Revealing spatially heterogeneous relaxation in a model. nanocomposite

Transcription

1 Revealing spatially heterogeneous relaxation in a odel nanocoposite Shiwang Cheng, 1 Stephen Mirigian, 2 Jan-Michael Y. Carrillo, 3,4 Vera Bocharova, 1 Bobby G. Supter, 3,4 Kenneth S. Schweizer, 2 Alexei P. Sokolov 1,5 1 Cheical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 2 Departent o Materials Science and Cheistry, Frederick Seitz Materials Research Laboratory, University o Illinois, Urbana, Illinois 61801, USA 3 Center or Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States 4 Coputer Science and Matheatics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States 5 Departent o Cheistry, Departent o Physics and Astronoy, University o Tennessee, Knoxville, Tennessee 37996, USA The detailed nature o spatially heterogeneous dynaics o glycerol-silica nanocoposites is unraveled by cobining dielectric spectroscopy with atoistic siulation and statistical echanical theory. Analysis o the spatial obility gradient shows no glassy layer, but the α-relaxation tie near the nanoparticle grows with cooling aster than the relaxation tie in the bulk, and is ~20 ties longer at low teperatures. The interacial layer thickness increases ro ~1.8 n at higher teperatures to ~3.5 n upon cooling to near T g. A real space icroscopic description o the obility gradient is constructed by synergistically cobining high teperature atoistic siulation with theory. Our analysis suggests that the interacial slowing down arises ainly due to an increase o the local cage scale barrier or activated hopping induced by enhanced packing and densiication near the nanoparticle surace. The theory is eployed to predict how local surace densiication can be anipulated to control layer dynaics and shear rigidity over a wide teperature range. Corresponding author. Eail: sokolov@utk.edu 1

2 I. Introduction Nanocoposite aterials are widely used in any advanced applications, including lightweight aterials, coatings, ebranes, and solar and uel cells 1-3 due to iproved echanical, theral, optical, electrical and catalytic properties. 1-7 It is believed that the large interacial area between nanoparticles and the host atrix plays a central role in the oten observed avorable acroscopic property changes which are nucleated by the odiication o atrix packing, relaxation and elasticity near the 3, 6, 8 particle surace. Thus, a clear understanding o the spatially heterogeneous structure and dynaics in interacial layers is crucial or the rational design o nanocoposites with perorances on deand. This undaental proble also arises in the geoetrically sipler, but technologically iportant, context o supported and capped thin ils Since the discovery o dierent obility doains in illed elastoers by NMR, 12, 13 additional experients, theories and coputational siulations 23, 26, have been applied to study the interacial properties in polyer nanocoposites PNCs. Owing to strong polyer-particle attractions required or iscibility and particle dispersion, a glassy or dead layer deined as an α-relaxation tie > 100 s ~1-6 n thick is oten suggested , 24, 25 However, this interpretation is controversial 20-22, 26 and the answer ay be cheically speciic. Current experiental liitations and inability to spatially resolve dynaics close to a nanoparticle surace provide signiicant obstacles in studies o the interacial obility gradient. Moreover, coputer siulations can probe only odestly slow activated dynaics ar above the 2

3 laboratory glass transition teperature. 35 Additional coplications in PNCs include the presence o ultiple interacial and conineent eects, 10, 19, 30, 36 e.g., changes o 16, 17, 34, 37, 38 entangleent density, strong physical adsorption o polyer chains resulting in extreely long equilibration ties 39 and pervasive nonequilibriu eects. These coplications render it very diicult to deinitively deterine the existence or absence o glassy layers, spatial gradients o relaxation and elasticity, thickness o dynaically perturbed layers, and their teperature variations upon approaching T g. In this article, we eploy the glycerol/silica nanocoposite GSNC as a odel aterial in order to exclude the coplications associated with chain conineent and entangleent eects, as well as the nonequilibriu chain adsorption phenoenon. The origin o atrix-particle attraction or iscibility is hydrogen bonding, which is representative o any practical PNCs. Based on a ore rigorous data analysis approach that allows the odel-ree extraction o the ull relaxation tie distribution in the nanocoposite, our broadband dielectric spectroscopy BDS easureents show clear evidence or a dynaic interacial layer. Key indings o the experiental analysis include a very broad distribution o long relaxation ties, a ean slowing down o roughly one order o agnitude relative to the bulk, and a ean layer thickness that grows ro ~1.8 n at T ~ 253 K to ~3.5 n near the bulk T g o glycerol. No indications o glassy layers are ound ro analysis o dielectric strength and teperature odulated DSC TMDSC data. In order to interpret our observations, and develop a real space understanding o spatial obility gradients inaccessible to direct experiental easureent, we synergistically cobine 3

4 atoistic siulations with a lightly coarse grained statistical echanical theory o activated relaxation that can bridge the high teperature siulation regie with the low teperatures relevant to experient. Overall, we obtain good agreeent between experient, theory and siulation, and a detailed physical picture o the spatial obility gradient in the interacial layer is constructed. The validated theory is then eployed to elucidate the role o aterial-speciic local densiication near the particle surace on slow dynaics and the oration o true glassy, high odulus layers. II. Details o experients and siulations A. Saple preparations and characterizations Pure glycerol, ethanol and ethanol were purchased ro Siga-Aldrich and were used as received. The SiO 2 nanoparticles NP o radius o R NP =D/2= 12.5 n were synthesized in ethanol at a concentration o 15 g/l by the odiied Stöber 40, 41 ethod. The glycerol/sio 2 nanocoposites GSNCs were prepared by the ollowing procedure: First, 0.3 g o glycerol was dissolved in 10 l ethanol. Then, dierent aounts o SiO 2 /ethanol suspension were added into the glycerol/ethanol solutions in a drop wise anner while stirring. Ater one hour o ixing, the GSNCs were then dried in the hood at roo teperature until ost o the ethanol evaporated. Ater that, all the saples were urther dried under vacuu conditions ~10-5 bar or another one week at 20 o C. Good dispersion o GSNCs were achieved as evidenced by the saple transparency and TEM Zeiss Libra 200 HT FE MC ater drying see Fig. 1. Detailed saple characterizations are shown in Table 1. The 4

5 loadings were easured by therodynaic gravitation analysis TGA Q50, TA Instruents. The volue raction was calculated by assuing the density o glycerol is ρ = 1.26 g/c 3 and the density o silica nanoparticles is gly ρ = 2.65 sio2 g/c 3. The T g values o GSNCs were easured by teperature odulated dierential scanning calorietry TMDSC Q1000, TA Instruents ro 0 o C to -100 o C with a cooling rate o 2 o C/in and odulation rate o ±0.5 o C/in. The average surace to surace interparticle spacing IPS d IPS was calculated as d = IPS R NP 3 4π 1/ 3 2, where is the volue raction o nanoparticles. Broadband dielectric spectroscopy easureents or these nanocoposites in the requency range between 10-2 Hz to 10 7 Hz and teperature range between 273 K to 153 K were carried out using a Novocontrol Concept-80 syste with an Alpha-A ipedance analyzer, a Quatro Cryosyste teperature controller and a ZGS saple cell. Table 1. Saple SiO 2 wt% SiO 2 vol% Characteristics o glycerol/silica coposites T g TMDSC K T g BDS K d IPS n GSNC GSNC GSNC GSNC

6 Figure 1: a Photograph o the glycerol/silica nanocoposite with loading o 23.6 vol%. b TEM iage o the sae coposite to show the dispersion o nanoparticles. Soe part o the iage involves ultilayers due to the diiculty o cryoicrotoing. The nanoparticles are individually seen and the saple is transparent, iplying a good dispersion o nanoparticles. B. Dielectric responses o heterogeneous aterials Figure 2: Geoetry o the two-phase odel TPM a and the interacial layer odel ILM b, the overall dielectric response at a given requency o the heterogeneous aterials is c. Dielectric spectra o nanocoposites are oten analyzed as a superposition o spectra o dierent coponents. This approach is not accurate, because the dielectric responses o heterogeneous aterials always contain contributions ro the intererence between dierent coponents 42, 43. For exaple, the siplest two-phase odel TPM with spherical particles the saple geoetry shown in Figure 2a 6

7 7 predicts the overall dielectric response c at requency calculated ro Maxwell s equations 42 by assuing the eective ediu boundary conditions: cross c = = = 1 where is the volue raction o nanoparticles; c, and are coplex dielectric unctions o the coposite, the atrix and the nanoparticle, respectively. Thus the spectru has not only contributions ro two coponents with their volue ractions, but also a cross ter that decreases the overall perittivity. I an interacial layer with dielectric unction l was added into the geoetry Figure 2b, the interacial layer odel ILM predicts even ore coplex dielectric unction 43 : S R S R l l l c = 2 with R = 2 l 3 l l l l l l d S = l d = = 1 l where l and are the volue raction o the interacial layer and the atrix, respectively. The above heterogeneous odels provide the irst order approxiation or the

8 overall dielectric response o our nanocoposites with spherical SiO 2 nanoparticles. In the analysis described below, = 3. 9 holds in our experiental teperature and requency range and the atrix is directly obtained ro experient on pure glycerol with a odiied dc-conductivity to atch the slightly higher dc-conductivity o our nanocoposites due to tiny aount o ipurities ro the NPs. C. Coputer siulations Atoistic olecular dynaics siulations o pure glycerol liquid and glycerol in contact with an aorphous silica substrate 44 were carried out using the class I orce ield, GAFF 45 and the MD code, LAMMPS 46 with GPU acceleration 47. Details o the orce-ield and the coparison o the siulation results or pure glycerol to published density data, 48 the static structure actor 49 and the interediate scattering unction 50 o published neutron scattering experients are provided in Appendix A. In our siulations, 5000 glycerol olecules are placed in contact with the substrate. The center-o-ass o the substrate was tethered at the origin by a haronic potential with interaction strength o 2000 kcal/ol. The siulation box is periodic in x,y and z directions and the silica substrate is replicated in the xy plane see Fig. A6. Initially, the syste was equilibrated in the isotheral-isobaric NPT enseble using a Nosé-Hoover therostat and barostat, where the barostat was only applied in the direction perpendicular to the substrate. The NPT equilibration proceeded up to 5 ns ollowed by a NVT enseble run or 100 ns. We tracked the parallel ean-squared-displaceent MSD o the center-o-ass COM o each glycerol olecule and averaged its MSD as a unction o distance ro the silica substrate,. 8

9 A bin size o 0.25 n was used and the counting ensured that both the initial and inal position o the COM o the olecule belonged to the sae bin. At long elapsed tie, MSD z 4D zδt such that the relaxation tie at z is coputed as τz 1/D z and τz/τ α =D,bulk /D z. To atch the experiental nanoparticle loading o 23.6%, <τgτ> is coputed based on data truncated at 3.8 n ro the surace o the substrate to atch the experientally deterined IPS. III. Elastic Collective Nonlinear Langevin Equation ECNLE theory Relevant technical aspects o prior ECNLE theory work 29, in the bulk and ree standing ils are suarized in Appendix B. Here, we only recall the physical ideas. We then discuss the new challenges associated with solid suraces and nanocoposites, and outline the zeroth order approach used in this article. A. Bulk Liquids and Free Standing Fils The oundational quantity to describe single olecule activated relaxation in the bulk liquid volue raction φ is a particle-displaceent-dependent r dynaic ree energy, F r = F r F r, which deterines the eective orce dyn ideal caging exerted on a oving tagged spherical particle due to its surroundings. 53 The localizing caging contribution, F caging r;φ,sk, captures the eect o interparticle interactions and local structure on the nearest neighbor length scale r < r 3d / 2, cage and is quantiied by the pair correlation unction, gr, or Fourier space static structure actor Sk. To execute a large aplitude local jup over the cage-scale barrier F B as predicted ro the dynaic ree energy requires a sall aount o extra space be 9

10 created which is realized via a spontaneous collective elastic luctuation o the liquid olecules outside the cage. The corresponding haronic strain ield decays as an inverse square power law o distance, and its aplitude is deterined by the predicted jup distance. The resultant elastic barrier F elastic is then deterined by the haronic stiness o the transient localized state and the eective jup distance. The total barrier or the activated event is thus F total = F B F elastic. To treat olecular liquids, a priori apping to an eective hard sphere luid is eployed based on the requiring that the cheistry and teperature-dependent volue raction exactly reproduces an equilibriu property o the real liquid, the diensionless copressibility. 52 The latter quantiies the aplitude o n-scale density luctuations which is the key structural order paraeter o ECNLE theory. Using the above eleents, Kraers theory is utilized to copute the ean barrier hopping tie which plays the role o the alpha or structural relaxation tie. 53 No adjustable paraeter applications o the theory to olecular liquids, including 52, 54 glycerol, reveals good agreeent with experients over 14 decades in tie. For a ree standing il, the bulk theory is odiied because o two distinct, but coupled, physical eects. 29 First, the local barrier is lower near the surace due to a loss o nearest neighbors, the raction o which can be analytically coputed as a unction o distance ro the cage center, α z. The dynaic ree energy is thus: F r; z = F r α z F r; φ 3 dyn ideal caging Second, the elastic cost or the re-arrangeent is reduced since no strain ield is present beyond the il interace. The presence o α z iplies all physical 10

11 quantities becoe position-dependent in the il, and as a consequence the elastic cuto eect is coupled to the sotened or ree interaces liquid near the surace and thus reduced caging eects are elt well into the il. The theory has been applied to predict obility gradients, ast surace diusion, T g shits, and other properties o ree standing ils which have been avorably copared to experient 29. B. Capped Fils and Nanocoposites Solid suraces introduce any new coplications that are diicult to theoretically treat. Here, we present a irst attept which has the virtue o allowing quantitative predictions to be ade that can be conronted with siulation and experient. The three key theoretical sipliications are as ollows. 1 Liquids layer near a solid surace, and their local interolecular structure also can change. Given the eleentary length scale in bulk ECNLE theory is the cage radius, such packing changes should be incorporated on this scale. A typical aniestation o an attractive substrate is an increase o local density in just the irst layer thickness, d o olecules 51 by a nonuniversal actor o λ > 1 copared to the bulk which depends on any physical and cheical actors. It odiies both the eective nuber o nearest neighbors o atrix olecules and the equilibriu pair correlations or Sk as a unction o distance ro the surace. 2 A tagged particle near a solid wall is issing atrix nearest neighbors, but experiences orces due to surace atos which will tend to approxiately copensate or this loss. We assue the net eect o these copeting orce eects in the deterination o the dynaic ree energy is solely 11

12 to densiy the irst layer o olecules. 3 The substrate atos in real aterials have variable degrees o vibrational otion or elastic stiness. We assue that this otion is not perturbed during a atrix relaxation event. Hence, there is no elastic energy penalty inside the solid substrate, and the cuto o the strain ield idea orulated or ree standing ils is adopted. This siplest and crude assuption does not require any extra paraeter to ipleent. Figure B1 shows that the increase in the local barrier due to near surace densiication is predicted to be the priary reason that relaxation slows down near the solid surace. Given the above sipliications, the dynaic ree energy as a unction o distance o a particle ro the solid interace, z, depends on a position-dependent volue raction, φz, as: F dyn r; z = F ideal r α zf caging r;sk;φz 4 Here, αz = φz / φ bulk quantiies the local densiication by a actor o λ > 1, and can be analytically coputed see Appendix B. The position-dependent changes o the eective hard sphere luid structure actor, Sk;φ z, are coputed using Percus-Yevick theory. As elaborated on in Appendix B, the cage scale barrier, F B, is increased near the adsorbing surace out to a distance r cage d ro the interace. The aplitude o the elastic strain ield and liquid stiness associated with the nonlocal coponent o the alpha relaxation event, and hence collective elastic barrier, are again unctions o the position in the conining geoetry, and quantiied based on idea 3. The relaxation tie spatial gradient then ollows ro Kraers theory. 53 Fro this, a slow layer thickness is identiied as the spatial region where the relaxation 12

13 tie is signiicantly above its bulk liquid value. We ind that this length scale, l layer, depends weakly on cheistry e.g., λ, but is essentially independent o teperature. One can then copute an average slow interacial relaxation tie as l layer 1 int er = dzτ α z τ 5 llayer 0 To copute the il-averaged distribution o relaxation ties, gτ, ro knowledge o the obility gradient, the relaxation process in each il layer is treated as a Poisson process controlled entirely by the ean relaxation tie τ α z. The probability density o relaxation ties is thus Pτ,τ α z = τ τ α z 2 e τ /τ α z. The experiental dielectric data was noralized using an integral over lnτ, and relevant theoretical quantity is thus τ Pτ,τ α z. For a il with thickness h one then has h τ g τ = 1/ h dzτp τ, τ α z 6 0 The local shear odulus in the absence o relaxation at a location z in the il, and the layer-averaged requency-dependent odulus, are coputed using the ethods eployed previously 51 see Appendix B. In our experiental syste, the nanoparticle and d IPS length are large copared to glycerol olecule d = 0.5~0.6 n and the coputed obility gradient spatial range. Moreover, D>>d iplies that locally the particle appears as a lat surace to the atrix olecules ignoring atoic scale surace corrugation. Both eatures 10, 36 sipliy theoretical analysis. This otivates the adoption o a well-known analogy between nanocoposites and capped ils by treating the eects o randoly dispersed particles on the atrix as identical to that o a lat surace. Thus, 13

14 or the purpose o describing glycerol dynaics, the nanocoposite is apped to a supported, sei-ininite il. Average nanocoposite quantities are calculated over a region extending ro the surace into the atrix a thickness, h = d IPS /2. The exact choice o h has no ipact on the calculations since h is larger than any relevant length scale. Obviously the range o validity o the 3 key sipliying ideas orulated above requires uch uture study. For our glycerol-silica syste, two speciic aspects should be entioned. First, our treatent does not explicitly account or glycerol-surace hydrogen bonding orces, but soe o their consequences are taken into account. In the successul treatent o bulk supercooled glycerol 50 with ECNLE theory, hydrogen bonding enters only via its signiicant odiication o the key order paraeter o the theory, the aplitude o n-scale liquid theral density luctuations. 52 This is in the spirit o sipliication 1 which assues the priary dynaical eect o hydrogen bonding is odiication o local structure. Concerning sipliication 3, recall that ECNLE theory couples the caging and the longer range elastic luctuation aspects. Our assuption that the strain ield is cut o at the surace ight be plausible given the high odulus o silica renders it hard to deor via sall displaceents o glycerol olecules. Also, since the two barriers or relaxation are coupled, the local densiication paraeter λ aects both o the, albeit in dierent ways see Fig.B1, and their relative iportance will depend on syste cheistry. IV. Results and Discussions 14

15 A. Dielectric spectra o nanocoposites with dierent loadings Figure 3 shows the experiental dielectric loss spectra o nanocoposites with dierent loadings in coparison with GSNC0. There are several notable eatures: 1 a signiicant increase o broadening in alpha relaxation peak upon increasing loading; 2 a reduction in alpha peak intensity relative to pure glycerol; and 3 a higher conductivity which ay arise ro a very sall aount o ipurities associated with the added nanoparticles. All these eects are siilar to results reported or PNCs The inset o Fig.3 shows the teperature dependence o the alpha relaxation tie estiated ro the peak position o the loss spectra peak τ α = 1/ peak. Good agreeent between our easureents and literature data 55 indicate our saples are ree o solvent ro saple preparations T = 233K τ α s GSNC GSNC GSNC 15.6 GSNC 23.6 Neat Glycerol Re " /T K GSNC 0 GSNC 8.8 GSNC 15.6 GSNC rad/s Figure 3: Dielectric loss spectra o coposites with dierent loadings in coparison with neat glycerol at T = 233 K. The inset shows the alpha relaxation tie estiated ro the peak position o the loss spectra o our experients and literature data o neat glycerol. 55 B. Signature o the interacial layer dynaics with slowing down 15

16 segental relaxation Several ethods have been proposed to interpret the dielectric broadening either by assuing an additional interacial process contributes to the spectral broadening, 18, 20 or by hypothesizing that only ree polyer is dielectrically active. 21 Such ethods treat the spectra in an additive way, neglect the cross ters characteristic o heterogeneous systes see section II.B, and are not accurate or quantitative analysis o the BDS spectra o a nanocoposite. 56 Here, we analyze our spectra in ters o the heterogeneous odel Section II.B which provides a uch ore accurate analysis. As shown in Fig. 4, or a NP volue raction o 23.6%, the TPM predicts a signiicant dielectric broadening black line even i the atrix dielectric relaxation is identical to pure glycerol. Thus, the observed spectral broadening can be ascribed partially to the heterogeneous nature o the nanocoposites. On the other hand, the experiental spectra are clearly broader than predicted by the TPM odel, which ay indicate atrix dynaics do change upon the addition o 19, 30, 36 nanoparticles. Since d IPS >>d even at the highest loading studied, spatial conineent eects should be negligible. Hence, we attribute the dierence between the TPM prediction and experient to the existence o a dynaically perturbed glycerol layer near the NP surace. On the other hand, the interacial layer odel that includes a perturbed layer describes well the BDS spectra both and over the entire requency range. We justiy the above analysis by eploying a odel-ree approach or calculating the relaxation tie distribution directly ro the dielectric unction, 42 which can 16

17 potentially separate two overlapping processes. 57 To rule out any NP contribution to the dielectric unction, we back calculate, using the TPM, the eective atrix dielectric unction e ro the easured coposite dielectric signal c. In this way, is replaced by e which contains only the atrix dynaics inoration. Figures 5a-5c show the real part e, derivative o the real part der = -π/2 e / ln, and iaginary part e o the eective dielectric spectru e o the GSNC 8.8 and GSNC 23.6 saples. The neat glycerol spectra are presented or coparison. The α-relaxation peak and the Maxwell-Wagner-Sillars MWS polarization peak can be clearly identiied in the derivative spectra Fig.5b. The MWS peak, according to the TPM, has the characteristic relaxation tie τ MWS 2 1 = 0. When σ >>σ, ~61 and ~3.9, it appears at a 2 σ 1 σ requency roughly o MWS σ 2π =, where the conductivity contribution to 0 42, 58 becoes coparable to. Since the e data contain no NP contributions, one can analyze the dynaics o the atrix ore accurately ro it. In linear response, the dielectric relaxation process can be described by a superposition o Debye unctions with dierent relaxation ties 42, 57 glnτ σ 0 : e = d lnτ i, where glnτ is the s 1 iτ 0 distribution o relaxation ties noralized as glnτdlnτ = 1, σ 0 is the dc-conductivity, 0 is the vacuu dielectric constant, the exponent s is ixed to be 1.0 in our analysis, and i is the iaginary unit. 57 We applied the generalized regularization ethod 59 to calculate this distribution as shown in Fig.5d, where the horizontal axis is 17

18 intentionally reversed since τ = -1. Figure 5d deonstrates that only the α-relaxation process is present in the glnτ spectru o pure glycerol, while the MWS process and an additional relaxation process appear in the nanocoposite spectra. The latter has a ean relaxation tie ~10-20 ties slower than the bulk α-relaxation and corresponds to the low-requency broadening o the spectra. We interpret it as direct evidence or the existence o a dynaically slower interacial layer. To the best o our knowledge, this is the irst clear experiental deonstration o the interacial layer dynaics o a nanocoposite. The overall relaxation tie distribution shown in Fig.5d still contains relaxation processes ro ions in the atrix: e.g. the MWS process and the dc-conductivity process, both o which are not relevant to the atrix segental relaxation. Thus, we can subtract the MWS and conductivity contributions ro the overall relaxation tie distribution. As shown in Fig.6a, the teperature-dependent interacial dynaics can be deterined ro the relaxation tie spectra o all saples over the wide teperature range o K. This distribution is a easure o the heterogeneous obility gradient near nanoparticles, albeit with no spatial resolution. The axia in the distribution unctions are taken as the characteristic relaxation ties o the α-process τ α, and the shallow peak at longer ties as the interacial layer process τ inter. Both τ α and τ inter are essentially independent o NP loading Fig.5d, as expected by experiental design. However, τ inter does exhibit a slightly stronger teperature dependence than τ α Fig.6a and inset o Fig.6b, with the ratio 18

19 τ inter /τ α odestly increasing ro ~ 8 to ~ 16 upon cooling or all nanocoposites. Signiicantly, there is a very broad tail o the distribution that extends to ~ 50 τ α with an aplitude that grows with cooling. Vogel-Fulcher-Taann VFT its o the ean relaxation ties in Fig.6b suggest the BDS T g o the bulk atrix and interacial layer are 190 K and 195 K, respectively, consistent with earlier studies o a thin glycerol il on a SiO 2 surace and under nanopore conineent Exp GSNC 23.6 ' T = 233K ', " ILM GSNC 23.6 ' Exp GSNC 23.6 " Exp GSNC 0 " TPM GSNC 23.6 " ILM GSNC 23.6 " rad/s Figure 4: Raw dielectric spectra o GSNC 23.6 and pure glycerol at 233 K sybols as well as the spectra predicted by the TPM black line and the ILM orange line or and purple line or at loading o 23.6%. The labels in the igure should be read as ollows: Exp GSNC 23.6 reers to the dielectric storage peritivity spectra ro Experiental easureents on the nanocoposite GSNC

20 ' e % 8.8% 23.6% a 233K 10 1 MWS b 'der 10-1 Alpha " e glnτ Figure 5: The eective dielectric unction and relaxation tie distribution ro the sae spectra o three dierent saples at 233 K: GSNC 0 red circles, GSNC 8.8 blue squares, GSNC 23.6 green diaonds. a the real part e ; b the derivative spectra der = -π/2 e / ln; c the iaginary part e ; d the relaxation tie distribution glnτ o the corresponding saples. The solid curves indicate a itting o the interacial segental process and the dashed curves show the it o the bulk α peak o two nanocoposites. Alpha c rad/s 10 2 Conductivity Interacial d 10 1 layer MWS Slow down τ s Alpha a glnτ alpha peak 23.6% 233K 23.6% 213K 23.6% 203K 0% 233K 0% 213K 0% 203K 10 0 Bulk layer <τgτ> Interacial layer 292K τ/τ α τ/τ α Theory 23.6% 207K 222K 252K Interacial peak Figure 6: a The noralized distribution o experiental relaxation ties at three dierent teperatures or pure glycerol illed sybols and nanocoposites with 23.6 vol% loading epty sybols. The inset shows the theoretical prediction o the total relaxation tie distribution open sybols, the contribution ro the bulk atrix solid 20 b log τ s Theory 23.6% 10 2 λ = τ inter /τ α Interacial layer λ = 1.04 λ = /T K -1 Bulk layer VFT itting Tg = 195 K VFT itting Tg = 190 K GSNC 0 τ α GSNC 8.8 τ α GSNC 23.6 τ α GSNC 8.8 τ inter GSNC 23.6 τ inter /T K -1

21 curves and the contribution ro the interacial layer dashed curves at our dierent teperatures 207 K red, 222 K blue, 252 K green and 292 K black. As described 2 τ in the text, the theory coputes τg τ = exp τ / τ z. b 2 α τ z Vogel-Fulcher-Taann VFT its dash curves o the bulk α relaxation tie τ α and interacial layer α relaxation tie τ inter suggest the BDS T g o the bulk and interacial layer to be 190 K and 195 K, respectively. τ α and τ inter are characteristic ties ro the relaxation tie distribution analysis. The inset shows a coparison o the teperature dependence o the ratio τ inter /τ α ro experient and theoretical calculations perored at three dierent values o the local densiication paraeter λ. The epty sybols in the inset have the sae eaning as in the ain igure. α C. Dynaics o the interacial layer: coparison between experients, theory and siulations The nanocoposite dynaical easureents above exhibit any eatures o heterogeneous dynaics. Although soe workers have suggested a obility gradient 15, 16, 32, 63 near the nanoparticles ro experients and siulations, explicit experiental studies that clearly reveal this eature and provide a undaental understanding o all eatures, especially in real space, are still issing. Moreover, no icroscopic and quantitatively predictive theory currently exists to treat capped thin il and nanocoposites. Hence, our goals are 4-old. i As described in section III and Appendix B, building on the ECNLE theory o bulk activated relaxation or olecular including glycerol and polyeric liquids, 64 along with its successul generalization to ree standing ils, 29 we orulate a zeroth order theory or hopping relaxation in capped ils and nanocoposites. ii To render the theory predictive or speciic aterials, we use results o our atoistic siulations section IIC at high teperatures where τ α < 100 ns. This provides local packing inoration required as an input to the dynaical theory, and serves as a benchark 21

22 to test the obility gradient predictions in the lightly supercooled regie. iii The theory is then used to study the deeply supercooled regie inaccessible to siulation, copare against our experiental data, and to suggest the physical nature o the spatial obility gradient that cannot be directly probed in experients. iv The theory is applied to gain ore generic insight concerning the connection between the aterial-speciic densiication at the particle-atrix interace and the degree o dynaical slowing, and elucidate the conditions required or oring glassy layers o speciied shear elasticity. Our theoretical predictions can be ade without adjustable paraeters i the local densiication paraeter λ introduced in section IIIB is known. To obtain the latter in an a priori anner, we use results o atoistic siulations o the glycerol-silica syste at relatively high teperatures T=293K. The icroscopic equilibriu density proile shown in Fig.7a displays the ailiar layering. Layer densities can be coputed by integrating under the density proile, and are ound to be equal to the bulk value per the adopted theoretical odel or all layers except the irst surace layer, where λ ~ 1.04 indicates denser packing. 65 The dynaic ean square displaceents o glycerol olecules have also been coputed, and the change o the ean τ α near the surace estiated. The sae Poisson process odel as described in section IIIB is eployed to obtain the ull distribution o relaxation ties. Figure 7b shows the gradient o reduced obility at 293 K deterined ro siulation. An order o agnitude slowing down is ound, and dynaics is suppressed over a ew olecular diaeters ro the silica surace, but there is no 22

23 dead layer τ α >100s. Figure 7b also shows ECNLE theory calculations o the obility gradient using λ = The results are in good agreeent with the siulation. This is especially signiicant since this value o λ agrees within the errors bars with estiates ro the equilibriu siulations. One can view this as a calibration step to encode cheically-speciic inoration into the sipliied structural odel eployed in the theory. In all subsequent coparisons o theory with our experiental data we ix λ = A good agreeent between siulation and theory results or the distribution o relaxation ties at 293 K inset o Fig.7b urther supports the useulness o the theory or local activated events that doinate at relatively high teperatures. Theoretical calculations or how the obility gradient evolves in the deeply supercooled regie are also shown in Fig.7b. Modest increase o the relaxation tie near the particle surace relative to the bulk behavior is predicted with urther cooling. However, the spatial range o the gradient o slow relaxation is essentially unchanged. Dissecting the theoretical calculations, and consistent with the 3 key sipliications invoked to orulate the theory in section IIIB, we ind that the obility gradient is largely deterined by cage scale physics and near surace densiication which enhances the local activation barrier. The sall undershoot o the obility gradient curve at low teperatures is a signature o cutting o the elastic ield at the solid surace, a eature that is sensitive to the sipliied theoretical approxiation adopted. The inset o Fig.7b shows that in the ore deeply supercooled regie, theory and experient are in overall good agreeent or the ull relaxation tie distribution. 23

24 This includes both the breadth o the long relaxation tie tail and its increasing aplitude with cooling. Quantitative deviations are evident deep into the high relaxation tie tail. Use o the icroscopic obility gradient proile allows one to unabiguously separate the ull relaxation tie distribution into its perturbed layer and unperturbed coponents. Representative results are shown in the inset o Fig.6a. One sees not only shiting o the slow layer distribution to longer ties with cooling, but signiicant broadening, odest aplitude growth, and other subtle shape changes that relect details o the teperature-dependent real space gradient. The inset o Fig.6b shows a coparison o theory and experient or the ratio o interacial layer ean alpha relaxation tie with respect to bulk ean alpha relaxation tie τ inter /τ α. Good agreeent is obtained or the sae value o λ = deduced ro coparison to our high teperature siulation. This provides additional support or the idea that the inluence o a atrix-particle interace on relaxation can be captured reasonably well based on the local densiication concept, at least or the glycerol coposite. The inset also shows that the theoretical predictions or the agnitude and teperature dependence o the ean slowing down are rearkably sensitive to the cheically-speciic degree o local densiication. For exaple, increasing the latter ro 3.8% to 4.5% results in a ean alpha tie enhanceent nearly 10 ties larger near the bulk T g. 24

25 Z n Figure 7: a Snapshot ro atoistic olecular dynaics siulation o glycerol in contact with a SiO 2 substrate and its equilibriu density proile at T =293 K; b The noralized with respect to the bulk α relaxation tie gradient as a distance Z ro the surace as coputed ro siulation and theory using λ = The dierent solid curves correspond to the theoretical obility gradient at 222 K, 252 K and 292 K. The illed triangles are ro siulations at T = 293 K. The inset shows relaxation tie spectra coparisons between theory, siulation and experient, where theory and siulation plot experients. τ g τ / τg τ in coparison with g lnτ / glnτ peak ro peak b τ/τ α <τgτ>/<τgτ> peak α relaxation Interacial peak Theory222K 252K τ/τ α Exp 0% 253K Exp 23.6% 253K Si 0% 293K Si 23.6% 293K Theory 0% 252K Theory 23.6% 252K Theory 0% 292K Theory 23.6% 292K 292K Siulation 293K D. Absence o glassy layers in glycerol/silica nanocoposites Fro the above analysis o the segental dynaics proile with spatial resolution, no indications o glassy dynaics τ α > 100 s were ound. To deterine the possible existence o a glassy layer, we study in detail the dielectric relaxation strength,, and heat capacity, C p, both o which are sensitive to the existence o the glassy layer. For the dielectric relaxation strength, we ephasize that the crosster Eq.1 reduces the noralized dielectric strength, c /1-, with increase in loading. This decrease can be clearly deonstrated in ters o the TPM: when, ' ~ ', ' ~ ' ; c 25

26 when 0, ' 0 >> ' 0, ' c 0 ~ 1 1 ' 0 < 1 ' 0. 2 Thereore, = ' 0 ' ~1 ' 0 ' < 1. In other c c c words, the noralized dielectric strength, c /1-, should not be a constant with loading, but rather should always be saller than the dielectric strength o the neat glycerol in our nanocoposites. The detailed analysis o experiental c that includes both bulk-like and interacial dynaics and c /1- reveals good agreeent with the predictions o the siple TPM or all loadings Fig.8a, suggesting that all the glycerol dipoles relax within our requency window. There is no sign o a easurable glassy or dead layer in the aplitude o the total dielectric signal. Siilar analysis can be done or the aplitude o the speciic heat jup at T g. The noralized speciic heat capacity o the atrix, C atrix p C C NC NP p p NP =, overlaps atrix with that o neat glycerol, and the step eature in DSC is copleted within 15K above T g or all the NP loadings, as shown in Fig. 8b. All these results conclusively deonstrate that there is no easurable glassy layer in our coposites. c ' T = 233K 10 2 GSNC 23.6 T = 233K 10 1 c Exp TPM rad/s 10 7 Experients TPM a Volue raction c /1- atrix C p J/g o C Figure 8: a Dielectric strength c let axis and noralized dielectric strength c /1- right axis at dierent loadings ro experients red sybols and TPM s % 8.8% 15.6% 23.6% C p J/g o C NC b Te o C T o C GSNC 0 GSNC 8.8 GSNC 15.6 GSNC 23.6 Nanoparticles

27 prediction blue sybols, where c is the step increase o and is the volue raction o the nanoparticles. The inset shows the estiates o c and the dielectric storage spectra o GSNC 23.6 ro experients and the TPM. All the data are collected at T = 233K. b Noralized speciic heat C atrix p C C NC NP = p p NP o the atrix glycerol atrix in Glycerol/SiO 2 nanocoposites and pure glycerol, where C, atrix p NC C p and NP C p are the speciic heat o the atrix, nanocoposite and pure nanoparticle, respectively; atrix and NP are the ass ractions o atrix and nanoparticles, respectively. The inset shows the raw data o speciic heat o pure glycerol, pure SiO 2 nanoparticles and nanocoposites with dierent loadings. No indications o a glassy layer are ound in our nanocoposites according to TMDSC data. E. Interacial layer thickness Based on pre-averaging the spatial obility gradient, the integrated dielectric spectral contributions in the ILM can be eployed to deduce a ean interacial layer volue raction, l. We ind l increases ro ~4% at 253 K to ~10% at 203 K or GSNC 8.8, and ro ~12% to ~28% or GSNC Fro this, the average interacial layer thickness l is estiated as l 1/ 3 l = 1 R NP, yielding a thickness that grows ro ~1.8 n at T = 253 K ~T g 58 K to ~3.5 n at T = 203 K ~T g 8 K, as shown in the Fig.9. These values are as expected independent o nanoparticle volue raction. Such a cooling-induced growth o the interacial thickness is consistent with the direct observation o an intensity drop o the ain loss peak inset in Fig.9; e.g., " ~ 3.6 at 253 K versus " ~ 7 at 203 K, corresponding to a ~17% and ~28% decrease in " relative to pure glycerol. Since N ~ ~ " d ln V, where is proportional to the nuber density, N/V, o olecules belonging to a speciic relaxation process and is the loss spectra 27

28 intensity, 42 the decrease o the intensity can be taken as indirect evidence o an increase o the glycerol population in the slower interacial layer which onotonically grows upon cooling. We ephasize that the estiated interacial layer thickness is consistent with a glycerol thin il study 60 which ound l ~ n at T = 233 K. A siilar trend upon cooling was observed or deeply supercooled salol 66 under surace conineent and 2-ethyltetrahydrouran 67 under nanopore conineent. The change o the layer thickness is also consistent with the logarithic growth o the nuber o dynaically correlated glycerol olecules in the sae supercooled teperature region as estiated using nonlinear dielectric easureents. 68 However, such an increase o the interacial layer thickness is not captured by our zeroth order theory. As indicated in Fig.7b, both siulation and theory ind a layer thickness o ~ n at high T = 293 K, consistent with experient. Upon cooling, the theory does not predict that the growth o the layer o retarded relaxation based on the real space obility gradient. In contrast, experients deduce a growing layer thickness with cooling, which is partially based on the increase o the long tie tail aplitude in the relaxation tie distribution in Fig.6a. The iplications o this apparent disagreeent is unclear given the theory does correctly predict the relaxation tie distribution aplitude grows with cooling insets o Fig.6a and 7b. One possibility is our inialist assuption sipliication 3 in section IIIB that the elastic contribution to the collective barrier is ully cut o at the solid surace is not quantitatively accurate at low teperatures. I distortion o the solid substrate is required or the atrix relaxation, we expect the elastic barrier will grow relative to 28

29 its bulk value with cooling resulting in an enhanced range o the spatial obility gradient. The theory also ignores surace corrugation, 66 possible structural ordering, 69 and perhaps other aspects o cooperative otion, 32 actors which ight becoe relevant at low teperature % % 23.6% l n " e o C Pure glycerol -20 o C " = % coposite " = % 17 % Hz Figure 9: Average interacial thickness as a unction o teperature based on the ILM or all three saples: 8.8% loading squares, 15.6% loading triangles and 23.6% loading diaonds. The inset shows a direct coparison o e o pure glycerol and nanocoposites with 23.6% loading at two teperatures. F. Predicting the interacial layer properties /T K -1 The present zeroth order theory can be utilized to suggest guidelines or controlling the degree o slowing down and teperature dependence o interacial relaxation, including the conditions required to realize kinetic glassy layers. Here, we peror odel calculations o the obility gradient, interacial layer ean relaxation tie and dynaic shear odulus, as a unction o the cheically-speciic degree o local densiication λ, and reduced teperature T/T g,bulk, or equivalently the pure atrix alpha relaxation tie. We eploy glycerol as a representative surrogate o olecular atrix aterials. The layer dynaic elastic shear odulus at ixed external requency is coputed using the relations given in Appendix B. 29

30 a <τ layer >/<τ bulk > T = 222 K τ α z;λ/τ α,bulk Z n 207 K 222 K 252 K λ λ Figure 10: a Average relaxation tie in the interacial layer noralized by the bulk liquid relaxation tie as a unction o the densiication paraeter λ or glycerol at 207 K blue, 222 K red, and 252 K yellow. For reerence, the theory predicts that the bulk T g is 202 K, and bulk relaxation tie is approxiately 5.8 s at 207 K, s at 222 K, and s at 252 K. The points arked on the lower teperature curves indicate the value λ = λ g where the interacial layer ean relaxation tie becoes 100 s representing one easure o the transition to a glassy layer. The inset shows the relaxation tie proile near the surace in units o its bulk analog at 222 K or λ = 1.04 blue circles, 1.06 red squares, 1.08 yellow diaonds, and 1.1 green triangles. The dashed line represents 100 s and serves to show that the existence o a glassy layer above a critical λ g and its growth in thickness with increasing λ. b Average dynaic shear odulus in the interacial layer o glycerol as a unction o the densiication paraeter λ at requencies = 0.01 blue, 0.1 red, and 1 yellow Hz. The solid dashed curves are at 207 K 222 K. The theoretical dynaic oduli or bulk glycerol are: i at 0.01 Hz, GPa at 207 K and 19 Pa at 222 K, ii at 0.1 Hz, 4.1 GPa at 207 K and 1.9 kpa at 222 K, iii at 1 Hz, 15.6 GPa at 207 K and 0.19 MPa at 222 K. The inset shows the b average dynaic shear odulus in the interacial layer with λ = 1.04 at = 0.01 blue circles, 1 red squares Hz. The dashed lines o corresponding color show the odulus in bulk glycerol at the sae requency. <G'> layer GPa K 1 Hz 207 K 0.1 Hz K 0.01 Hz <G'> layer GPa 0.01Hz 1 Hz 207K 222 K 222 K Hz 1 Hz 0.1 Hz T K The ain rae o Fig.10a shows the theoretical ean layer relaxation tie relative to its bulk analog as a unction o the densiication paraeter at three teperatures. The latter are chosen to roughly correspond to the onset o the deeply supercooled regie τ α = 520ns, iddle o the deeply supercooled regie τ = 2.7s, and close to the bulk T g τ = 5.8s α α. One sees a strong, supra-exponential growth o the interacial layer relaxation tie with increasing surace densiication paraeter, which becoes ore draatic at lower teperatures. 30

31 At 207 K and 222 K, glassy layers eerge above the bulk T g beyond a critical value o surace densiication. The inset shows exaples o the ull obility gradients at a ixed teperature o 222K bulk τ α = 2.7s or our values o λ. As local densiication grows ro 4% to 10%, the relaxation tie near the surace sharply increases by 10 orders o agnitude. The thickness o the glassy layer when it exists grows with increasing surace densiication, and is ~ 1 n at λ = 1.1. The overall interacial layer thickness, deined as the distance ro the surace required to recover the bulk relaxation tie, also grows onotonically with λ ro ~ 1.5 n to ~ 3 n ~2.5-5 glycerol diaeters. Figure 10b shows representative interacial layer dynaic shear odulus calculations as a unction o λ at two teperatures 207 K, 222 K and three low requencies o 0.01, 0.1 and 1 Hz. The corresponding shear oduli o bulk glycerol are stated in the caption. At 0.01 Hz, with increasing surace densiication the interacial layer shear odulus grows by over 2 decades at 207 K, and by nearly 9 orders o agnitude at 222 K. At both teperatures, and all three requencies studied, a glass-like odulus o 5-10 GPa is attained at suiciently high local surace densiication. At ixed teperature and λ, layer shear oduli grow with increasing requency, as expected. The inset shows an exaple at two requencies o how increasing interacial layer shear rigidity eerges upon cooling at ixed surace densiication λ = Massive increases are ound in all cases, ollowing a roughly identical teperature-dependent or. The calculations suggest that the average echanical stiness o the layer can be tuned ro liquid-like to glass-like over a 31

32 narrow range o teperature. V. Conclusions We have cobined broadband dielectric spectroscopy, atoistic siulation, and a novel activated relaxation theory to unravel the nature o slow dynaics in the interacial layer o a odel glycerol-silica nanocoposite. Due to its relative siplicity, the glycerol-silica syste serves as a odel syste to study the coplex interacial eects in nanocoposites, and the new data analysis ethods presented here can be applied to probe dynaics in ore coplex polyer nanocoposites. Our analysis reveals that the relaxation near the particle surace slows down relative to bulk behavior, and increasingly so upon cooling. We ascribe the observed slowing down o the interacial dynaics ainly to the densiication o the liquid near the attractive surace and its inluence on the cage scale local barrier to activated hopping. The thickness o the interacial layer increases upon approaching the bulk T g. However, no indications o glassy layers are ound ro analysis o dielectric relaxation strength or ro the step eature in the DSC easureents. In addition, we have orulated a zeroth order statistical echanical theory o activated, spatially-resolved interacial dynaics near a solid surace under lightly and deeply supercooled conditions. Once the near surace densiication paraeter, required as input to the theory, is deduced ro equilibriu atoistic siulations at high teperature, the theory provides quantitative predictions or dynaics and echanical properties o the interacial region. Estiates o the densiication 32