Maxon and roton measurements in nanoconfined 4 He
MMaxon and roton measurements in nanoconfined He M. S. Bryan ∗ and P. E. Sokol Department of Physics, Indiana University, Bloomington, IN 47408, USA (Dated: May 14, 2018)We investigate the behavior of the collective excitations of adsorbed He in an ordered hexagonalmesopore, examining the crossover from a thin film to a confined fluid. Here we present the inelasticscattering results as a function of filling at constant temperature. We find a monotonic transitionof the maxon excitation as a function of filling. This has been interpreted as corresponding to anincreasing density of the adsorbed helium, which approaches the bulk value as filling increases. Theroton minimum exhibits a more complicated behavior that does not monotonically approach bulkvalues as filling increases. The full pore scattering resembles the bulk liquid accompanied by a layermode. The maxon and roton scattering, taken together, at intermediate fillings does not correspondto a single bulk liquid dispersion at negative, low, or high pressure.
I. INTRODUCTION
The structure and density of liquids in confinement isof great interest in a wide range of phenomena, rang-ing from the liquid-liquid phase transition in water ,molecular layering in thin films , supercritical fluids inconfinement , and confined glass formers . In bulk liq-uid He, the structure and density are closely related tothe excitation spectrum , the phonon-maxon-roton spec-trum, which has been extensively measured . Liquidhelium, a simple atomic liquid with well-known interac-tions, is a model system to explore the effects of confine-ment on liquid structure using the energies of the excita-tions as a spectroscopic probe of the local fluid densityin a restricted geometry.Previous studies of confined helium in large pore sys-tems (diameter > , xerogel ,and Vycor have observed excitation spectra in agree-ment with the bulk liquid. However, for small pores a re-cent neutron spectroscopy study concluded that liquid inpartially filled pores corresponded to bulk liquid heliumat negative absolute pressures . Albergamo et al. ob-served that the energy of the maxon is shifted downwardrelative to the corresponding bulk value, and as the porespace is filled with further liquid, the energy of the maxonapproaches the bulk value from below. By extrapolatingthe behavior of the bulk liquid to the nanoconfined case,they reported negative pressures down to -5.5 bar canbe realized in pores of diameter 47 ˚A. Negative pressurestates are metastable against cavitation, and have beenmeasured extensively in He . A realization of negativepressure in porous media would constitute a significantamount of negative pressure liquid available for study,with much longer lifetimes, than in typical ultrasonicstudies. Unfortunately, their measurements did not ex-tend to the roton region of the spectrum and so the fillingdependence of the roton was not reported.Recently, Prisk et al. measured the excitation spec-trum as a function of temperature and filling in anothersmall pore system . They identified two regions - a lowdensity phase and a high density phase, correspondingto a thin film and bulk-like excitation spectrum. Withinthese regions the maxon and roton energy was indepen- FIG. 1. Previously reported phase diagram . The superfluidtransition line (blue triangles) is taken from previous torsionaloscillator measurements of Ikegami et al . Inelastic neutronscattering with no well defined excitations are indicated byblack circles. Scattering corresponding to a low density thinfilm (red stars) are shown to the left of the crossover region.Bulk-like scattering, in agreement with the bulk liquid scat-tering under vapor pressure (dark red diamonds), is shown tothe right of the crossover region. Inelastic neutron scatteringmeasurements reported in this paper are shown as white cir-cles, moving horizontally through the shaded gray crossoverregion. dent of filling. They interpreted this in terms of twodifferent phases with a crossover regime between themas shown in Fig. 1. The low and high density phaseswould correspond to pressures of -7.5 and 0 bar, respec-tively, based on the maxon energy. However, this studydid not address the intermediate filling range studied byAlbergamo et al .In this paper we present inelastic neutron scatteringmeasurements of the phonon-roton spectrum of super-fluid helium within an ordered mesoporous silica knownas Folded Sheet Material (FSM-16). We report simulta-neous measurements of the maxon and roton energies as a r X i v : . [ c ond - m a t . o t h e r] M a y a function of pore filling in the crossover region, betweenthe thin film and bulk-like regions. Following Alberg-amo et al. , we have attempted to interpret the pore fill-ing dependence of these excitations by extrapolating thedensity dependence of the bulk fluid to the nanoconfinedcase. However, we find that a self-consistent interpreta-tion of the scattering data cannot be obtained when theextrapolation is simultaneously applied to the maxon andthe roton. We therefore infer that the structure of super-fluid helium, when partially filling pore spaces only a fewnanometers in diameter, is not consistent with viewingthe confined liquid as a bulk fluid under negative pres-sure. II. EXPERIMENTAL APPROACH
FSM-16, a monodisperse porous silica with hexagonalpores arranged on a triangular lattice, provided the con-fining media for He in these measurements. A detaileddescription of the synthesis process of FSM-16 has beenreported elsewhere . This sample was used in a pre-vious study and has been characterized via N and He adsorption isotherms. The nitrogen isotherm wastype IV and yielded a BET surface area of 1015 m /gand a relatively narrow pore-size distribution centeredat 27 ˚A with a full width half-max of approximately 2˚A. The helium isotherm yielded a monolayer coverageof 21.9 mmol/g and a full pore filling at approximately43 mmol/g. The structure of dry FSM-16 was character-ized by x-ray diffraction which was consistent with a 2Dtriangular lattice (space group symmetry p6mm ) with alattice constant of a ≈
45 ˚A. The FSM-16 was heatedto 100 ◦ C for 24 hours and pumped to P < − torrto remove surface contamination, primarily water vapor,prior to the measurements.Inelastic neutron scattering was performed at the ColdNeutron Chopper Spectrometer (CNCS) located at theSpallation Neutron Source . CNCS is a direct geometrytime of flight spectrometer that views a cold modera-tor. An incident energy E i = 3.65 meV was selected toavoid Bragg scattering from the Al sample cell. This inci-dent energy yielded a dynamic window of approximately0 < Q < − momentum transfer and 0 < E < µ eV. Standard data reduction routines wereused to reduce the raw time-of-flight data to S(Q,E) ,which was subsequently analyzed using the DAVE soft-ware package .The FSM-16 sample was contained in a cylindrical Alu-minum sample can with 5.71 cm height and 2 cm diam-eter. A continuous flow helium cryostat with an OxfordHelioxVT He insert was used to cool the sample. Gaswas loaded into the cell via a small capillary using stan-dard volumetric techniques. A small heater was placedon the capillary near the sample cell to eliminate super-fluid in the fill line which would give an undesirable heatload on the cell.
III. RESULTS
Neutron scattering measurements were carried out atT = 350 ± n = 21.9 mmol/g, as de-termined by helium adsorbtion isotherms, was also per-formed. Previous neutron scattering studies have indi-cated that this first adsorbed layer is solid andtorsional oscillator measurements indicate that the on-set of superfluidity occurs only after the completion ofthe first solid layer . The measured monolayer scatter-ing was broad and featureless, consistent with an inertadsorbed solid.The dynamic structure factor S(Q,E) for three fillingsare shown in Fig. 2. The bulk liquid phonon-roton curveis also shown for comparison, as a white dotted line. Themonolayer filling measurement was used as backgroundand subtracted from the data. This includes scatteringfrom the porous material and sample cell, as well as thebroad, featureless scattering that results from monolayer He itself. At all fillings studied the excitation spectrumexhibits the characteristic phonon-maxon-roton behav-ior of the superfluid phase, accompanied by additionalscattering known as the layer mode . The layermode is a relatively broad excitation that spans the ro-ton region in Q , with energies lower than the bulk rotonenergy. The energies of the bulk-like excitations, even faraway from the roton minimum and layer mode, are dif-ferent than the bulk values at low fillings. As the filling isincreased the overall intensity of the scattering increasesand the observed excitation energies move closer to thebulk excitation energies. Finally, at the highest fillingstudied the excitation energies across the entire dynamicwindow are in good agreement with the bulk values.The highest filling has a weak feature at roton energies(approximately 0.75 meV) across all Q , particularly nearthe maxon Q , known as the ghost roton. The ghost rotonis produced from multiple scattering that includes oneroton and one elastic scattering event, from the FSM-16or adsorbed He. This Q -independent feature was fit to acommon energy and width across all Q , then subtractedfrom the data. The ghost roton is approximately 100times less intense than the scattering near the bulk roton,and is not noticeable in that region of intense scattering.In the phonon region of the dispersion, the He scatteringis much weaker and as a result, a direct observation ofthe speed of sound is not possible with this data set.Constant Q cuts through the scattering at the maxon( Q =1.2 ˚A − ) are shown in Fig. 3 for three fillings, withthe ghost roton and background subtracted. Points faraway from the maxon peak are generally centered nearzero, demonstrating a reasonable background subtrac-tion. At the lowest filling studied (38.0 mmol/g) themaxon energy is significantly below the bulk value. Asfilling increases, the center of the peak moves to higherenergies and approaches the bulk liquid value. Themaxon intensity slightly decreases initially from 38.0 to FIG. 2. Dynamic structure factor S(Q, E) for 38.0, 40.0, and 43.0 mmol/g with monolayer background subtracted. All plotsshare a logarithmic intensity scale, with the bulk phonon-roton spectrum shown as a white dotted line.FIG. 3. A cut along energy of the maxon, Q = 1.2 ˚A − .Data for 38.0, 40.0, and 43.0 mmol/g fillings are shown in redcircles, green triangles, and blue squares respectively. Corre-sponding fits are shown as a colored line. The peak centerof the bulk liquid scattering is shown as a vertical dashedline. Background and the ghost-roton have been subtracted,leaving only the maxon peak scattering and corresponding fit. Q =1.95 ˚A − ) are shown in Fig. 4. Again, theghost roton and background have been subtracted andhave yielded a reasonable background subtraction. Atthe lowest filling the scattering is very broad and asym-metric with a peak below the bulk roton energy. Asthe filling increases, a sharp peak near the roton energyemerges while the broad component remains relativelyconstant. The broad component has been interpreted asa 2D roton at the pore wall while the sharp componenthas been interpreted as a more bulk-like liquid in thepore center . At the highest filling studied thebulk-like roton energy approaches that of the bulk whilethe 2D roton exhibits little filling dependence. FIG. 4. A cut along energy of the roton, Q = 1.95 ˚A − . Pointsand fit lines are correspond to identical fillings as the previ-ous figure. The bulk-like roton and layer mode are summedto make the total fit shown. Two Gaussian components, thebulk-like mode and the layer mode, are required to fit theasymmetric peak in the roton region. The bulk-like scatteringis centered near the bulk liquid value, marked with a dashedline. The layer mode is centered at lower energies, is lessintense, and broader than the bulk-like mode. As filling in-creases, the layer mode scattering remains mostly unchanged,while the bulk-like scattering approaches the bulk liquid en-ergy and becomes more intense. Detailed values for the excitation energies were ob-tained from fitting constant Q cuts of the data. It wasconvenient to split the data into two regimes. The firstis the maxon region from 0.8 < Q < − . The sec-ond is the roton region from 1.75 < Q < − . Inaddition to the fits to the excitations, described below,an additional broad, low-intensity Gaussian was used tofit any additional background and multiphonon scatter-ing. This additional background is fixed by scattering faraway from the phonon-maxon-roton scattering, and cen-tered above 1.5 meV. The additional background is typi- FIG. 5. Bulk-like collective excitations as a function of filling.Separate fillings are offset in both energy (0.15 meV each)and momentum transfer (0.05 ˚A − each). The layer modehas been removed for clarity. At each filling, a dashed lineindicates the bulk spectrum. cally an order of magnitude less intense than the maxon.While there has been some effort to understand the de-tailed higher energy (E > , the scattering in confinement appears broadand featureless at energies above the phonon-roton ener-gies.The excitations in the maxon region could be fit witha single Gaussian peak. The single Gaussian form gavean excellent fit to the observed scattering, as can be seenin Fig. 3, at all Q (cid:48) s in the maxon region and at allfillings. The observed peaks were slightly broader thanthe calculated instrumental resolution indicating theremight be some intrinsic lifetime for the maxon.The roton region required a sum of two functions toadequately describe the observed scattering as has beenused previously . A narrow Gaussian near thebulk roton energy, the bulk like excitation, and a broaderGaussian well below the bulk, the layer mode, were used.Previous high resolution measurements of the bulk-likepeak using neutron backscattering found the width to beextremely narrow at low temperatures , much smallerthan our current resolution. Thus, a Gaussian with awidth determined by the instrumental resolution wasused to fit this component. The layer mode roton scatter-ing is broad, symmetric, and less intense than the bulk-like mode. This could also be modeled by a Gaussian,but with a much broader width and lower energy. Allfits overlap with the data within error bars. Typical fitsare shown in Fig. 4 where it can be seen that they provideand excellent description of the experimental scattering.However, at low fillings the two peaks are less distinct andthe parameters of the two Gaussians are more highly cor-related. As filling increases, the low energy componentincreases slightly in intensity, but not in center or width. FIG. 6. Fitting parameters for the maxon region dispersionas a function of filling. The dashed line indicates the bulkvalue of the maxon energy. Previous measurements of 37.7mmol/g are shown as open circles. The high energy component increases rapidly in inten-sity and the center changes with filling. At 43.0 mmol/g,the highest filling measured two identifiable, but overlap-ping peaks appear with the bulk-like component centerof 0.743 ± .The resulting peak centers of the maxon region and theroton region’s bulk-like Gaussian are plotted in Fig. 5 asa function of filling. At the lowest filling (38.0 mmol/g),the entire excitation spectrum is similar to the bulk liq-uid spectrum, plotted as a black dashed line, but lower inenergy across all Q . This is in contrast to previous mea-surements at lower fillings , which measured a reducedmaxon energy but an enhanced roton energy. As fillingincreases, the maxon region of the dispersion monoton-ically approaches the bulk curve. The roton region atthe highest fillings studied is identical to the bulk liquiddispersion. However, as filling increases, the roton dis-persion does not monotonically approach the bulk dis-persion.An inverted parabola was used to fit to the measuredexcitation energies as a function of Q in the maxon re-gion to obtain the maxon energy and Q max . These areplotted as a function of filling in Fig. 6. Previous mea-surements of 37.7 and 43.0 mmol/g are also plotted forboth maxon Q max and energy. The resulting peak ener-gies as a function of filling monotonically approach thebulk value (indicated by a dashed line in Fig. 6) and thepreviously measured 43.0 mmol/g value.The bulk-like roton dispersion can be characterized bythree parameters, ∆, Q R , and µ which fit the roton dis-persion as E = ∆ + ¯ h ( Q − Q R ) / µ , known as theLandau dispersion. These parameters are plotted as afunction of filling in Fig. 7. The effective mass µ de-creases monotonically as a function of filling. Previously FIG. 7. Landau dispersion parameters for the roton as afunction of filling. Dotted line indicates bulk value. Previousmeasurements of 37.7 and 43.0 mmol/g are also shown inpink and blue diamonds, respectively. Q R initially increasesuntil 39.0 mmol/g, then decreases to the bulk value and doesnot change significantly from 40.0 mmol/g upwards to 43.0mmol/g. ∆ initially decreases until 39.0 mmol/g, then in-creases towards the bulk value from below. measured values of µ at 37.7 and 43.0 mmol/g indicatea general trend of a monotonically decreasing effectivemass, with the current µ at 43.0 mmol/g in agreementwith the previously measured value. A monotonic behav-ior of µ including current and previous results is observed.However, this is not the case with ∆ and Q R . At the low-est filling studied, ∆ is significantly below the bulk value.As filling is increased, ∆ decreases and then increases to-wards the bulk value from below. The value of Q R varieswithin ± − of the bulk value, with previous thinfilm measurements indicating a Q R of 1.782 ˚A − whichis more than 0.14 ˚A − lower than the bulk value. Thecurrent measurements of µ , ∆, and Q R are in agreementwith the previously measured values at 43.0 mmol/g.Significant scattering below the bulk-like excitationspectrum, the layer mode, is present in this data at allfillings measured. As evident in Fig. 4, the layer mode ismuch broader than the bulk-like mode, and the intensitydoes not change significantly as a function of filling at Q R . As with other studies of helium confined in porousmedia, the layer mode dispersion only exists across a lim-ited region in Q , near the bulk roton minimum .The layer mode does evolve with filling, with a small increase in intensity in the region with energies belowthe bulk dispersion, at Q > Q R . This change occursat low fillings, and 40.0 mmol/g and above there is nochange in the layer mode scattering. The low filling dis-persion is nearly a horizontal line, and evolves into amore parabolic shape at high filling, as well as increasingin energy from 0.49 ± ± . Based on those predictions, weestimate a layer mode density of 0.063-0.072 ˚A − . IV. DISCUSSION
The density dependence of the excitation spectrum iswell known in bulk liquid He. Several experimentalstudies have measured the bulk liquid from saturated va-por pressure up to pressures above 20 bar . Asdensity increases, the maxon energy increases and theroton energy decreases. Theoretical studies have also ex-amined the bulk liquid below saturated vapor pressuredensity using Density Functional Theory (DFT) andpath integral Monte-Carlo . These studies have exam-ined the behavior of the maxon and roton from bulk liq-uid density (0.0218 ˚A − ) down to nearly the spinodalpoint (0.0161 ˚A − ) where the liquid is no longer stableagainst separation into a two phase fluid. The theoret-ical results for densities below the SVP bulk liquid area smooth continuation of the trends observed at higherdensities. That is, the maxon energy decreases and theroton energy increases with decreasing density. Theseprevious results are shown in Fig. 8.The smooth evolution of the roton and maxon energieswith density can be captured using a simple analyticalfunction. We have used the form E = E + B ( ρ + ρ ) ,for excitation energy E and density ρ , with constants E ,B, and ρ , to fit the behavior of both the maxon and ro-ton energies over the entire density range covered in Fig.8. The DFT results were used for densities below SVPsince they agree well with the bulk values and exist atseveral densities below SVP bulk density. Equal weightwas given to all measurements since the uncertainties ofthe theoretical points were not known. The fits do anexcellent job of capturing the measured behavior at bulkdensities and above as well as the theoretical values atdensities below the SVP liquid. The parameters E , B,and ρ are listed in Table 1.It has been proposed that the maxon energy could beused to directly measure the density of the core liquid.Lauter et al. studied helium films on grafoil and ob-served maxon energies below the bulk values. They in-terpreted these as due to a low density liquid resultingfrom the mismatch with the confining potential of the FIG. 8. Maxon and roton energy as a function of density.Both theoretical and experimental work is included, with theprevious MCM-41 measurements of Albergamo et. al (red di-amonds), and the cubic fit to the maxon energy (black dottedline) as a function of density also from that work. Bulk liq-uid maxon energies from previous measurements include neu-tron scattering data from Andersen (white square) , Gibbs(green circle) , and Graf (black diamonds) . DFT resultsof the maxon (white hexagons) and roton (black hexagons) with Path Integral Monte Carlo maxon energies (white tri-angles) and roton energies (black triangles) are theoreticalcalculations. Bulk liquid roton measurements are includedfrom Gibbs (yellow squares) , Dietrich (orange diamonds) ,and Stirling (purple squares) . Our fit, as discussed in thetext, is shown as a blue dashed line and green dashed line forthe maxon and roton, respectively.TABLE I. Parameters of the energy-density relation used inthis work for the maxon and roton excitations, with energyin units of meV and density in units of number of atoms per˚A . The function used is of the form E = E + B ( ρ + ρ ) . E B ρ Roton 0.924 -16800 0Maxon 1.35 59300 -0.0358 substrate. More recently Albergamo et al. have mea-sured the maxon energy in a small porous system (4.7nm diameter), MCM-41. They also observed a reducedmaxon energy which they attributed to the core liquidbeing at a negative pressure, stretched apart by the sub-strate. A functional form for the density dependence ofthe maxon in MCM-41 was proposed by Albergamo etal. , which is also shown in Fig. 8 for comparison. How-ever, it should be noted that in neither of these studieswas the density of the liquid measured independently.We have used the maxon-density relation from Alberg-amo et al. to determine the density of the confined liquidin our studies. These should be qualitatively comparablesince the MCM-41 used by Albergamo and the FSM-16 used here both have silica-based hexagonal mesopores FIG. 9. Roton energy plotted as a function of density. Here wehave used the observed energy of the maxon and the maxonenergy-density curve of Albergamo et al. to determine thedensity for our recent measurements. The black hexagons arefrom the DFT results of Maris and Edwards . Measurementsof He in confinement include previous FSM-16 measurements(open red circles) , and current measurements (red squares).Bulk liquid roton measurements are included from Gibbs (yel-low squares) , Dietrich (orange diamonds) , and Stirling(purple squares) . The green dashed line is a fit includingprevious experimental and theoretical work, discussed in thetext. The measurements presented here do not scale withdensity as expected and deviate significantly from the bulkliquid trend. with comparable pore sizes. While the maxon energy-density relation of Albergamo, shown in Fig. 8, differsslightly from the one we have determined at the lowestdensities, they are nearly identical in the region coveredby these studies.Fig. 9 shows our measurements of the roton energyas a function of density as determined by the maxonenergy. Also shown are the theoretical DFT results aswell as several measurements in the bulk liquid. As canbe seen, our measured roton energies in confinement donot follow the smooth evolution with density evident inboth the previous bulk measurements and the theoreticalcalculations. Our measurements observe a roton energythat initially decreases with decreasing density, oppositeto the expected behavior. As the density, determined bythe maxon energy, is further lowered the roton energy be-gins to increase and approach the theoretically predictedvalues. We note that the density here is predicated onthe assumption that the confined liquid simply behavesas a bulk liquid at densities below the bulk SVP density,i.e. a liquid under negative pressure. The large discrep-ancy between the measured and expected roton energiescalls into question this assumption.This assumption that the confined liquid behaves likethe bulk liquid at negative pressure can be examined di-rectly, by using the maxon and roton energies indepen- FIG. 10. Inferred He density as a function of filling, forthe maxon shown in red and the roton shown in black. Theenergy-density relationship is shown in the previous figure(Fig. 8) and was used to generate inferred density for eachmaxon and roton energy measured. Approximate first liq-uid layer completion ( n f ) is marked with a vertical dashedline. Previous measurements of this sample are included fromPrisk , as open black and open red points. dently to determine an inferred density for the confinedliquid. If the confined liquid behaves as a bulk liquidat negative pressure then the densities inferred from themaxon and roton should agree and follow the same be-havior as the pores are filled. The inferred density fromboth the maxon and roton, using our fits to the bulk val-ues and DFT calculations, are shown in Fig. 10 as a func-tion of pore filling. As can be seen, at the highest fillingsmeasured the inferred density from the maxon and rotonare similar and approach the bulk density. However, atlower fillings these inferred densities differ dramatically.The inferred density from the maxon decreases with de-creasing pore filling, as has been observed in previousstudies and consistent with the proposal that the con-fined liquid is a bulk-like liquid at negative pressure. Theroton, however, exhibits a dramatically different behav-ior. The inferred density from the roton increases withdecreasing pore filling in direct contradiction with boththe expected behavior and the density inferred from themaxon energy. At lower pore fillings the inferred den-sity decreases and then remains constant for single layerfilms. However, the densities inferred from the maxonand roton never come into agreement at low fillings.The previous studies of Prisk et al. reported that themaxon and roton energies were consistent with valuesof bulk liquid value for high pore fillings. They alsofound that for fillings within the first liquid layer (21 . Intermediate fillings of superfluid helium confined inFSM-16 have been measured with inelastic neutron scat-tering, as the thin film transitions into a confined fluid.The scattering of the maxon region indicates a monotonictransition in density from the lower density of the thinfilm phase to a bulk-like density and spectrum at higherfillings. However, Landau equation parameters and theroton scattering exhibit a more complicated behavior asa function of filling, and do not monotonically increase.The intermediate filling behavior observed in the maxonand roton region does not correspond to a bulk-like dis-persion at negative, low, or high pressure. Only at thehighest filling measured does the spectrum correspond toa bulk-like dispersion at saturated vapor pressure whenfull pore filling has been reached. VI. ACKNOWLEDGEMENTS This research used resources at the Spallation Neu-tron Source, a DOE Office of Science User Facility op- erated by the Oak Ridge National Laboratory. 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