Chain Arrangement and Glass Transition Temperature Variations inPolymer Nanoparticles under 3D-ConfinementDaniel E. Martínez-Tong,† Michelina Soccio,†,* Alejandro Sanz,† Carolina García,§ Tiberio A. Ezquerra,†
and Aurora Nogales†,*†Instituto de Estructura de la Materia, IEM-CSIC, C/Serrano 121, Madrid 28006, Spain§Instituto de Química Física Rocasolano, IQFR-CSIC, C/Serrano 119, Madrid 28006, Spain
*S Supporting Information
ABSTRACT: Polymer nanospheres with different size dis-tributions of poly(ethyl methacrylate) are prepared by twodifferent methods, with and without the aid of a surfactant.The calorimetric trace of these spheres shows an increase ofthe glass transition temperature that has been evaluated bymeans of an entropy model. This 3D-confinement, imposed bythe nanospheres, leads to a limiting number of repeatingpolymer units in the sphere and thus to a reduction of thepossible configuration states of the polymer chains, which isultimately related to variations in the bulk value of the glasstransition temperature. Our model is evaluated against our calorimetric measurements as well as with the data available in theliterature. Good agreement between data and model is found for many cases, proving that confinement is related to reductions inentropy for these systems.
1. INTRODUCTION
Current interest in the properties of polymers confined intonanometer scale is very intense, both from the fundamental andpractical perspectives.1 Polymers are widely used in nano-fabrication processes like wires of nanometer-scale diameters,2
nanoimprinting,1 and nanoscale polymeric particles.3 Confinedpolymers are central to a broad range of advanced materials andemerging nanotechnologies,2c,4 with applications includingbiomaterials,5 micro- and optoelectronics,6 and energycapture/storage,7 among others. Besides cutting-edge fabrica-tion strategies, control over the changes in properties inducedby nanoscale confinement is a central issue to be taken intoaccount.For several years, modification of the physical properties due
to size effects when approaching nanometer lengths in glassforming systems in general, and in polymers in particular, hasbeen strongly debated. Among the different properties,variations of the glass transition of confined polymers havemotivated a good number of experimental works.8
Confinement experiments in glass forming systems ingeneral, and in polymers in particular, have been considered avery elegant way of probing the existence of a correlation length(ξ) of cooperative motions that produces the observed slowing-down of the dynamics upon cooling, as the temperatureapproaches the liquid to glass transition temperature.8a As theconfining length approaches the order of the correlation length,modification of the properties is expected. However, in realexperiments, besides pure finite size effects, other factors likethe enhanced role of interfaces appear.9 Moreover, veryrecently it has been pointed out that, finite size and interfacial
effects are not the only ones to be considered in confinementexperiments to univocally determine the deviation from bulkbehavior. In thin films, recent models and experimentsevidenced, that other parameters, like the interfacial freevolume,10 should also be taken into account. Another fact toconsider that particularly applies to the case of polymer films isthe changes in polymer chain conformations when subjected toconfinement. In this respect, different confinement geometrieswould imply different conformations. In thin films (1dimension is confined)9b and nanowires (2 dimensions arerestricted)11 the motions of the polymer molecules aremodified in at least one of the directions of the space.Therefore, the observed effect will by anisotropic. Also,ultrastable glasses of poly(methyl methacrylate) (PMMA)have been reported,12 with a glass transition temperature insome cases up to 40 °C higher than that of the bulk polymer.Those ultrastable glasses are formed by the assembly of nearlyspherical polymer nanoglobules. In this geometry, the globularnature of individual polymer chains is preserved whileconfinement is imposed by the size of the globules. Veryrecently, some works on the glass transition and the dynamicalproperties of polymer nanospheres have appeared, but again,there seems to be some controversy about the impact of theconfinement on the observed physical properties. In this workwe show how spherical polymer particles that contain less thana threshold number of repeating units exhibit an increase of the
glass transition temperature. We propose a model to establishthe key parameter that controls the shift of the glass transitiontemperature in globular 3D-confinement.
50K, PDI = 1.11), is used as received. For the formulations withsurfactant, Sodium dodecyl sulfate (SDS, Sigma-Aldrich) was used asreceived..Nanoparticles Preparation. PEMA nanospheres were prepared
by two different methods: the miniemulsion technique widelydiscussed by K. Landfester in the past decade3,13 and a modificationof this technique that make no use of surfactants.Method 1. Surfactant-Aided Miniemulsion. A polymeric solution
is dispersed into a nonsolvent + surfactant one. In the present case,PEMA was dissolved in chloroform. The polymer solution was addedto an aqueous SDS solution. Pre-emulsification was obtained bystirring (15−30 min, case depending) at room temperature.Ultrasonication of the stirred mixture allows obtaining a miniemulsion.Afterward, evaporation of the polymer solvent under stirring yields astable dispersion of polymeric nanospheres in a nonsolvent media.Details of the concentration of each solution employed and times ofeach preparation step are reported in Table 1. To eliminate the excess
of SDS suspensions were dialized against water using a Viskingmembrane (Visking DTV by Medicell Int Ltd. Cutoff range 12−14kDa (12000−14000 g/mol)). The size of the obtained nanospheres isgoverned principally by the concentration of the polymeric solutionand by the concentration of the surfactant. (PEMA1 and PEMA3 wereprepared following this method).Method 2. Precipitate of Nanospheres without Surfactant. By
this method, a stable precipitation of PEMA micro to nanospheres inwater is obtained. The proposed method is nearly similar to the onepresented above, with the only different of the absence of surfactant. Inorder to obtain different distribution sizes, two different approacheswere applied. The same procedure as the one reported abovedeveloped by Landfester et al.13a without the aid of surfactant yields toa precipitate in the form of spheres with mean diameters around 230nm. PEMA4 is prepared in this way. To obtain smaller spheres withoutsurfactant, the ultrasonication of the PEMA in CHCl3 solution wasperformed in an open flask, and it was extended until completeevaporation of CHCl3. In this way PEMA2 was prepared. To avoidcontamination from possible bulk precipitate in these two preparationmethods, once the respective protocols are complete, the samples werefiltered with a filter paper.Atomic Force Microscopy. The size distribution and shape of the
obtained nanoparticles were characterized by Atomic Force Micros-copy (AFM) images. A Multimode microscope equipped with aNanoscope V Controller by Bruker, was used in the ScanAsyst mode,and with ScanAsyst-Air probes. The samples for AFM analysis werespun coated on (100) silicon wafers, previously treated with a piranhasolution for 30 min.Differential Scanning Calorimetyr (DSC). Calorimetric meas-
urements were carried out by means of a Perkin-Elmer DSC8500instrument equipped with an Intracooler 2 sub ambient device andcalibrated with high purity indium standards. In order to measure theglass transition of the samples under investigation, the external blocktemperature was set at −100 °C. To ensure that there are no effects on
the measurements by the presence of remaining water, nanoparticleswere lyophilized. The resulting fine powder was encapsulated inaluminum pans and heated from −50 to +150 °C at a rate of 60°C·min−1. This rate was selected so a good calorimetric response couldbe obtained even when the amount of sample was in the detectionlimit (∼3 mg). The glass-transition temperature was taken as themidpoint of the heat capacity increment ΔCp associated with the glass-to-rubber transition which gave the same value as the one taken fromthe maximum peak of the heat flow derivative.
3. RESULTS AND DISCUSSIONMiniemulsion process is an interesting approach to producepolymer spheres with nanometric dimensions.3 Here, poly-(ethyl methacrylate) (PEMA) nanospheres of different sizeswere prepared by surfactant assisted and nonsurfactant assistedminiemulsion techniques. The surfactant chosen was sodiumdodecyl sulfate (SDS). The size distribution of the spheres,prepared with different polymer concentrations, ultrasonicationtime, surfactant concentration, etc., was characterized by atomicforce microscopy (AFM). The impact of the size confinement,imposed by the spheres, on the polymer chain and morespecifically on their glass transition was studied by differentialscanning calorimetry (DSC).
3.1. Morphology of the Nanoparticles. AFM measure-ments confirmed that the particles prepared are spheres withsubmicrometer diameters: No other geometrical structureswere found. Figure 1 presents AFM height images of selectedsamples prepared with and without surfactant, namely PEMA1,PEMA3 and PEMA2, PEMA4, respectively, together with theirrespective size distributions. Quantitative analysis of the sizedistribution was performed in several regions of spun coatedsilicon wafers (Figure 1, f and e) and details are given in Table2. The AFM experiments revealed that preparations with andwithout surfactant lead to morphologically similar nano-particles. No pronounced ripening effects in the absence ofsurfactant for this specific polymer are observed. The finaldiameter distribution is governed principally by the concen-tration of the polymer solution and by the interplay betweenthe ultrasonication and the evaporation of chloroform times. Inboth preparation methods (with and without surfactant) theconditions were chosen to obtain particles with diameter in thesubmicrometer range, and spheres with diameter below 100 nmreferred hereafter as submicrometer and nanometer sizeparticles, respectively. For formulations with SDS, the resultsare comparable with those available in the literature for othertypes of polymers.13a,14
3. 2. Influence of Confinement on the Tg of PEMA.Calorimetry is an effective analytical tool to characterize theglass transition and phase transformations under confine-ment.15 Differential scanning calorimetry traces from bulk,submicrometer (PEMA3, PEMA4) and nanometer sized(PEMA1, PEMA2) spheres prepared by miniemulsiontechnique with and without surfactant are presented in Figure2a. Figure 2b shows the heat flow derivative (dH/dT), in whichthe Tg appears as a peak.This is an alternative and much more sensitive method to
analyze changes in the heat flow and allows observing possiblebroadening due to size effects in the system. As evidenced inFigure 2b, dH/dT is affected by the volume restriction imposedby the spheres. In the case of submicrometer spheres preparedby the miniemulsion technique with the use of SDS as asurfactant, the Tg appears in dH/dT as a broader peak than thatof the bulk, and with its maximum displaced toward highertemperatures. On the other hand, nanometer sized spheres also
Table 1. Preparation Parameters for the Different StudiedSamples
show a displacement in the maximum position of the peak, toeven higher temperatures, and the peak is narrower. Theseresults point toward an increase of the glass transitiontemperature of PEMA with decreasing particle size, trend thatcan also be found in the analysis of the fictive temperature(Table 2), useful where the calorimetric trace shows enthalpyovershoots as in PEMA3. It is worth mentioning that in ourresults Tg = Tf within the experimental error. In the particularcase of submicrometer spheres, the peak in dH/dT presents abimodal shape, that may indicate that there is a population ofspheres of enough size to exhibit a bulk like behavior (peak atlow temperatures) and other population of spheres that exhibit
a confined like response (high temperature). This is inagreement with the distribution of particles sizes that havebeen observed in the AFM images, (Figure 1), where besidesthe larger particles, a non-negligible population of small spheresis observed.Similar results were obtained in the case of nanospheres and
submicrometer spheres prepared without surfactant and thegeneral trend is that an increase on the Tg is observed as thesize of the nanoparticle is reduced (Figure 2, parts a and b). Inthe submicrometer spheres prepared by the nonsurfactantmethod, the calorimetric trace in the Tg region appears to bevery similar to that of the bulk. This can be related to thediameter distribution of the spheres. However, dH/dT for thissubmicrometer samples shows a small peak at high temper-atures that can be correlated to the response of small sizedspheres that are readily seen in the AFM images. This smallpeak was verified in several DSC scans of samples with similarsize distribution corroborating that this was not an artifact.Regarding the strength of the calorimetric transition, it is worthpointing out that all the nanosphere samples, independently oftheir size distribution, exhibit ΔCp values similar to the one ofthe bulk (see Table 2).The shape of the nanospheres is lost as the temperature
increases above Tg for each sample. Subsequent DSC runs showthat in all the nanostructured samples, the Tg returned to itsbulk value after the first scan (Figure 3, parts a and b, as anexample). This effect can be understood by considering thepossible coalescence of different spheres once the system hasenough mobility to allow interdifussion of chains from differentspheres. In order to test this hypothesis, AFM pictures weretaken before and after heating PEMA in the initial form ofnanometer sized spheres well above its Tg (Figure 3c). Afterheating, the samples become unstructured in the form of largedrops (Figure 3d), which explains the bulk like Tg.All of our results indicate that the three-dimensional
nanostructuring of PEMA in nanometer size spheres leads toan increase of the glass transition temperature with respect tothat of the bulk. In this line, it has been reported very recently,an increase of 40 °C with respect to the bulk Tg of poly(methylmethacrylate) (PMMA) spherical nanoglobules with sizesranging between 20 and 500 nm.12 Li and co-workers,16 alsoshowed that for nanoparticle aggregates of atactic PMMAconfined to nanospheres of 20−40 nm in diameter, there is anincrease of 20 °C in the Tg. The authors suggest the existenceof a hard amorphous region, where the cooperative motion ofthe repeating units was restricted. The increase of the Tg, due tothe geometrical confinement, is not restricted to methacrylates,for example, similar increase in Tg has been reported forpolystyrene nanoparticles.17 This behavior was attributed to the
Figure 1. AFM images and histograms for PEMA nanoparticles. Left.SDS assisted PEMA particles (a and c), with 0.2 and 5 wt %concentration respectively, (PEMA1 and PEMA3 from Table 1) andtheir corresponding diameter distributions (e). Right. SDS free PEMAparticles (b and d), with 2 wt % concentration in each case (PEMA2and PEMA4 from Table 1) and its corresponding size distribution (f).
Table 2. Characteristics of the Distribution and Thermal Properties of the Studied Samples
sample size distribution (nm) width of the distribution (nm) mean diameter (nm) Tg (°C) Tfa (°C) ΔCp (J/g°C)
aThe fictive temperature (Tf) is reported in this table since it is the most appropriate way to characterize a glass on heating in the case of largeenthalpy overshoots.
hypothesis that when the polymer chains are in spheroid form,the chains are more compressed than on the bulk state.17a
However, the literature on the subject shows the samecontroversy in the shift of Tg in confined polymer systemsinto nanospheres or nanodroplets, as well as in polymer thinfilms.8c,18 In this work, we propose that the 3D-confinementimposed to the polymer chains can be rationalized taking intoaccount thermodynamical considerations and that it should beindependent of the polymer system. Precisely, we propose thata decrease of entropy, due to the less possibilities ofarrangement of the polymer chains imposed by geometricalconfinement, is responsible for the observed positive shift in theglass transition temperature of the nanospheres. The nano-
spheres assemblies prepared in this work provide an idealsystem to study these effects, since interaction with anysubstrate has been minimized by the fact that most of theexistent interphases are polymer air, and, on the other hand, theimposed confinement is 3-dimensional, that allows to avoidpossible chain stretching effects that appears in thin films, andthat for sure have an impact on the glass transition temperatureof the systems, besides size confinement.There are several reviews available in the literature that
discuss the issue of confinement entropy and its impact onTg.
8c,19 First, based on the configurational entropy theory of theglass transition, developed by Gibbs and Di Marzio20 andextended by Adam and Gibbs,21 taking into account simply size
Figure 2. (a) DSC normalized traces of bulk PEMA (black) and PEMA spheres of different diameters. Red lines correspond to submicrometerdiameter spheres, whereas blue curves correspond to nanometer-sized spheres. (b) Derivative of the heat flow with respect to T of the heat flowshown in part a.
Figure 3. AFM scans (a) before and (b) after a DSC scan of SDS-PEMA nanoparticles (PEMA1). (c) DSC normalized traces of the 1st and 2ndscans and (d) corresponding derivative of the heat flow. Similar results are obtained for SDS-free PEMA nanoparticles.
effects, Tg should increase in confined polymers. Several yearsago, Mi et al presented a thermodynamic model for accountingthe increase of Tg in nanosized single chain polymericparticles22 made of polymers with high molecular weight. TheTg behavior related to single chain polymers (SCP) have beenstudied extensively in the past decades, and the results fordifferent systems have shown an increase in the glass transitiontemperature.17b,23 According to the proposed model,22a theincrease of the Tg in SCP is related to a decrease of the entropyin the system, due to a different arrangement of the polymericchains, imposed by the geometrical confinement. In fact, themodel proposes that the behavior of the Tg in the single chainparticles (Tgs) is related to the bulk Tg (Tgb) by the followingequation:
≅Δ
+− −T
TkN
CN Nln ( ln 2)
p
xgs
gs
/3 2
(1)
where k is the Boltzmann’s constant, N the degree ofpolymerization, ΔCp the specific heat change of the bulkmaterial, and x a constant that equates 5/3 for a self-avoid chainand 2 for a random-walk chain. Note that since the right side ofthe equation is always positive, the glass transition temperatureof the single-chain polymer is always higher than the bulk. It isimportant to notice that to obtain the single chain polymers, allof the cited authors have worked with systems with highmolecular weight (∼106) that resulted in nanoglobules ofsimilar size in comparison to this work. However, it is clear thatin our case, as well as in other works,12 the final nanoparticlesystem is not a single chain polymer, although the behaviorseems to be related when dealing with the changes suffered bythe Tg due to confinement. Here we generalize this entropicmodel for SCP to a confined multiple chain polymeric system(MCP). The first consideration is how the number ofconformations (C) of the polymer chain in the nanosphere isaffected by the confinement. The number of conformations of achain molecule can be calculated following the relation24
μ≈ −C nu SCP (2)
where μ ≈ 2−6 is the number of rotational isomers, determinedby the type of monomer, and nu‑SCP is the number of repeatingunits in a single chain, that is equal to the parameter N in eq 1.Therefore, the main factor controlling the number ofconformations of a SCP is the total number of repeatingunits. To generalize this relation to multiple polymer chains thenumber of possible conformations can be obtained by means ofsimple combinatory mathematics as
μ≈ ·C n nu u (3)
where nu the number of repeating units in each chain and nc isthe number of chains in the system. The relation for SCP (eq2) is recovered if nc = 1. From eq 3, it follows that the numberof conformations of the system is controlled by the productnu·nc, which is just the total number of repeating units in thewhole. According to the results in the literature, the advantageof having a SCP system is that there are less possible states forthe organization of the chain (highly ordered system25) sincenu‑SCP is low, and thus there is a reduced entropy. Similarly, forMCP the total number of repeating units in the system is thekey parameter that controls the possible conformational states,and thus the entropy. Nanospheres offer a framework forobtaining MCP systems with a reduced number of repeatingunits, regardless the molecular weight of the polymer. Providedthat the product (nu·nc) ≈ nu‑SCP, the MCP system should havethe same conformational possibilities as the SCP, which leadsto a similar entropy behavior. Possible interaction amongdifferent chains should not be different from the interactions ofa chain with itself since the entanglement of the polymericsystems, do not allow a chain to distinguish its own repeatingunits from the rest. Also, in our hypothesis, the influence of theend-groups is not taken into account since its proportion withrespect to the total repeating-units is negligible (about 0.5% inthe worst case scenario).With these ideas in mind we may postulate that, if a polymer
system is confined to a volume where the number of repeatingunits (nu·nc) is below a certain value, the available conforma-tional states are reduced and consequently the arrangement ofthe chains must be different from that of the bulk. Thus, theentropy of the system is reduced, leading to an increase of Tg.Above this number of repeating units threshold, the systembehaves in the same manner as in the bulk state.In order to probe this postulate, we have analyzed the results
available in the literature together with our present results forPEMA nano and submicrometer spheres. The number of chainsin a polymeric nanoparticle can be calculated following thediscussion presented by Pilcher and Ford,23b and using theformula:
ρ=n
V NMc
A
w (4)
where V is the volume of the nanoparticle, ρ its bulk density,NA the Avogradro’s number and Mw the molecular weight ofthe polymer. The degree of polymerization (N) can becalculated as:
Table 3. Changes in Tg in This Work and in the Literature and Their Corresponding Number of Repeating Units per Particle inEach Case
reference polymer nanosphere diameter (nm) change in Tg nc ntotal
where MO is the molecular weight of the monomeric unit.Notice that the degree of polymerization (N) is just thenumber of repeating units per polymeric chain, and thus equalto nu. Finally, the total number of repeating units in ananoparticle system follows the relation:
ρ= · =n n n
V NMtotal c u
O
A
(6)
We have applied this set of equations to the data available inthe literature, for amorphous polymers subjected to 3D-confinement. The results are summarized in Table 3, andarranged with increasing ntotal.A clear trend for the onset of induced confinement effects in
polymeric nanospheres is observed in Table 3. There are twomarked regions, determined by a threshold value of ntotal around105, where the confinement effects are clearly distinct. For ntotal< 105, all the gathered evidence points toward an increase of Tg,that is based on the decrease of entropy in the nanospheres dueto less possible conformational states. Notice that this behaviordepends only on the total number of repeating units thatconforms the spheres instead of the number of chains. Thisobservation supports the idea that the value of ntotal is a keyparameter when analyzing the possible conformational statesfor any polymeric system.On the other hand, when ntotal is increased over 105, there
seems to be no change in Tg, only contradicted in oneparticular case. According to the equations previouslypresented, the larger the values of ntotal, the higher the entropyof the system since there are more available conformationalstates of the polymeric repeating units. When this value isgreater than some threshold (∼105), although the system is innanoscale geometry, the sample behaves just like in the bulk.There are some exceptions to the trend presented here, like thecase of polystyrene presented by Ding and co-workers.26
However, this work and several others have been recentlyreviewed by Zheng and Simon,28 who managed to prove thatproblems during the preparation steps lead to a plasticizer effectthat ultimately generates the decrease in the Tg. There are otherworks, as reviewed by Mi et al.,22b where the authors evidencedan increase of the entropy in single chain polymeric particles.However, in those works, some key parameters, such as the sizeof the particles were not presented, so it was impossible toquantify the value of ntotal. Also, in the case of the work bySasaki et al.,27 although no change in the Tg position could bemeasured, a clear decrease in the ΔCp of the polymernanospheres, with decreasing diameter of the nanospheres,could be found. Changes in ΔCp can be related to changes inentropy of the system, thus agreeing with our model. Anyhow,as reviewed by Koh,21 the change in the Tg position does notalways relates to a specific ΔCp trend, making the use of thisparameter really difficult as a fundamental quantity fordescribing the changes due to confinement.In a work by Mackay et al.22b cross-linked PS nanoparticles,
with very low diameters (<10 nm) and with a value of ntotal∼103, exhibited no change in Tg. Other authors, like Cherianand co-workers, found an increase of Tg for cross-linkedpolycarbonate nanoparticles29 with comparable physicalparameters to those of Mackay. However it is difficult toestablish a comparison between those nanoparticles and thecorresponding bulk, since the polymers used were chemically
modified and this may have somehow affected the behavior ofpolymeric system in confinement. This is the reason why theseresults are not included in Table 3. Recent works by Zhang30
and Feng,31 analyzed the Tg behavior in PS nanoparticlessuspended in water, under several confinement conditions.Both groups found that for surfactant free PS nanoparticles,with diameter from 700 to 20 nm, a decrease in Tg wasmeasured by different DSC approaches. The results wereexplained using a free surface model by comparing the freenanoparticles with capped PS nanoparticles, with inorganic30
and surfactant31 shells, where no thermodynamic change tookplace. The existence of a free surface layer deals primarily withthe interaction between the polymer system and thesurrounding media (water in these specific cases) instead ofwith the arrangement of the polymer within the nanoparticles,as in our model. Also, it is worth mention new studiesconducted by Boucher32 and Zhang33 that suggest a decouplingbetween cooperative segmental mobility and the glass transitiontemperature, in thin polymer films and organic nanospheres.The authors propose that molecular mobility could not be one-to-one related to the calorimetric Tg due to the presence ofsurface effects in nanoconfined systems.Recent works in thin films geometries point toward the extra
free volume induced at the interfaces in these geometries asresponsible for perturbation in the glassy dynamics, inparticular, modifications in monomer density at the interface.34
In that framework, our results and analysis indicate that inPEMA nanospheres with diameters below 100 nm, the chainsare packed in such a way that free volume is considerablyreduced compared to the bulk. In this sense, the present systemof polymer nanospheres could serve as a platform to testproximity effects for recently proposed models like the latticemodel of dynamic heterogeneity and kinetic arrest in glassforming liquids.10
4. CONCLUSIONS
An increase of the glass transition temperature (Tg) was foundfor PEMA nanoparticles, in a dimension range well below 100nm. By a systematic preparation and discrimination of thedifferent variables that may affect the glass transition of thedroplets of few polymer chains we have demonstrated that themain contributing factor that affects the shift or no shift of theTg is the number of possible conformations, which is given bythe number of repeating units in the droplet. These results wereexplained by invoking a reduction of entropy in the system, dueto the decrease of the possible conformational states, whichultimately lead to a better arrangement of the polymeric chainsin the nanosystem.Our model correlates well with a significant amount of results
available in the literature for the past 2 decades, dealing withdifferent polymers and preparation methods.
■ ASSOCIATED CONTENT
*S Supporting InformationCalorimetric trace of SDS. This material is available free ofcharge via the Internet at http://pubs.acs.org.
NotesThe authors declare no competing financial interest.
■ ACKNOWLEDGMENTSFinancial support by MAT2008-03232, MAT2011-23455, andMAT2012-33517 from MINECO is gratefully acknowledged.D.E.M and M.S. thank CSIC for the tenure of JAE-Prefellowship and JAE-Doc contract, respectively, and the FondoSocial Europeo (FSE) for cofinancing the JAE Program. A.S.acknowledges CSIC for a research associate contract.
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