. 8 Fragility and elastic behavior of a supercooled liquid 0 0 2 Madhu Priya and Shankar P. Das n a School of Physical Sciences, J 9 Jawaharlal Nehru University, ] t New Delhi 110067, India. f o s Abstract . t a m A model for the supercooled liquid is considered by taking into account its solid like properties. - d We focus on how the long time dynamics is affected due to the coupling between the slowly n o decayingdensityfluctuationsandthelocaldisplacementvariablesinthefrozenliquid. Resultsfrom c [ our model agree with the recent observation of Novikov and Sokolov [Nature (London) 431, 961 1 v (2004)] thatthefragility indexmofaglassformingmaterial islinearly relatedtothecorresponding 7 7 ratio K/G of the bulk and the shear moduli. 3 1 . 1 0 8 0 : v i X r a 0 I. INTRODUCTION Understandingthebasicmechanism forsolidificationofasupercooledliquidintoanamor- phous structure constitutes an area of much current research interest in Condensed matter physics. As the liquid is increasingly supercooled its shear viscosity η increases and the dy- namics slows down enormously. The glass transition temperature T is characterized by the g generic value of η 1014P with the corresponding relaxation time reaching the laboratory ∼ time scales. Experiments show that the nature of relaxation in different supercooled liquids approaching the glass transition is not universal. Liquids have been classified as fragile or strong depending on their dynamical behavior in the vicinity of T . Strong liquids show a g steady increase of viscosity with the lowering of temperature. In fragile systems the viscosity first increases slowly in the temperature range higher than T and this trend is followed by g a much sharper increase of η near T . This classification of different glass forming materials g is facilitated in terms of the so called fragility parameter defined [1] as, m = ∂lnη/∂(T /T) . (1) g |Tg More recently evidence of a dynamic crossover in the value of the viscosity induced by pressure change has also been reported [2]. An associated characteristic of the supercooled stateisitssolidlikebehavior[3]. Thisismanifestedintheelasticresponseofthesystemtoan applied stress. The elastic behavior persists for times shorter than the structural relaxation time. The frozen solid with amorphous structure has transverse sound modes in addition to the longitudinal modes which are present in the normal liquid state. Recently Sokolov and Novikov [4] have demonstrated that the fragility parameter m of a liquid is linearly related to the corresponding ratio K/G of its bulk and shear moduli, i.e., m 17 = 29(K/G 1). − − The dependence of the fragility and the vibrational properties of the liquid on the basic interactionpotentialwasalsotestedrecentlywithcomputersimulations[5]. Thisstrengthens further the possibility of understanding the vibrational and relaxation properties of the frozen liquid from a common standpoint. In the supercooled state of the liquid, a tagged particle makes rattling motion and is tem- porarily trapped in the cage formed by its neighboring particles. The cage eventually breaks giving rise to continuous particle motion. The time for which the tagged particle is localized grows with supercooling. The time correlation function of the collective density fluctuations develops a plateau over intermediate times and eventually decays to zero. Ergodicity thus 1 persists in the supercooled liquid over longest time scales. This behavior of the density cor- relation function is explained in the Mode coupling theory (MCT)[6] of supercooled liquids. The basic theoretical scheme used in this approach involves a formulation of the dynamics in terms of a few slow variables. In this paper we present using the mode coupling approach, a model which includes the solid like properties of a supercooled liquid. We study the effects on the dynamics due to the couplings of slowly decaying density fluctuations with the extra slow modes present in the amorphous solid. Our results conform to the above linear relation between m and K/G reported in Ref. [4]. The paper is organized as follows. In the next section we present the model for the dynamics using an extended set of slow modes for the liquid state. In section III we present the results from the model and demonstrate how it explains the observed data for simple liquids. We end the paper with a discussion. II. MODEL STUDIED A. Nonlinear dynamics of Slow modes The dynamics of the liquid is formulated in terms of a small set of slow modes. The MCT is a basic step in this direction for studying the dynamics of a strongly interacting liquid (at high density) by taking into account the effects of coupling the slow modes in the system. The simplest version of MCT deals with the conserved densities of mass and momentum respectively denoted by ρ(x,t) and g(x,t). We assume the system to be a collection of N classical particles each of mass m¯. ~r (t) and p~ (t) are respectively the position and the α α ˆ momentum of the α-th particle at time t. For the density ρˆ and the momentum density ~g we use the standard prescription [7], N ρˆ(~x,t) = m¯ δ(~x r~ (t)) , α − α=1 X N gˆ(~x,t) = pi (t)δ(~x r~ (t)) . (2) i α − α α=1 X For the solid like state in which the particles vibrate about their mean positions, the above set of slow modes is further extended to include the new collective variable [8] u(x,t). This is defined in terms of the displacements u (t) of the α-th particle, (α = 1 to N) from their α 2 respective mean position denoted by Ro [9] such that R (t) = Ro + u (t). We adopt the α α α α definition, N ρˆ(~x,t)uˆ(~x,t) = m¯ ui(t)δ(~x r~ (t)) , (3) i α − α α=1 X The metastable positions of the atoms Ro (α = 1 to N) in the glassy system remain α unaltered for a long time. In a crystal with long range order these positions are independent of time. The equation of motion for u is obtained using standard procedure[9, 10] in the form of a generalized Langevin equation, ∂u g g δF i i + u = Γ +f (4) i ij i ∂t ρ ·∇ ρ − δu j where indices i,j refer to the different spatial coordinate axes. The Gaussian noise f i is related to the bare damping matrix Γ , through the fluctuation dissipation relation, ij < f (x,t)f (x′,t′) >= 2k TΓ δ(x x′)δ(t t′), where k T = β−1 is the Boltzmann factor. i j B ij B − − The average thermal speed of a liquid particle of mass m¯ is v = 1/√βm¯. F[ρ,g,u] is the effective Hamiltonian such that the probability of the equilibrium state is given by e−F. For the isotropic solid F is obtained in the general form [9, 11], dx g2 F = +A(δρ)2 +2Bδρs +Ks2 +2Gs˜ s˜ (5) 2 " ρ T T ij ji# Z whereδρ = ρ ρ isthedensityfluctuationandρ istheaveragemassdensity. Thequantities o o − A and B in (5) are related to the static structure factor ( correlation of density fluctuations at equal time ) for the amorphous solid. The symmetric strain tensor field s is defined in ij terms of the gradients of the displacement field u(x), 2s = ( u + u ) u u ij i j j i i m j m ∇ ∇ − ∇ ∇ ( such that s = s ). The trace and the traceless parts of s , respectively defined as ij ji ij s = s and s˜ = s δ s /3, appear in the expression (5) for free energy of the T i ii ij ij ij T − isotropPic solid. The equation of motion for ρ is the continuity equation, ∂ρ + g = 0. (6) ∂t ∇· For the momentum density g the dynamics is given by the generalized Navier-Stokes equa- tion, ∂g i + σ = θ , (7) j ij i ∂t ∇ j X 3 where θ ’s are the gaussian noises related to the bare or short time transport properties of i the liquid [12]. The symmetric stress tensor σ is obtained as a sum of the reversible part ij σR and the irreversible (dissipative) part σD, such that [9] ij ij g g σR = i j +Pδ 2Gs +2s [K¯s +Bδρ]+4Gs s (8) ij ρ ij − ij ij T im jm where K¯ = K 2G/3. The quantity P in (8) is identified with pressure in conventional − hydrodynamics and is obtained as local functional of the hydrodynamic fluctuations, δρ2 s2 P = (Aρ B)δρ+(Bρ K¯)s +A K¯ T Gs s . (9) o o T lm ml − − 2 − 2 − The dissipative part of the stress tensor σD is expressed in terms of the bare viscosities. ij 2 σD = η [ v + v δ ( .v)] ζ δ ( .v) (10) ij − 0 ∇i j ∇j i − 3 ij ∇ − 0 ij ∇ where η and ζ are the bare shear and bare bulk viscosities respectively and v g/ρ. 0 0 ≡ B. Renormalization of transport coefficients The time correlation functions of the slow modes are obtained by averaging over the noises in the nonlinear equations of motion using standard field theoretic methods [13]. The correlation functions between two slow modes ψ and ψ are defined as, α β G (12) =< ψ (1)ψ (2) > (11) αβ α β The dispersion relations for the various hydrodynamic modes in the system are obtained from the pole structures of the correlation functions. The corrections to the correlation functions due to the nonlinearities in the equations of motion for the slow variables are expressed in terms of the self-energy Σ defined through the Dyson equation G−1(q,ω) = [G0(q,ω)]−1 Σ(q,ω) . (12) − G0(q,ω) refers to the matrix of the correlation functions corresponding to the linear dynam- ics of the fluctuations. Assuming the amorphous solid state to be isotropic the correlation function can be separated in terms of a longitudinal and a transverse part as G (q,ω) = qˆqˆGL (q,ω)+(δ qˆqˆ)GT (q,ω) (13) αiβj i j αβ ij − i j αβ 4 Similarly the self-energies are split into their respective longitudinal and transverse parts denoted by Σ and Σ . Using the Dyson equation (12) the renormalized viscosity of the L T liquid taking into account the nonlinearities in the dynamics is obtained in the form 1 Γ(q,ω) = Γ + γL(q,ω) (14) 0 2k T gˆgˆ (cid:20) B (cid:21) where ΣT,L(q,ω) = q2γT,L(q,ω) (15) gˆgˆ − gˆgˆ The functional forms of the self energies are computed with the diagrammatic techniques of the MSR field theory [14]. Of particular interest in the context of glassy dynamics is the density auto correlation function φ(q,t). The Laplace transform of φ(q,t) (which is normalized with respect to its equal time value) is obtained as a two step continued fraction in terms of the memory function Γ(q,z) [12], 1 φ(q,z) = . (16) z Ω2/[z +Γ(q,z)] − o Ω is the microscopic frequency of the liquid state[15]. For the case of a normal liquid with 0 only the standard set of conserved densities as a slow variable, the dominant nonlinearity in the momentum equation comes from the coupling of the density fluctuations in the Pressure functional in (8). This obtains [16] the standard mode coupling model in which memory function includes only products of density correlation functions [17, 18]. The present formu- lation involving an extended set of hydrodynamic variables which include u ( in addition to the conserved densities) obtains the longitudinal as well as the transverse sound modes in the amorphous solid. Their speeds are respectively obtained as, c2 = M/ρ +Aρ 2B and L o o− c2 = G/ρ [11], where M = K+4G/3is the longitudinal modulus. By including the coupling T o of density fluctuations δρ with the displacement field u ( see eqn. (8)), a new contribution to the memory function Γ(q,t) is obtained. At the one loop level this is obtained as, Γ(q,ω) = Γ +Γ(1)(q,ω)+Γ(2)(q,ω) (17) 0 Γ(1)(q,ω) and Γ(2)(q,ω) are given by the expressions 8B2 d3p dΩ Γ(1)(q,ω) = p2[u4GL (p,Ω)+u2(1 u2)GT (p,Ω)]G (q p,ω Ω) k T (2π)3 2π uu − uu ρρ − − B Z Z (18) 5 and A2 d3k dΩ Γ(2)(q,ω) = G (k,ω)G (q k,ω Ω) (19) ρρ ρρ k T (2π)3 2π − − B Z Z The role of convective nonlinearities is assumed to be absorbed in Γ which is the bare 0 part or short time part of the transport coefficients. The contribution Γ(1) is obtained from the first diagram of fig. 1 resulting from the coupling of the displacement field u with the density fluctuation δρ. Γ(2) is the contribution from the second diagram in fig. 1 coming fromthecoupling ofdensity fluctuations. Inevaluating these diagrammaticcontributions we make the approximation that for time scales ( short compared to the structural relaxations) over which the supercooled liquid displays elastic behavior, the longitudinal and transverse correlations of the displacement field are frozen ( constant in time ), i.e., φL,T δ(ω). uu ∼ Therefore as a result of the solid like behavior over intermediate time scales, GL and GT uu uu are then obtained as k T k T GL (k,t) = B , GT (k,t) = B (20) uu Mk2 uu Gk2 where the k−2 dependence of the correlation function arises from the static structure factor for the displacement fields. Substituting these approximate forms for the displacement cor- relations and evaluating the integrals in (18) and (19), we obtain for the density correlation function and the memory function the simple form of coupled nonlinear integral equations, 1 φ(z) = (21) z Ω2/[z +Γ(z)] − o Γ(t) = c φ(t)+c φ2(t) (22) 1 2 where in eqns. (21) and (22) we have dropped the wave vector dependence of φ and Γ for simplicity. The density correlation function is approximated in the form G (q,t) = ρρ χ (q)φ(t) with χ being the equal time correlation of the density correlation function ρρ ρρ determined bythethermodynamicpropertiesliketemperatureanddensity. Theconstantsc 1 andc in(22)aredeterminedfromanevaluationofthevertexfunctionsinthisapproximation 2 ( of wave vector independence) as, 8 λ ∆ λ 0 σ o c = [1+f(σ)], c = , (23) 1 3 1−∆σ! 2 (1−∆σ)2 6 where ∆ = ∆ (1 2σ)/(2 2σ), and f(σ) = 2/5(1 2σ) with σ being the Poisson’s σ o − − − ratio σ = (3K 2G)/[2(3K + G)]. We have used in the expressions (23) the definitions − ∆ = B2/(AG) and λ = (Λ3/6π2n )(v/c )2 in terms of parameters dependent on the o o o L thermodynamic state of the system. The equilibrium number density of particles in the fluid is n (ρ = m¯n ). Λ is the upper cutoff of wave vector representing the shortest length o o o up to which the fluctuations are considered. In considering the mode coupling expression for the viscosity we have ignored here the presence of very slow vacancy diffusion mode and its coupling to density fluctuations. Finally, it is useful to note here that we are considering the simple form of the model in which all processes giving rise to ergodic behavior in the asymptotic dynamics have been ignored. III. RESULTS The central focus in the present analysis is the time dependence of the density autocorre- lation function φ(t). From the coupled set of eqns. (16) (inverse Laplace transformed in the time space) and (22), we obtain a nonlinear integro-differential equation for the dynamics of φ(t), t ¨ ˙ ˙ φ(t)+φ(t)+φ(t)+ dsΓ[φ(t s)]φ(s) = 0 (24) − Z0 where the dots refer to derivative with respect to time. This equation is solved numerically to obtain the behavior of the density correlation function. It is clear that the dynamics of the density correlation function φ(t) is driven here by the memory function Γ(t) which is expressed in terms of φ itself. This constitutes a nonlinear feed back mechanism [17, 18] causing a dynamic transition of the liquid to a nonergodic state in which the long time limit of the density correlation function φ(t ) = f. The quantity f is termed as the non → ∞ ergodicity parameter (NEP). In the plane of c and c the dynamic transition occurs along 1 2 the line c = 2√c c [19]. 1 2 2 − In fig. 2 we display the phase diagram with c and c showing the ideal glass transition 1 2 line. Thoughthisdynamic transitionisfinallycutoffduetoergodicity restoring mechanisms, it marks a cross over in the dynamical behavior of supercooled liquid. The liquid state is characterized by the density correlation function following initial power law relaxations, followed by a final stretched exponential decay φ exp[ (t/τ)β] [19, 20]. In the glassy ∼ − state the density correlation function decays to a nonzero value equal to the non-ergodicity 7 parameter. In fig. 2 we have also shown the values of c and c corresponding to the 1 2 nonergodic states in which the dynamics is studied in this paper (to be explained below). For presenting our results in the following, we express time in units of Γ /(ρ c2) where o o L Γ = ζ + 4η /3 is the bare longitudinal viscosity of the liquid related to its short time o o o dynamics. We keep the λ appearing in the expression (23) for the coupling constants of the o memory function fixed at a constant value (=.4) throughout the calculation. This numerical value is reached by treating λ as an adjustable parameter here for comparison of the results o of the present model with experimental data. This essentially implies that the cutoff Λ of wave vector integration is being treated as an adjustable parameter in the coarse grained model we present here. A key quantity of interest in the present analysis is the fragility parameter m which by definition is related to the final relaxation behavior of the supercooled liquid near T . g But over this temperature range, the MCT approach has not been very successfull[6] in explaining the relaxation behavior. A useful observation[4] in this respect is that the slope of the viscosity vs. inverse-temperature plot in the high temperature range can be linked with the fragility. This is justified as follows : For very fragile systems (large m) the slope of the η vs. T /T curve at temperatures near T is large and hence it must be correspondingly g g small at the other end of the Angell plot, i.e., for temperatures much higher than T . This g is because the curves for different m meet at both ends on the T /T axis. By examining g experimental data, it was pointed out [4] that the slope of the η vs. T /T curve on the g high temperature side is inversely related to the fragility m. Since MCT is a valid theory for the slow dynamics well above T , we have investigated the relaxation behavior which g follows from the present model in this high temperature range. We focus on the growth of the relaxation time τ instead of that of the viscosity η. In our model, increasing the parameter ∆ brings the system closer to the ideal transition ( since it results in an increase o of τ ) and hence this parameter is treated like the inverse of temperature, ∆ T /T. The o g ∝ dependence of the relaxation time τ on ∆ at various (fixed) K/G values is displayed in fig. o 3. For large K/G the observed variation is similar to the η vs. T /T curve of a fragile liquid. g Although according to MCT model the power law behavior describes the relaxation time data in this range better, we fit τ, following Ref. [4], to an Arrhenius form with activation energy κ, i.e., τ exp(κ∆ ). By the property of the Angell plot referred to above, we o ∼ assume m˜ = 1/κ to be proportional to the fragility m. ∆ is proportional to T /T with o g 8 the proportionality constant being independent of K/G. Thus we assume that constants A and B in the expression (5) for the free energy are only functions of temperature and the two elastic constants K and G are to be treated as independent entities. Fig. 4 displays m˜ against the variation of K/G showing an approximately linear behavior. Therefore the present model agrees qualitatively with the linear correlation predicted by Novikov and Sokolov. It should be noted that the inverse nature of the relations between fragility and the slope of the viscosity vs. inverse-temperature plot respectively at the high and low temperature sides is a property of the Angell plot itself. However the reciprocal relation i.e. κ 1/m as used above can only be justified in a qualitative manner. In order to ∝ demonstrate the sensitivity of the results on the values of the parameters ∆ and λ we o o display in fig 4. the m˜ vs K/G curves for different values of λ ranging from λ = 0.3 to 0 0 λ = 0.8. The nature of the curves are qualitatively similar to what the best fit value for λ 0 o obtains. Next we consider the dynamics on the other side of the transition when the density correlation function freezes to a constant until ergodicity restoring processes take over. In the glassy state, the NEP f is estimated from the plateau which φ(t) reaches over long time. Fig. 5 displays the variation of f with the ratio K/G. The nonergodic behavior and the elastic response of the supercooled liquid as seen here are valid over initial time scales for which the system is frozen and is solid like. However estimating the fragility parameter m directly involves computing the temperature dependence of viscosity or relaxation time near T . As already pointed out, the long time dynamics in the deeply supercooled state close to g T is still beyond the scope of MCT. Our calculation of the NEP at low temperatures here g only refer to intermediate time scales up to which the present description in terms of the supercooled liquid is valid. Therefore in order to link fragility m with the elastic properties of the supercooled liquid we make use of the dependence of the fragility m on the NEP f as inferred from analysis of experimental data [21]. In fig. 6 the data points and best fit curve for the experimental results linking NEP f with the fragility m, i.e., m = 167 176f, − is shown. The NEP values noted here are taken at the glass transition temperature T g [21]. To test this empirical relation further we compare the α = (1 f)/f vs. m in the o − inset of fig. 6 with another set of experimental data [22] from X-ray Brillouin scattering. Similar qualitative behavior is seen validating the link between the NEP with the fragility. The dependence of m on the ratio K/G of elastic constants is then obtained through their 9