time in order to determine the time evolution of 0/aa . Mass conservation At the start of the foaming proce ss, the chemical potential of the CO 2 molecules in the nucleated voids is lower than chemical potential of CO 2 molecules in the PMMA -CO 2 solid. Consequently, CO 2 gas molecules migrate from the PMMA -CO 2 solid into the voids . The concentration of CO 2 gas mol ecules ( , )C r t at time t and position r (for a r b ) can be obtained by solving Fick’s second law of diffusion [27]: 2 2C D Crtrr r   =    (14) in the deformed configuration, where D is the diffusion coefficient for CO 2 in PMMA. Measurements of D at temperatures and pressures typical for solid -state nanofoaming of PMMA by CO 2 are available in the literature as follows . Guo and Kumar [28] measured D based on desorption measurements and found that D range s from 12 2 -12.5 10 m sD−= to 11 2 -13.65 10 m sD−= for temperatures ranging from -30˚C to 100 ˚C at a CO 2 pressure equal to 5 MPa . Li et al. [29] measured D by a sorption technique and found that D is in the range of 11 2 -16 10 m s− to 11 2 -19.5 10 m s− for temperatures ranging from 30˚C to 70 ˚C, and pressures ranging from 6 MPa to 18 MPa. Now, introduce a characteristic diffusion time D : 2 D()DL D= (15) wher e DL is a diffusion length which is approximated for the void growth problem by: 0 d1 3 43 DNLb  (16) Observations of cell nucleation densities of PMMA nanofoams ( 20 d10N m-3) imply DL < 133 nm [1]. Upon assuming 12 210 m /sD−= , we obtain D  20 ms via Eq. (15), which is two Preprint of http://doi.org/10.1098/rspa.2019.0339 11 orders of magn itude lower than typical observed cell growth times for solid -state nanofoaming of PMMA by CO 2 as reported by Martín -de León et al. [9] and reported above. We conclude that the CO 2 concentration profile ( , )C R t is spatially uniform at all times : ( , ) ( )C R t C t= . Consequently, we do not need to solve the diffusion equation to predict void growth during solid -state nanofoaming of PMMA by CO 2. We shall assume that the mass of gas molecules in th e void s and in the surrounding solid is constant ; leakage of gas molecules to neighbo uring voids or the sample’s environment is neglected . The resulting mass conservation statement for CO 2 reads: () ()p 3 3 g 3 p 3 3 g 3 0 0 0 0 0 + a C b C b a a a    − = − + (17) where p is the density15 of the PMMA -CO 2 solid and g is the density of the CO 2 in the void s. The relations Eq. (12) and Eq. (17) form a coupled pair of equations which can be solved to obtain p as a function of cavity expansion 0/aa . Additional assumptions are detailed below. Depression of the g lass transition temperature by CO 2 The dissolution of CO 2 into a linear, amorphous polymer such as PMMA reduces the glass transition temperature gT of the PMMA -CO 2 solid . This plasticization effect is attributed to the increased mobility of PMMA chains due to lubrication by the CO 2 molecules , and the decrease of the intermolecular bond strength , as the C O2 molecules increase the spacing between the PMMA chains [30,31] . A range of experimental techniques have been used in the literature to determine the glass transition temperature Tg of PMMA as a function of CO 2 mass concentration C . Chiou et al. [32] made use of DSC to measure 0 gg/TT as a function of C , where 0 gg ( 0) T T C== . Likewise, Wissinger and Paulaitis [33] measured the dependenc e of 0 gg/TT upon C via creep compliance measurements. Guo and Kumar [28] made use of solid - state foaming experiments to observe the relation betw een 0 gg/TT and CO 2 for a PMMA -CO 2 15 We assume that the density of the PMMA -CO 2 solid is equal to the density of PMMA absent CO 2 at standard conditions (i.e. -31190 kg mp= ) based on the measurements of Pantoula and Panayiotou [40] and Pantoula et al. [41] who observed that the relative incr ease in volume of a PMMA -CO 2 mixture is close to the relative increase of the mass of a PMMA - CO 2 mixture for a CO 2 pressure up to 30 MPa . Preprint of http://doi.org/10.1098/rspa.2019.0339 12 mixture. The measured 0 gg/TT versus C data, for PMMA -CO 2, as reported by Chiou et al. [32], Wissinger and Paulaitis [33], and Guo and Kumar [28] are shown in Figure 5. Chow [34] used statistical thermodynamics to predict 0 gg/TT as a fun ction of C and introduce d a parameter  where : gp w w1M C zM C=− (18) Here, p wM is the molecu lar weight of the polymer repeat unit ( p wM = 100.12 g mol-1 for a methyl methacrylate monomer) , g wM is the molecular weight of the gas ( g wM = 44.01 g mol-1 for CO 2), and z is a lattice co ordination number equal to 2 , as suggested by Chow [34]. In addition, Chow [34] define d a parameter  : p wp CzR M= (19) where R is the universal gas constant and pC is the change in specific heat capacity of the polymer at the glass transition temperature at constan t pressure. The normalized glass transition temperature is then predicted by: ()() ( )g 0 gexp 1 ln 1 lnT T    = − − + (20) Equation (20) is curve fitted to the measured 0/ggTT versus C data shown in Figure 5 by a suitable choice of pC . The fitted value for pC equals 355 11J kg K−− which is slightly higher th an the value of pC for PMMA as measured by DSC , see Chiou et al. [32] and Li et al. [35]. Constitutive model for the PMMA -CO 2 solid We assume that the effective stress e of the PMMA -CO 2 solid at a given strain e , strain rate e and normalized temperature g/TT is the same as that given by PMMA in the absence of CO 2: the effect of CO 2 is accounted for by a shift in the value for Tg. The d eformation mechanisms for PMMA in uniaxial tension close to the glass transition temperature have been Preprint of http://doi.org/10.1098/rspa.2019.0339 13 reviewed recently by Van Loock and Fleck [19] an deformation mechanism maps were constructed by performing a series of uniaxial tension tests on the high wM PMMA over a range of temperatures near the glass transition and over two decades of strain rate . The operative deformation mechanism depends upon th e temperature g/TT , the strain rate e , and strain e . We shall make use of the constitutive models as calibrated by Van Loock and Fleck [19] for the high wM PMMA: the Ree-Eyring equation and a rubbery -flow model. For the low wM PMMA it is necessary to construct an alternative deformation mechanism map. This is reported in the Appendix. For this grade, the relevant deformation mechanisms are Ree -Eyring and viscous flow. The Ree -Eyring equation relates e in the glassy and glass transition regime to temperature g/TT and strain rate e : ee 0-sinh expv q kT kT  =  (21) where 0 is a reference strain rate, q is an activation energy, v is an activation volume, and k is Boltzmann’s constant . Visco -elastic effects are neglected in this finite strain regime. Van Loock and Fleck [19] also fitted an empirical equation to relate e to g/TT and e in the rubbery regime for the high wM PMMA : e e 0 R e gR1nTET     =− (22) where 0 RE is a reference modulus, R is a temperature sensitivity coefficient, R a reference strain rate, and n a strain rate sensitivity c oefficient. Note that the rubbery regime above the glass transition is absent f or PMMA grades of relatively low molecular weight , i.e. wM < 150 -1kg mol [36]. Instead, a linear, viscous flow rule can be used to describe the constitutive behavior of a low wM PMMA for g/1TT : ee3 = (23) where  is a temperature -dependent viscosity [37,38] : Preprint of http://doi.org/10.1098/rspa.2019.0339 14 1g 0 2 g g- ( / -1)ex-p/ /1TC T T C T T =+ (24) in terms of a reference viscosity 0 at g/TT = 1; 1C and 2C are fitting constants. The dependence of the effective stress e upon normalized temperature /gTT and strain rate e is assumed to be governed by Eq. (21) and Eq. (22) for the high wM PMMA and by Eq. (21) and Eq. (23) for the low wM PMMA . The fitted parameters for the constitutive laws for the high wM PMMA are taken from Van Loock and Fleck16 [19] and are summarized in Table 3. An addit ional series of tensile tests have been performed on the low wM PMMA at temperatures close to the glass transition in order to calibrate Eq. (21) and Eq. (23) for the low wM PMMA as detailed in the Appendix . The resulting calibrated parameters for Eq. (21) and Eq. (23) for the low wM PMMA are included in Table 3. Gas laws The equilibrium concentration C of CO 2 in PMMA is a function of CO 2 pressure p and of temperature. Here, we assume that Henry’s law suffices such that [39–42]: H C K p= (25) where Henry’s law coef ficient HK is assumed to be independent of both temperature and pressur e. Assume that the concentration of CO 2 at the surface of the cavity ( 0 Ra= ) is in equilibrium with the CO 2 pressure within the void via Eq. (25). Take 91 H7.74 10 Pa K−−= for both the low wM and the wM PMMA gr ades, based on the measured C = 0.24 equilibrium concentration of CO 2 in PMMA at a pressure p equal to 31 MPa and temperature T = 25 ˚C, as detailed in section 2.2. Also, assume that the CO 2 gas in the void statisfies the ideal gas law : g g wRTpM= (26) Temperature -time profile during void growth 16 We assume that the de pendence of the effective stress e of the PMMA -CO 2 solid upon pressure is small as a first order approximation for the void growth problem . Preprint of http://doi.org/10.1098/rspa.2019.0339 15 During the rapid release of pressure at the end of the saturation phase, the samples cool down from the saturation temperature equal to 25 ˚C to a temperature17 0T = -15 ˚C due to adiabatic cooling of the expanding gas. The samples are subsequently placed in a thermal bath at a maintained foaming temperature fT . Upon submersion in the foaming bath, assum e that the temperature profile ()Tt is of the form : ()()( ) 00 f 1 exp - / TT t TT − = + − (27) where  is a time constant associated with the heat conduction into the PMMA, as measured by a thermocouple . Void growth simulations Void growth during solid -state foaming is simul ated by solving t he equilibrium Eq. (12) and the mass conservation statement Eq. (17) simultaneously , with due account of the dependence of gT upon C via Eq. (20), the dependence of the effective stress e of the PMMA -CO 2 solid upon e, e and g/TT (Eqs. (21) to (23)), the gas l aws (Eqs. (26) and (25)), and the time - temperature profile as captured by Eq. (27). The resulting system of equations is solved by numerical integration18. The values of the processing parameters and the material properties are summarized in Table 4. Note t hat the initial porosity 0f is: 3 0 0 0fa b=  (28) and is estimated19 to equal 310− for both the low wM and high wM PMMA nanofoams . The initial void radius 0a is estimated by : 17 Measured by placing a thermocouple on the sample after pressure release at the end of the saturation phase. 18 The numerical integration was conducted within the Matlab computing environment by means of the ode15s function . 19 The initial porosity 0f is estimated by saturating low wM and high wM PMMA precursors with CO 2 at p = 31 MPa and T = 25 °C. Upon release of the pressure to atmospheric pressure, the samples were immediately immersed in liquid nitrogen to prevent the growth of the nucleated voids. The porosity of the samples was measured by the method detailed in section 2 after the CO 2 was completely desorbed. The measured porosity was assumed to be representative for 0f . Preprint of http://doi.org/10.1098/rspa.2019.0339 16 0 0 d1 3 3 4faN  (29) where the cell nucleation density dN equals 20 32 10 m− for the low wM PMMA nanofoams (see Table 1) and dN equals 20 320 10 m− for the high wM PMMA nanofoams (see Table 2). 5. Results and d iscussion of the void growth predictions Consider the deformation mechanism maps for the low wM PMMA (see Figure 6a) and for the high wM PMMA (see Figure 6b). We superpose the predict ed trajectory of the effective stress at the surface of the c avity e by the void growth model as a function of g/TT for foaming temperatures fT = 25 °C and fT= 80 °C, and for a foaming time up to 600 s . Note that both the temperature T and glass transition tempe rature gT evolve in time during foaming. For both the low wM and high wM PMMA , at the start of foaming , T equals 0T and g/TT is close to 0.9; at this instant e is close to 0.8 MPa for the low wM PMMA and e is close to 0.3 MPa for the high wM PMMA . When the temperatu re increases from 0 TT= to f TT= , g/TT rises to almost unity and e rises steeply . The void growth simulations suggest that during solid -state foaming of PMMA, the normalized temperature g/TT remains between 0.9 and 1 and consequently void growth does not occur within e ither the viscous regime (low wM PMMA) or within the rubbery regime (high wM PMMA). The measured porosity f is plotted as a function of foaming time ft for fT = 25 °C to fT = 80 °C, and compared with the predicted f versus ft curves for the low wM and high wM nano foams , in Figure 7a and Figure 7b, respectively. There is reasonably good agreement between the measured and the predicted f- ft curves for fT = 25 °C and fT = 40 °C. The void growth model overestimates the porosity at fT = 60 °C and at fT= 80 °C, where porosities close to maxf are observed. Observations of SEM micrographs suggest that cell walls tear , leading to open -celled micro structure s. This is confirmed by open cell content measurements by gas pycnometry : nanofoams with the highest observed porosities have predo minantly open -celled Preprint of http://doi.org/10.1098/rspa.2019.0339 17 microstructures , see Figure 3. At increased foaming temperature s (i.e. fT = 100 °C) collapse of the foam ed open -celled microstructure is observed leading to measured porosities below the maximum observed porosities at fT = 80 °C, as shown in Figures 2 c and 2d. We proceed to explor e two alternative hypotheses for cell wall failure which could lead to open -celled microstructures as observed for the PMMA nanofoams: (i) achievement of a critical hoop strain at the void, or (ii) achievement of a minimum (critical) value of ligament thickness between neighbouring voids. (i) Critical hoop strain Assume t hat the solid surrounding the expanding void is incompressible. Then, by Eq. (3), 3 3 3 3 00 -- b b a a= (30) Recall that the initial (as -nucleated) porosity 0f equals 3 00( / )ab as defined in Eq. (28) and the current porosity f equals ()3/ab . Now, rearrange Eq. (30), to express f as a function of 0f and the true hoop strain s at the surface of the void, where ()()s0 = ln / r a a a== : ()()-1 -1 s0 1 exp - 1 3 -f f += (31) Tearing of the cell wall occurs when s equals the /gTT -dependen t20 true tensile failure strain f . The critical porosity ff corresponding to this ductility -governed failure criterion reads: ()() 0 f-1 -1 f 1 exp - 1 3 -f f += (32) (ii) Critical ligament size The alternative failure hypothesis assumes that there is a minimum number of confined polymer c hains separating individual cells to prevent rupture of the solid between the cells. Define the smallest distance betwee n two neighbouring cells h as: () 2 - h b a= (33) 20 We assume f to be insensitive to strain rate [19,46] . Preprint of http://doi.org/10.1098/rspa.2019.0339 18 Then, upon making use of the expressions 3 0 0 0( / )f a b= , ()3/ f a b= , and Eq. (31), we obtain : 1 13 3-1-0 -1 0-1-1- 1 2ffh af  =  (34) Write ch as the critical cell wall thickness , and assume that it is independent of the value of /gTT . The corresponding critical value of porosity cf is given by Eq. (34) with c hh= . The ductility -governed porosity limit ff as given by Eq. (32) is plotted in Figure 7 based on the predicted hoop strain s during void growth. Note that w e make use of the measured response of f versus g/TT (Eq. (A .2) for the low wM PMMA and Eq. (A .1) for the high wM PMMA as detailed in the Appendix) and assume that the initial porosity 0f equals 310− . The measured values of final porosity f and the predictions of the void growth model exceed the porosity limit as given by ff . We now plot the porosity limit cf in Figure 7 via Eq. (34) for 3 010f−= by taking c0/ha = 3 (low wM PMMA) and c0/ha = 4.2 (high wM PMMA) in order to match to observed values of the maximum observed porosity maxf of the nanofoams. Recall that the initial void size 0a of the low wM PMMA nanofoams is estimated to be close to 10.5 nm, whereas 0a is close to 5 nm for the high wM PMMA nanofoams. Consequently, the estima ted corresponding critical cell wall dimension ch equals 32 nm for the low wM PMMA nanofoams , whereas ch equals 21 nm for the high wM PMMA . These values for ch are of the same order of magnitude as root-mean -square end-to-end distance eeR of the PMMA chains , i.e. eeR ≈ 20 nm for the low wM PMMA and eeR ≈ 110 nm for the high wM PMMA based on an idealized equivalent freely jointed chain calculation [43]. This is in agreement with the results of Crosby and co- workers who conducted a series of uniaxial tensile tests on thi n polystyrene (PS) films with wM = 136 000 g mol-1 [44,45] . They found that the tensile failure strain f decrease s with decreasing film thickness t in the regime t = 15 nm to t = 77 nm ; these values are close to the estimated value for eeR = 25 nm of the PS ch ains. Preprint of http://doi.org/10.1098/rspa.2019.0339 19 Concluding remarks Solid -state nanofoaming experiments are performed with two grades of PMMA of markedly different molecular weight ( wM = 92 500 g mol-1 and wM = 3 580 000 g mol-1). It was found that the molecular weight of the PMMA has a profound eff ect upon the microstructure of the PMMA nanofo ams. When subjected to identical foaming conditions, t he observed cell size l  35 nm of the high molecular weight PMMA nanofoams is an order of magnitude less than that of the low molecular weight PMMA nanofoa ms, l  250 nm . This is consistent with the observation that the nucleation density , dN 20 320 10 m− of the high molecular weight PMMA nanofoams is an order of magnitude higher than that of the low molecular weight PMMA nanofoams 20 3 d2 10 m N− . In addition, a limit in attainable p orosity maxf was observed: maxf equals 0.65 for the high molecular weight PMMA and maxf equals 0.75 for the low molecular weight PMMA. The microstructure of the PMMA nanofoams transitions from closed -celled to open -celled at a porosity close to maxf . A vo id growth model has been developed to simulate cavity expansion during solid -state nanofoaming of PMMA by CO 2. Experimentally calibrated constitutive laws for the PMMA grades close to the glass transition are used in the simulations . The predicted porosity of the nanofoa ms versus foaming time , at selected foaming temperature s, are in good agreement with the measured responses for porosities well below the maximum observed porosity. There is also close agreement between the predicted and observed sensitivity to molecular w eight . This suggests that the observed difference in constitutive response close to the glass transition between the two PMMA grades leads to the measured difference in porosity. Moreover, cell wall tearing account s for the observed limit in final porosity . Our analysis suggests the existence of a limiting minimum cell wall thickness of magnitude close to that of the end -to- end distance of the polymer chains . When the cell wall thickness approaches this minimum value during foaming , ruptur e of the cell wall s occurs ; resulting in an open -celled structure , and to a limit on foam expansion . Acknowledgements Financial support from the Engineering and Physical Sciences Research Council (UK) award 1611305 (F. Van Loock) and the FPU grant FPU14/02050 (V. Bernardo) from the Spanish Ministry of Education is gratefully acknowledged. N. A. Fleck is grateful for additional financial support from the ERC project MULTILAT. Financial assistance from SABIC, Preprint of http://doi.org/10.1098/rspa.2019.0339 20 MINECO, FEDER, UE (MAT2015 -69234 -R) are acknowledged too. The autho rs would also like to thank Dr. Martin van Es from SABIC for the technical assistance. Preprint of http://doi.org/10.1098/rspa.2019.0339 21 List of table captions Table 1: Measured porosity f, average cell size l , standard deviation of observed cell size s , cell nucleation density dN , and open cell content vO of the low wM PMMA nanofoams as a function of foaming time ft and foaming temperature fT . Foams collapsed at fT = 100 C , and so no open cell content values are reported for nanofoams produced at fT = 100 C. Table 2: Measured values for the porosity f , the average observed cell size l , the standard deviation of the observed cell sizes s , the cell nucleation density dN , and the open cell content vO of the high wM PMMA nanofoams as a function of foaming time ft and foaming temperature fT . Foams collapsed at fT = 10 0 C , and so no open cell content values are reported for the nanofoams produced at fT = 100 C . Table 3: Fitted parameters for the constitutive laws for the low wM PMMA ( Eq. (21) and Eq. (23)) and the high wM PMMA obtained from Van Loock and Fleck [18] , see Eq. (21) and Eq. (22). Table 4: Summary of the assumed processing parameters and material properties for the void growth predictions. Preprint of http://doi.org/10.1098/rspa.2019.0339 22 Tables Table 1: Measured porosity f, average cell size l , standard deviation of observed cell size s , cell nucleation density dN , and open cell content vO of the low wM PMMA nanofoams as a function of foaming time ft and foaming temperature fT . Foams collapsed at fT = 100 C , and so no open cell content values are reported for nanofoams produced at fT = 100 C. ft (s) fT ( C) f l (nm) s (nm) dN ( 0 3 210 m− ) vO 60 25 0.45 219 87 1.50 0.12 180 25 0.47 228 79 1.50 0.08 300 25 0.51 283 112 0.91 0.08 600 25 0.51 235 85 1.48 0.08 60 40 0.52 262 102 1.22 0.07 180 40 0.61 250 125 1.70 0.02 300 40 0.64 254 105 1.27 0.15 600 40 0.66 233 103 2.11 0.14 60 60 0.56 234 89 2.34 0.07 180 60 0.66 297 111 1.72 0.33 300 60 0.68 279 122 1.76 0.40 600 60 0.68 284 109 1.63 0.36 60 80 0.72 333 134 1.16 0.63 180 80 0.74 288 138 1.83 0.90 300 80 0.75 297 125 1.75 0.78 600 80 0.73 274 109 2.08 0.93 60 100 0.64 297 122 1.21 - 180 100 0.68 253 110 1.81 - 300 100 0.62 246 103 1.75 - 600 100 0.51 291 125 0.76 - Preprint of http://doi.org/10.1098/rspa.2019.0339 23 Table 2: Measured values for the porosity f , the average observed cell size l , the standard deviation of the observed cell sizes s, the cell nucleation density dN , and the open cell content vO of the high wM PMMA nanofoams as a function of foaming time ft and foaming temperature fT . Foams collapsed at fT = 100 C , and so no open cell content values are reported for the nanofoams produced at fT = 100 C. ft (s) fT ( C) f l (nm) s (nm) dN ( 0 3 210 m− ) vO 60 25 0.22 36 14 14.9 0.30 180 25 0.28 23 10 40.0 0.22 300 25 0.29 30 12 9.0 0.28 600 25 0.31 36 18 6.9 0.21 60 40 0.33 28 13 54.2 0.19 180 40 0.42 32 16 32.3 0.07 300 40 0.45 37 14 7.8 0.08 600 40 0.47 45 29 26.0 0.09 60 60 0.45 37 14 20.4 0.08 180 60 0.55 39 17 24.0 0.03 300 60 0.57 40 17 31.8 0.28 600 60 0.57 41 19 25.8 0.03 60 80 0.58 39 20 21.8 0.51 180 80 0.60 39 19 27.8 0.73 300 80 0.60 38 19 36.6 0.95 600 80 0.59 44 22 46.6 0.88 60 100 0.59 34 15 35.4 - 180 100 0.53 27 14 80.4 - 300 100 0.50 37 18 24.9 - 600 100 0.45 34 12 32.6 - Preprint of http://doi.org/10.1098/rspa.2019.0339 24 Table 3: Fitted parameters for the constitutive laws for the low wM PMMA ( Eq. (21) and Eq. (23)) and the high wM PMMA obtained from Van Loock and Flec k [19] , see Eq. (21) and Eq. (22). low wM PMMA high wM PMMA 3 (nm ) v− 2.5 1.8 (J)q 197.31 10− 197.31 10− -1 0 (s ) 561.5 10 561.5 10 0 (Pa s) 62.8 10 - 1C 3.2 - 2 (K)C 17.3 - 0 R (MPa)E - 65.8 R - 0.8 1 R (s )− - 1.58 n - 0.173 Table 4: Summary of the assumed processing parameters and material properties for the void growt h predict ions. low wM PMMA high wM PMMA 0p (MPa) 31 31 ap (MPa) 0.1 0.1 (s) 20 20 p3 (kg m )− 1190 1190 ° g ) (CT 114.5 116.5 f0 10-3 10-3 0a (nm) 10.5 5 Preprint of http://doi.org/10.1098/rspa.2019.0339 25 List of Figure captions Figure 1: SEM micrographs of the low wM nanofoams at (a) f C60T= , (b) f10 C0 T= and of the high wM nano foams at (c) f C60T= and (d) f10 C0 T= . Figure 2: Nanofoaming experiments on the low wM and high wM PMMA grades: (a) measured average cell size l versus foaming time ft for fT = 60 °C, (b) measured porosity f versus foaming time ft for fT = 60 °C and fT =100 °C, (c) measure d porosity f versus foaming temperature fT for the range of explored foaming times ( ft = 60 s to ft = 600 s) for the low wM nanofoams, and (d) measured f versus fT for the range of explored foaming times ( ft = 60 s to ft = 600 s) for the high wM nanofoams. Figure 3: Measured open cell content vO as a function of porosity f for (a) the low wM PMMA nanofoam and (b) high wM PMMA nanofoam. Figure 4: Spherical void in (a) undeformed configuration with initial radius 0a and initial outer radius 0b and (b) deformed configurati on at time t of the void with radius a , outer radius b and gas pressure p. Figure 5: The normalized glass transition temperature 0 gg/TT of PMMA as a function of CO 2 mass concentration C , as reported by Chiou et al. [30], Wissinger and Paulaitis [31], and Guo and Kumar [26]. The 0 gg/TT versus C curve is given by the calibrated version of Eq. (20). Figure 6: Deformation mechanism maps for (a) low wM PMMA and (b) high wM PMMA (for a reference strain ref0.05= ), for contours of effective strain rate e . The predicted effectiv e stress at the surface of the cavity e is plotted as a function of /gTT for foaming temperatures fT = 25 °C and fT = 80 °C and for a foaming time up to 600 s. Figure 7: Predicted and measured p orosity f versus foaming time ft , for fT = 25 °C to fT = 80 °C for (a) low wM nanofoams and (b) high wM PMMA nanofoams. The ductility -governed porosity limit ff is plotted via Eq. (32) for an initial porosity 3 010f−= . The minimum cell wall thickness -governed porosity limit cf is plotted via Eq. (34) for 3 010f−= and c0/ha = 3 (low wM PMMA) and c0/ha = 4.2 (high wM PMMA). Preprint of http://doi.org/10.1098/rspa.2019.0339 26 Figures Figure 1: SEM micrographs of the low wM nanofoams at (a) f C60T= , (b) f10 C0 T= and of the high wM nanofoams at (c) f C60T= and (d) f10 C0 T= . Preprint of http://doi.org/10.1098/rspa.2019.0339 27 Figure 2: Nanofoaming experiments on the low wM and high wM PMMA grades: (a) measured average cell size l versus foaming time ft for fT = 60 °C, (b) measured porosity f versus foaming time ft for fT = 60 °C and fT =100 °C, (c) measured porosity f versus foaming temperature fT for the range of explo red foaming times ( ft = 60 s to ft = 600 s) for the low wM nanofoams, and (d) measured f versus fT for the range of explored foaming times ( ft = 60 s to ft = 600 s) for the high wM nanofoams. Preprint of http://doi.org/10.1098/rspa.2019.0339 28 Figure 3: Measured open cell content vO as a function of porosity f for (a) the low wM PMMA nanofoam and (b) high wM PMMA nanofoam . Preprint of http://doi.org/10.1098/rspa.2019.0339 29 Figure 3: Spherical void in (a) undeformed configuration with initial radius 0a and initial outer radius 0b and (b) deformed configuration at time t of the void with radius a , outer radius b and gas pressure p. Preprint of http://doi.org/10.1098/rspa.2019.0339 30 Figure 5: The normalized glass transition temperature 0 gg/TT of PMMA as a function o f CO 2 mass concentration C , as reported by Chiou et al. [32], Wissinger and Paulaitis [33], and Guo and Kumar [28]. The 0 gg/TT versus C curve is given by the calibrated version of Eq. (20). Preprint of http://doi.org/10.1098/rspa.2019.0339 31 Figure 6: Deformation mechanism maps for (a) low wM PMMA and (b) high wM PMMA (for a reference strain ref0.05= ), for contours of effective strain rate e . The predicted effective stress at the surface of the cavity e is plotted as a function of /gTT for foaming temperatures fT = 25 °C and fT = 80 °C and for a foam ing time up to 600 s. Preprint of http://doi.org/10.1098/rspa.2019.0339 32 Figure 4: Predicted and measured p orosity f versus foaming time ft , for fT = 25 °C to fT = 80 °C for (a) low wM nanofoams and (b) high wM PMMA nanofoams. The ductility -governed porosity limit ff is plotted via Eq. (32) for an initial porosity 3 010f−= . The minimum cell wall thickness -governed porosity limit cf is plotted via Eq. (34) for 3 010f−= and c0/ha = 3 (low wM PMMA) and c0/ha = 4.2 (high wM PMMA). Preprint of http://doi.org/10.1098/rspa.2019.0339 33 Appendix Calibration of constitutive laws for the low wM and high wM PMMA Constitutive laws are calibrated for the low wM PMMA grade21 close to its glass transition temperature. We follow the procedure of Van L oock and Fleck [19] who constructed deformation and failure maps for the high wM PMMA grade22 in uniaxial tension close to the glass transition temperature. A series of u niaxial tensile tests were performed on the low wM PMMA grade for a range of temperatures (T = 90 ̊C to T = 170 ̊C) and at a nominal strain rate 2 -15.9 1 s0 e−= . The dogbone specimen geometry and the measurement procedures are detailed in Van Loock and Fleck [19]. Note that the low wM PMMA dogbone specimens are machined from the foaming precursor sheets . The true stress versus true strain response s of the low wM PMMA dogbone specimen s are plotted in Figure A1 .a for g 0.94 / 1.01 TT and in Figure A .1b g 1.04 / 1.14 TT . The true stress versus true strain curves of the high wM PMMA grade are included in Figures A .1a and A.1b. Loading -unloadin g uniaxial stress stress versus strain curves for the low wM PMMA and high low wM PMM are shown in Figure A .2. At g/TT = 0.93, t he elastic unloading of the low wM and the high wM PMMA occurs in the manner of an elasto -viscoplastic solid, with remnant a finite strain at zero load . The qualitative stress versus strain response of the low wM and the high wM PMMA is different when the temperature is increased to g/TT = 1.06. The elastic rubbery regime is entered for the high wM PMMA and the unloading curve is almost coinc idental with the loading curve ; there is negligible hysteresis and negligible remnant strain. No rubbery regime is observed for the low wM PMMA above the glass transition. At g/TT = 1.06 and g/TT = 1.12 , the stress versus strain response of the low wM PMMA in unia xial 21 Altuglas V825T with g114.5 CT= and wM = 92 500 g mol-1. 22 Altuglas CN with g116.5 CT= and wM = 3 580 000 g mol-1. Preprint of http://doi.org/10.1098/rspa.2019.0339 34 tension is linear viscous. Unloading is accompanied by a finite remnant strain. The high wM PMMA transitions from the rubbery regime to a viscous regime at g/TT = 1.16. First, consider the elasto -viscoplastic regime. The dependence of the measured flow s trength y of the low wM and high wM PMMA grades upon g/TT is shown in Figure A .3 for -2 -15.9 10 s e= . A single transition Ree -Eyring equation, Eq . (21), is fitted to the yσ versus T/Tg response o f the low wM PMMA in the glassy and glass transition regime ( corresponding to 0.94  T/Tg  1.04). We assume that q equals 197.31 10 J− and 0 equals 56 -11.5 10 s= for both the low wM and the high wM PMMA , as reported by Van Loock and Fleck [19]. The activation volume v equals -32.5 nm for the low wM PMMA, and -31.8 nmv= for the high wM PMMA [19]. The resulting curve fits are i ncluded in Figure A .3. Second, consider the viscous regime for the low wM PMMA . We fit a linear, viscous constitutive law, Eqs. (23) and (24), to the measured y versus g/TT curve s of the low wM PMMA in the regime of 1.06  T/Tg  1.14 and -2 -15.9 10 s e= . The fitting values are 6 02.8 10 Pa s=   , 13.2 C= , and 217.3 K C= . The resulting curve fit is adequate, see Figure A .3. Third, consider the rubbery regime of the high wM PMMA. The constitutive description, Eq. (22), is adequate upon making use of previously measured values ( 0 R65.8 MPa E= , R= 0.80 , -1 R= 1.58 s , and 0.173n= [19]), as shown in Figure A .3. Tensile ductility of the low wM and high wM PMMA Van Loock and Fleck [19] measured the true tensile failure strain , that is ductility , f of the high wM PMMA grade by testing a dogbone geometry at g/TT < 1 and an hourglass -shaped specimen geometr y at g/TT ≥ 1. The measured values for f of the high wM PMMA grade are plotted as a function of the normalized temperature g/TT for a nominal strain rate 215.9 1 s0 e−−= in Figure A .4. The f versus g/TT failure envelope is adequately fitted by a linear relation [19]: Preprint of http://doi.org/10.1098/rspa.2019.0339 35 f g3 7.3 6.T T −= (A.1) An additional series of uniaxial tensile tests have been conducted on the low wM PMMA grade by using the same measurement methods as that detailed in the work of Van Loock and Fleck [19]. No failure was observed at T  145 ̊C prior to the attainment of the ma ximum cross -head extension. The measured f versus g/TT curve is shown in Figure A .4. The failure envelope of the low wM PMMA grade close to the glass transition is also fitted by a linear relation: f g13. 1 3 1.7T T −= (A.2) Preprint of http://doi.org/10.1098/rspa.2019.0339 36 Figure A.1: Measured true tensile stress  versus true tensile strain  curves for the low wM and high wM PMMA grades in uniaxial tension for a nominal strain rate 215.9 1 s0−−= and for temperatures ranging from (a) T = 90 ˚C to T = 120 ˚C and (b) T = 130 ˚C to T = 17 0 ˚C. A cross at the end of the curve denotes specimen failure. Preprint of http://doi.org/10.1098/rspa.2019.0339 37 Figure A.2: Loading -unloading true stress versus true strain curves for the low wM PMMA and high wM PMMA grades in uniaxial tension, at selecte d values of g/TT , for a nominal strain rate 415.9 1 s0−−= . Preprint of http://doi.org/10.1098/rspa.2019.0339 38 Figure A .3: Deformation mechanism maps of the low wM and high wM PMMA grade s. Flow strength y (= e ) versus g/TT is plotted , with the curve fits of the constitutive models included for a reference strain ref0.05= . Preprint of http://doi.org/10.1098/rspa.2019.0339 39 Figure A.4: The measured true tensile failure strain f as a function of normalized testing temperature g/TT for th e low wM and high wM PMMA grades, at a nominal strain rate 21 e5.9 0 s1−−= . Preprint of http://doi.org/10.1098/rspa.2019.0339 40