Downloaded by guest on September 27, 2021 www.pnas.org/cgi/doi/10.1073/pnas.2010787117 system the of matrix length confinement the varying upon changes liquid to on based us elasticity liquid wavevector allows to framework contributions various resulting the combined elasticity decompose The liquids, of materials. theory of amorphous microscopic theory the of phonon in developments the recent on with ideas Frenkel’s by 7). liquid. order the (6, the of coworkers scale on length and weaker, confinement Noirez is of liquids work and of of pioneering 4) elasticity the (3, low-frequency with al. The long et starting the Derjaguin of research), view of liquid (in recently of fairly history shear identified to been low-frequency related has Conversely, is elasticity time. which charac- Maxwell time, a Frenkel viscoelastic at the the scale, vanishing time their internal it and teristic links phonons which acoustic theory, Frenkel’s transverse by to described (8). well gigapascals is of order behavior the The of moduli cor- elastic range shear megahertz to responding the in techniques ultrasonic with flow devel- measured submillimeter full and the micro-, for nano-, limitation technologies. small-scale a not of is is phenomenon opment This This very understood. confinement. at stress in currently shear albeit support frequency, to liquids low of remark- later property a However, found solid-like and (2). able (3–7) view view accepted this challenged diffu- an have behave become the experiments liquids has frequency) and This absent (high solids. is times motion as liquid short the at of that component sive is high. is here sufficiently liquid is idea a oscillation The mechanical of pro- of solid, response frequency amorphous the the an vided that of that shown from has indistinguishable basically (1) Frenkel by work T liquids across liquids of elasticity scales. the length of and description time different basis general the more provides length and a confinement results, for the experimental suc- of with agreement function theory in a L, as The liquids rigidity. of modulus an shear shear law to scaling liquid leading the the predicts thus cessfully of response, mechanical increase soft effective responsi- nonaffine are can that for liquid modes emerg- ble a low-frequency The to the state. confinement suppress liquid applying effectively that the nonaffine shows of the framework theory combine ing matter Frenkel condensed we the disordered with Here for systems valid since. dynamics ever lattice models of state theoretical theory elas- to liquid the challenge of a the 0.01 understanding posed of of have basic order and our liquids the question of on into ticity call scales Hz, frequency con- 0.1 low in 2020) to very 27, rigidity May on review shear liquids, for (received unexpected fined 2020 of 15, July observations approved and Experimental NJ, Princeton, University, Princeton Debenedetti, Kingdom G. United Pablo 4NS, by E1 Edited London London, of University Mary Queen Astronomy, and Kingdom; Physics United 0AS, CB3 Cambridge Cambridge, a Zaccone Alessio liquids of confined elasticity shear low-frequency the Explaining eateto hsc A oteoi”Uiest fMln 03 ia,Italy; Milan, 20133 Milan, of University Pontremoli,” “A. Physics of Department olwn rvosltrtr 9,w nrdc h Hessian the introduce we (9), literature previous Following inspired elasticity liquid of description a provide we Here typically is liquids of response mechanical High-frequency 1 eeatct flqisi eludrto ntehg fre- high pioneering the where in response, understood mechanical well the of is limit liquids quency of elasticity he − | 10 ofie liquids confined 3 a n ssrnl eedn ntesubmillimeter the on dependent strongly is and Pa, k n hst dniyhwtesermdlso a of modulus shear the how identify to thus and , a,b,c,1 | H rheology ij n otaTrachenko Kostya and = −∂ | 2 mrhu materials amorphous U G /∂ 0 ∼ ˚ q L i −3 ∂ ˚ q j o h low-frequency the for c n h fn force affine the and aeds aoaoy nvriyo abig,CmrdeC30E ntdKndm and Kingdom; United 0HE, CB3 Cambridge Cambridge, of University Laboratory, Cavendish d L. ˚ dntdwt h igntto) whereas notation), ring the with (denoted ha) o iud h Hessian the liquid, Cartesian a (i.e., to deformation For refer macroscopic shear). indices the hence Greek of deformation, field. components macroscopic force ”affine” to name subject the frame atom initial on the acting in force the sents q h silto rqec fteetra tanfil,whereas field, Also, strain limit. external the high-frequency of the frequency oscillation in the survives what is, where state ence 10) (9, written be can coordinates, mass-rescaled modes equi- normal nonfully (9). instantaneous frequencies) over (imaginary include averaging to from configurations obtained librated state reference a field 1 under distributed BY).y is (CC article access open This interest.y competing no declare paper.y authors the The wrote K.T. and and A.Z. A.Z. research; and performed data; A.Z. analyzed research; K.T. designed K.T. and A.Z. contributions: Author complex the for expression 10): following (9, elas- constants the elastic of obtains definition from one the stress, applying decomposition tic and (10), eigenmode eigenfrequency to and time transformation Fourier Fur- (9). work previous in thermore, shown as Hamiltonian particle-bath atom each position. is force brings displacement the displacement This relax nonaffine tion. to through- additional order atoms in an all required deformation, on equilibrium the mechanical out tensor keep to strain order external in the force by phys- net (noncentrosym- scribed atom A side disordered motions: on right-hand nonaffine the acts the to of leading on environment effect term metric) the The represents potential. ically anharmonicity pair long-range by the from mediated of atoms arises between which coupling dynamical coefficient friction microscopic where b owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To i eateto hmclEgneigadBoehooy nvriyof University Biotechnology, and Engineering Chemical of Department ssonpeiul,teeuto fmto fatom of motion of equation the previously, shown As h bv qaino oincnb eie rmamodel a from derived be can motion of equation above The stecodnt fatom of coordinate the is Ξ C i η ,κχ αβκχ C Born steGenSitVnn tantno and tensor strain Venant Green–Saint the is αβκχ {x ˚ q = i olwn tnadmnpltos hc involve which manipulations, standard Following . i ∂ d eoe h fn ato h lsi osat that constant, elastic the of part affine the denotes (t dt f i (ω 2 =˚ ) i x 2 /∂η nteafiepsto ie,tepsto pre- position the (i.e., position affine the in i = ) + q κχ i C ν (t αβκχ where , dx Born ) dt − i ˚ dt q − i sa xaso rudarefer- a around expansion an as }, i + V 1 nteiiiludfre frame undeformed initial the in η H i raieCmosAtiuinLcne4.0 License Attribution Commons Creative κχ X nteafiepsto,ta is, that position, affine the in ij H n x stesri esr Here, tensor. strain the is f ij j i ω Ξ = η cigi h fn posi- affine the in acting p 2 stpclyeautdin evaluated typically is κχ NSLts Articles Latest PNAS Ξ ˆ ,n i n .A consequence, a As ). i − ,κχ ,αβ oanw(nonaffine) new a to f ω i η Ξ ˆ = 2 κχ n + ∂ ,κχ , U i ων κχ /∂ ω d , q colof School = i denotes | xy ν repre- f3 of 1 i sa is in , [2] [1] for ω p

APPLIED PHYSICAL BRIEF REPORT SCIENCES denotes the internal eigenfrequency of the liquid [which results, Differently from the gapless longitudinal dispersion relations e.g., from diagonalization of the Hessian matrix (9)]. We use the and generally from phonon dispersion relations in solids, liquids notation ωp to differentiate the eigenfrequency from the external have the gap in k-space in the transverse phonon sector. This oscillation frequency ω. follows from the dispersion relation (15), Born In liquids, a microscopic expression for G∞ ≡ C is pro- xyxy r vided by the well-known Zwanzig–Mountain (ZM) formula (11), 1 ω (k) = c2k 2 − , [6] in terms of the pair potential V (r) and the radial distribution p,T 4τ 2 function g(r). The sum over n in Eq. 2 runs over all 3N degrees of freedom (for a monoatomic liquid with central-force pair where τ is the liquid relaxation time and c is the transverse speed interaction). Also, we recognize the typical form of a Green’s of sound. function, with an imaginary part given by damping and poles ωp,n Eq. 6 follows from the Maxwell–Frenkel approach to liquids which correspond to the eigenfrequencies of the excitations. where the starting point of liquid description includes both elas- As usual when dealing with eigenmodes, the sum over n (label- tic and viscous response (15) and implies that transverse modes 1 ing the eigenmode number) can be replaced with a sum over in liquids propagate above the threshold value kg = 2cτ , thus set- 2 2 2 2 wavevector k, with k = kx + ky + kz , and kx = πnx /L. We then ting the gap in momentum space, as ascertained on the basis of recall that the numerator of the Green’s function, which is given molecular dynamics simulations in liquids (16). At the atomistic by the eigenfrequency spectrum of the affine force field, can be level, the Frenkel theory attributes τ to the average time between ˆ ˆ 2 molecular rearrangements in the liquid (1). In the limit of large expressed as Γ(ωp ) = hΞn,xy Ξn,xy in∈{ωp ,ωp +δωp } ≈ Aωp , where 2 τ or viscosity, Eq. 6 becomes gapless and solid-like. A ≈ (1/15)κR0 with κ the spring constant for the intermolecu- In a large system, kg sets the infrared cutoff in a sum or integral lar bond and R0 the bonding distance, as proved analytically in ref. 12. This parabolic law holds up to high eigenfrequencies as over k-points. In a confined system with a characteristic size L, shown in simulations (9). the lower integration limit becomes We thus rewrite Eq. 2 in terms of a sum over k as follows:  1  kmin = max kg , . [7] 2 L ∗ A X ωp,k G (ω) = G − , [3] ∞ V ω2 − ω2 + iων k p,k Then,

where A is a numerical prefactor. Z kD ω2 (k) G∗(ω)=G − B p,L k 2dk [8] In isotropic media, eigenmodes can be divided into longitudi- ∞ 2 2 1 ωp,L(k) − ω + iων nal (L) and transverse (T) modes. Therefore we can split the sum L in Eq. 3 into a sum over L modes and a sum over T modes, Z kD ω2 (k) − B p,T k 2dk, ω2 (k) − ω2 + iων 2 kmin p,T ∗ X ωp,kλ G (ω) = G − A , [4] ∞ ω2 − ω2 + iων where the lower integration limit for the longitudinal modes kλ p,kλ in the second term is given by the system size L. The lower where λ = L, T . Furthermore, we introduce continuous vari- integration limit for the transverse modes in the third term is k 7 G∗ ables for the eigenfrequencies ωp (k), by invoking appropriate given by min in Eq. . We take the real part of which G0 dispersion relations ωp,L(k) and ωp,T (k) for L and T modes, gives the storage modulus and focus on low external oscil- respectively (as discussed below). Hence, the discrete sum over lation frequencies ω  ωp used experimentally. In both integrals eigenstates can be replaced, as is standard in solid-state physics, numerator and denominator cancel out, leaving the same expres- P V R 3 sion in both integrals. Therefore, as anticipated above, the final with a continuous integral in k-space, k ... → (2π)3 ... d k: low-frequency result does not depend on the form of ωp,L(k), nor of ωp,L(k), although the latter, due to the k-gap, plays an Z kD ω2 (k) ∗ p,L 2 important role (see Eq. 7) in controlling the infrared cutoff of G (ω)=G∞ − B 2 2 k dk [5] 0 ωp,L(k) − ω + iων the transverse integral. In the experiments where the size effect Z kD 2 ωp,T (k) 2 − B 2 2 k dk, 0 ωp,T (k) − ω + iων 100000 the upper limit of the integral is set by the Debye cutoff wavevec- tor kD , which, in any condensed matter system (be it solid or 10000 liquid), sets the highest frequency of atomic vibration. One should note that while k is in general not a good quan- 1000

tum number in amorphous materials (as the connection between ]

energy and wavevector is no longer single-valued as it is in crys- a P [ 100

tals where Bloch’s theorem holds), it still can be used to provide ’

successful descriptions of the properties of amorphous materials G

and liquids (13). 10 We now discuss the dispersion relations for longitudinal and transverse excitations in liquids. For the longitudinal modes, one can resort to the Hubbard–Beeby theory of collective modes in 1 0.01 0.1 1 liquids (14), which has been shown to provide a good description L [mm] of experimental data, and use equation 43 in ref. 14. As will be 0 shown below, the final result for the low-frequency G does not Fig. 1. Low-frequency (≈ 0.01 Hz) storage modulus G0 as a function of depend on the form of ωp,L. However, for the mathematical com- confinement length L. Experimental data refer to short-chain liquid crys- pleteness of the theory it is important to specify which analytical talline (LC) polymer liquids PAOCH3 (in the isotropic state) well above Tg (6), forms for the dispersion relations can be used. whereas the solid line is the prediction from Eq. 10.

2 of 3 | www.pnas.org/cgi/doi/10.1073/pnas.2010787117 Zaccone and Trachenko Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 1 .Zazg .D onan ihfeunyeatcmdl fsml fluids. simple of moduli elastic High-frequency Mountain, D. R. Zwanzig, R. 11. Lema A. 10. c,i a ensoni e.1 htteafieterm affine the that (here, 19 term ref. nonaffine in shown negative been the mechan- been has statistical have it equilibrium and versions ics, (18)] two equivalent be [the to of formalism shown version response stress-fluctuation nonaffine last the the the using is size equilibrium, system modynamic the on depends while which term, term only the Here Eq. G where obtain we and survives, equation above the the in only term confinement, third submillimiter under liquids for Therefore, that such exactly, out seen, is confinement of lofrplmrmlsi e.9.I i.1w opr the temperature transition compare glass the we above well state), 1 ( LC-polymer Fig. confined of In data 9). modulus ref. length storage in the melts for trend polymer for Eq. also that (note state isotropic the in length finement low-frequency contains that diameters. integral molecular few an after as zero given is it mula acn n Trachenko and Zaccone .M .Ln,G un .L ono,Ioofiuainleatccntnsadliq- and constants elastic Palyulin V. V. Isoconfigurational 9. Johnson, L. liquid W. in Duan, behaviour G. elastic Lind, low-frequency L. a M. of non-entangled 8. Identification of Baroni, response P. rheological Noirez, Solid-like L. Noirez, 7. L. liquids. Baroni, wetting P. nanoconfined Mendil, of H. dynamics viscoelastic 6. Nonlinear of Riedo, elasticity E. Shear Li, Tsidypov, D. D. T. B. Lamazhapova, 5. D. K. of Bazaron, modulus B. shear U. complex Derjaguin, V. The B. Budaev, O. 4. Zandanova, K. Bazaron, U. Derjaguin, B. liquids. glass-forming of 3. models elastic and transition glass The Colloquium: Dyre, C. J. 2. Frenkel, J. 1. ∞ enwcmaeEq. compare now We Phys. temperature. zero at solids disordered dynamics. lattice non-affine alloy. from polymers forming glass metallic bulk a (2006). of fragility uid water. state. molten the in polymers Lett. Rev. Phys. frequencies. low at liquids low-viscosity liquids. small-molecule and polymeric Phys. Mod. Rev. edn to leading 7, osntdpn on depend not does 4447 (1965). 4464–4471 43, β L .Py.Cnes Matter Condens. Phys. J. G 0 te .Mlny u ue o h us-ttcadvsoeatcrsos of response visco-elastic and quasi-static the for rules Sum Maloney, C. ˆ ıtre, rdce yEq. by predicted = 0 iei hoyo Liquids of Theory Kinetic = 3 β/ α aaee-repeitoso h iceatcrsos fglassy of response viscoelastic the of predictions Parameter-free al., et G 012(2008). 106102 100, and 5–7 (2006). 953–972 78, G ∞ sanmrclpeatr tsol entdthat noted be should It prefactor. numerical a is L 0 − β fcnndlqisa ucino h con- the of function a as liquids confined of sn h aao h Csotcanpolymer short-chain LC the of data the using α r ueia rfcos nalqi nther- in liquid a In prefactors. numerical are Z G 1/L u.Py.J E J. Phys. Eur. k k 0 L D (ω g 711(2012). 372101 24, G 10 ihwl-otoldexperimental well-controlled with 10,  eas n o xml,teZ for- ZM the example, for in, because k 0 → 2 Ofr nvriyPes xod 1946). Oxford, Press, University (Oxford ≈ oaalbeeprmna aaof data experimental available to dk PAOCH L Polymer 1 .Sa.Phys. Stat. J. )=0 = 0) β G hs e.A Rev. Phys. 1) and (17), = 0 0 L otMatter Soft 78 (2006). 77–85 19, safnto fconfinement of function a as G −3 2 713(1989). 97–103 30, a enscesul tested successfully been has ∞ − , for 3 − α 3 iud i h isotropic the (in liquids ) 1–5 (2006). 415–453 123, 2525 (1990). 2255–2258 42, k L α 3 D 3 k 4588 (2018). 8475–8482 14, hs e.Lett. Rev. Phys. min aclec other each cancel ) k ∞ → dV D 3 = + (r L T 1 β 3 bl liquids). (bulk )/dr g L codn to according ae from taken , −3 hc is which , . G 015501 97, ∞ .Chem. J. [10] and [9] 1 .Cli,P atnt,Dnmcmcocpchtrgniisi eil linear flexible a in heterogeneities macroscopic Dynamic Martinoty, P. Collin, D. 21. solid-like a Highlighting Gebel, G. Chancelier, L. Noirez, L. Baroni, P. Mendil-Jakani, H. 20. relaxation of dependence weight Molecular Winter, H. H. Rosa, De E. M. Jackson, K. J. 17. nonaffine the of description analytical Approximate Scossa-Romano, E. Zaccone, A. 12. 8 .Mzn,L .Slet .Sel .Msa .L art uofnnierte nthe in Poli nonlinearities P. Cutoff Xu, Barrat, H. L. J. Wittmer, P. Mossa, J. S. Sperl, 19. M. Silbert, E. L. Mizuno, H. 18. the of evolution and Emergence Trachenko, liquid K. Brazhkin, the V. V. of Dove, T. thermodynamics M. Yang, and C. modes 16. Collective Brazhkin, V. V. liquids. Trachenko, in K. motion 15. Collective Beeby, L. J. Hubbard, J. McDonald, R. 14. I. Hansen, P. J. 13. oafierlxtosta otntersos.Tepredicted The response. the soften G that long-wavelength relaxations suppressing by nonaffine rigidity shear the increases tively size is, that liquids, for atomic result 2(b known by figure in probed limit data the water the in as nanoconfined such even microscopy force polymer and nonentangled (21), (20), the liquids liquids predicted ionic in the the include systems with These follow compatible experimental scaling. data well Other also experimental work. are the this literature that in evident predicted is law It 6. ref. eerhOfie otatW1N-9205.Po.L orzi gratefully is Noirez L. discussions. and Prof. input for W911NF-19-2-0055. acknowledged contract Office, Research ACKNOWLEDGMENTS. Availability. Data (5). nanoscale and of micro manipulation controlled the the at for liquids avenues new up open may and of cutoff (long-wavelength) infrared an over the to integral an leads as confinement expressed of contribu- is and negative, phonon-like mod- is momentum shear ulus low-frequency different their the to of into contribution affine/relaxational terms nonaffine liquid the in the decompose tions defor- to of atomic us nonaffine allows elasticity on approach based This liquids mations. of modulus shear the demonstrated rigorously as first liquids the 19. equilibrium ref. while other in in each vanishes cancel exactly side respectively) nonaffine, out right-hand and the (affine terms on two term third the 0 ncnlso,w aedvlpda nltclter of theory analytical an developed have we conclusion, In oye melt. polymer Lett. bis(trifluoromethanesulfonyl)imide. Chem. tri-octylmethylammonium in behavior and temperature, condition, boundary time. of sampling Effects systems: model glass-forming lated flexible linear monodisperse of behavior flow polymers. and entanglement the for spectra time states. supercritical and liquid of spectra in gap solids. amorphous of response o-eprtr irtoso lse n crystals. and glasses of vibrations low-temperature state. (1969). 556–571 ∼ k hsepan h euigtecnnmn size confinement the reducing why explains This L. L itga.wihi nesl rprinlt confinement to proportional inversely is which -integral. e.Po.Phys. Prog. Rep. −3 Macromolecules a sfloe ymn ifrn xeietlsystems experimental different many by followed is law 7537 (2013). 3775–3778 4, L hs tt eh Appl. Mech. Stat. Phys. h bv qainEq. equation above the ∞, → .Ce.Phys. Chem. J. l td aaaeicue ntearticle. the in included are data study All 152(2015). 016502 79, ..akolde nnilspotfo SArmy US from support financial acknowledges A.Z. 4623 (1994). 2426–2431 27, sa .Wysr .Bshae,Sermdlso simu- of modulus Shear Baschnagel, J. Weysser, F. nska, ´ hoyo ipeLiquids Simple of Theory 253(2013). 12A533 138, hs e.B Rev. Phys. 3–4 (2003). 235–248 320, G 825(2011). 184205 83, 0 0 = hs e.Lett. Rev. Phys. k ic h vrl non- overall the Since . tlwfeunybecause frequency low at hs e.E Rev. Phys. NSLts Articles Latest PNAS Esve,Asedm 2013). Amsterdam, (Elsevier, .Py.CSldSaePhys. State Solid C Phys. J. 10 eoestewell- the recovers 0 frf .Also, 5. ref. of ) 152(2017). 215502 118, 434(2016). 043314 93, k h effect the , G L 0 | ∼ effec- .Phys. J. f3 of 3 L L −3 −3 2, k

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