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Structure–entropy relationship in repulsive glassy systems
Physica A 298 (2001) 371–386
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Structure–entropy relationship in repulsive glassy systems Shalom Baer ∗ Department of Physical Chemistry, The Hebrew University, 91904 Jerusalem, Israel Received 5 March 2001
Abstract The entropy of glass can be evaluated from the experimental structure data and given laws of intermolecular forces. The method is based on the functional relation @2 S=@E 2 = − (-E)2 −1 , which connects the entropy function S = S(E) to structure via the energy E and the spatial energy /uctuations (-E)2 . This method, previously applied to a model cohesive system, is extended to strong repulsive systems. In cohesive systems at low thermal temperature, E is mainly potential energy which can be determined from pair potentials and molecular pair distributions. In contrast, in strong repulsive systems, characteristic of systems subject to high external pressure, E is mainly kinetic and its dependence on structure can be derived only by quantum mechanics which relates the strong repulsive forces to an e4ective volume available for molecular motion. This dependence has a form peculiar to the wave nature of the particles, and is illustrated by a cell model treatment of a disordered dense packed hard spheres system. In the low thermal temperature limit, it leads to an entropy independent of Planck’s constant and of the particle mass. To integrate the above equation we use a model of the radial distribution g(r) in the form of an analytic function, g(r) = g(r; L; D), where L is a set of parameters specifying a lattice characterizing the dominant local con9gurations of atoms and D is a “structural di4usion” parameter providing a measure of the degree of spatial decay of coherence between local structures in the amorphous system and the degree of structural disorder. The model provides a representation of structure by a point in the low dimensional parameter space {L; D}. Integration is performed along a path connecting the ordered state (L; 0) to (L; D). Whereas S = S(D) increases with D, for strongly repulsive systems E = E(D) decreases with D, leading to an ordered state with highest energy. This implies a transition from an ordered to a more stable amorphous phase, in accordance with the observed phenomenon of high pressure induced amorphization, a transic 2001 tion under high pressure and low temperatures from a crystalline to an amorphous state. Elsevier Science B.V. All rights reserved. PACS: 05.70: Ce; 61.43:−j; 05.10:−a Keywords: Entropy; Glass structure; High pressure ∗
Tel.: +972-2-658-5269; fax: +972-2-651-3742. E-mail address: [email protected] (S. Baer).
c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 2 7 3 - 4
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1. Introduction The problem of 9nding the entropy of a system arises whenever it is required to determine the relative stability of di4erent states of the system. This problem enters for instance in the determination of the tie line between /uid and solid branches of the pressure obtained in separate molecular dynamics runs [1,2], or in the interpretation of heat capacity data of a supercooled liquid and glass compared to that of a crystal [3]. In the former case, it is solved by imposing constraints on the system which enable to follow the entropy (free energy) change along a path from the solid state to a state of dilute gas constrained to single occupancy of cells of volume V=N , whose entropy is known. However, in the case of glass the application of calorimetry has led to erroneous conclusions on its entropy [3–9], since glass has a frozen-in structure which is not in internal equilibrium, hence the contribution of the disordered structure to the entropy cannot be probed by heat exchange with an external bath. In a previous paper [10] a scheme was presented for obtaining the entropy from structure data, provided the molecular interaction potentials are known. It is based on the following general relation between the entropy as a function of energy, S = S(E), and the spatial energy dispersion (-E)2 : @2 S 1 =− : (-E)2 @E 2
(1)
This relation holds for any macroscopic system irrespective of whether it is in internal equilibrium or not. It follows from the property of extensivity (additivity) of a macroscopic system, i.e., the system can be viewed as composed of many subsystems, themselves macroscopic [11,12], and the energies of the subsystems add up to the total energy of the system. Moreover, the subsystems constitute an ensemble of systems in di4erent microstates, and this ensemble uniquely speci9es the macrostate of the physical system. Hence by the Central Limit Theorem of probability theory [13], the energy distribution in this ensemble is asymptotically a sharply peaked Gaussian around the average energy: J 2 (E − E) : (2) p(E) ∼ C exp − 2(-E)2 Using the information theoretic de9nition for the entropy [14,15] S = kB log (E);
(E) = number density of microstates ;
(3)
this distribution can be obtained by maximizing (3), subject to the constraint of a J producing a Boltzmann distribution, i.e., a canonical given average energy, E = E, ensemble representation of the system, p(E) ˙ (E)e−E :
(4)
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Comparing (4) with (2), one can put now for the entropy, replacing EJ by E, S = kB log (E) ;
(5a)
@S : (5b) @E As de9ned in (5b), = (E) = (kB T )−1 is a reciprocal canonical temperature uniquely determined by the ensemble specifying the macrostate of the system at a given energy E, and is derivable directly from the entropy. However, only when the system can be brought into complete thermal equilibrium with an external heat bath at a temperature Tth , can one have an equality T = Tth . When the thermal motion of atoms contributes negligibly to the disorder in the system (see below) one can refer to S and T as structural entropy and temperature, respectively. Given the energy, the system is understood to be constrained by additional parameters. For instance, keeping the volume V 9xed in a homogeneous system in complete internal equilibrium, its entropy function is S = S(E; V ). The change of entropy when such a parameter is varied can be followed if the parameter can be coupled to a suitable force which performs measurable work [9]. More sophisticated constraints can be realized in computer experiments, such as single cell occupancy [1,2,16]. However, other constraints, only implied by their indirect e4ect [17], must be e4ective in the case of glasses and amorphous solids, keeping the systems practically unchanged in time. Denoting the particular set of constraints by c, we have strictly (E) ≡ (E; c), (S ≡ S(E; c)), and c is kept constant in the di4erentiations in (1) and (5). Since both E and (-E)2 are known functionals of molecular distributions and interaction potentials [18–20], one can integrate (1) to obtain the entropy if the structure of the system can be characterized by a small set of parameters and a path can be found in this parameter space, connecting the given state to a reference state of known entropy. Such a path is indeed provided by a modeling of molecular distributions by parametrized analytic functions, based on a general model for structure in condensed systems [21]. It turns out that experimental radial distribution data, g(r), can be represented accurately by an analytic expression g(r) = g(r; L; D) depending on a small set of parameters (L; D). L is an underlying “local lattice” (or set of di4erent lattices) representing dominant spatial con9gurations of neighboring atoms, and D is a “structural di4usion” parameter (or set of parameters) representing the decay with distance of the coherence (matching) between atomic con9gurations at di4erent localities. In its simplest version, the model leads to the expression n 1 (r − a − a + a )2 1 g(r) = g(r; L; D) = exp − ; n (4W )3=2 4W 0 =
; =1
(6) where the -sum extends over the positions a of all unit cells of L, the -sums run over all (n) points in a unit cell, and is the particle density in L. The brackets 0 denote averaging over all the orientations of L, or of r: W = W (r) has the asymptotic forms W ∼ D(r − ) (see the appendix). Since the methodology leading to g(r; L; D)
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also leads to speci9c expressions for higher order distributions [22] in terms of L and D, one can view the set (L; D), having been chosen optimally to represent g(r) data, as uniquely specifying the structure (macrostate) of the system, and providing the desired parameter space determining both E and (-E)2 . Now, when D = 0, the right-hand side (rhs) of (6) becomes practically a sum of -functions representing g(r) of a perfect lattice structure L. This is a completely ordered system [23] whose entropy must be S(EL ) = 0, where EL is the lattice energy. Hence, integrating (1) along the path 0 6 D 6 D;
L 9xed ;
(7)
one obtains the entropy S(E) = S(L; D) of the system with the given structure (L; D). Applying the above outlined method, the entropy of a model liquid and solid amorphous metal was evaluated [10] using a given radial distribution g(r; L; D) with L = FCC and a typical liquid D as structure data. The interaction between the ions was assumed to be a Coulomb potential, including the interaction with a background of opposite charges, and a very short range strong repulsive pair potential was added according to a scheme devised by Born [24]. The energy was expressed as a function of L and D by 1 E = E(L; D) = N u(r)g(r; L; D) d 3 r ; (8) 2 where u(r) is an e4ective pair potential. Because of the cohesive forces, (8) is practically the total energy at liquid and lower temperatures. The kinetic energy of thermal motion of the atoms is then negligible compared to the interaction energy and the pressure required to keep the system in a condensed state (which is the vapor pressure of the system) is practically small and vanishes in the limit Tth → 0. On the other hand, in a repulsive system, where the forces between the particles are purely repulsive, a positive pressure is required to counteract the kinetic pressure of the atoms and to con9ne the system to a 9nite volume V , down to the Tth → 0 limit. Thus, the foregoing scheme for evaluating the entropy from structure data requires modi9cations taking into account the positive kinetic energy in the Tth → 0 limit. This is illustrated in the following by the extreme case of a hard sphere system, where the interaction energy (8) vanishes. We will make use of a quantum cell version, related to the given g(r) data, to evaluate E and (-E)2 and will apply it to obtain the entropy of a strongly repulsive system as outlined above. The g(r) data we choose are typical of an amorphous solid (see Fig. 1), whose atoms are thermally practically unexcited, i.e., the solid is at Tth = 0, but at a nonzero canonical temperature, T ¿ 0. For glass, S(Tth = 0) ¿ 0, and the energy E(Tth = 0) is not the energy of an ordered solid. 2. Quantum cell model and the structure of a glassy repulsive system In a repulsive system, where the forces between the particles are purely repulsive, the pressure required to con9ne the system to a 9nite volume V is positive and does not vanish in the Tth → 0 limit. Contrary to the classical ideal gas whose energy
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Fig. 1. Radial distribution g(r) = g(r; L; D): solid line, D = 0:006; dashed line, D = 0:010; dotted line, D = 0:014; (L = FCC; r0 = 1:3714 (see the appendix)).
E = (3=2)NkB T vanishes at T = 0, quantum mechanics requires a positive, volume and structure dependent “zero point” energy which maintains a positive internal pressure. These properties of the repulsive system are illustrated by a model of a purely repulsive system with only exclusion forces between particles, e.g. a system of hard spheres (HS). Such a system has zero potential energy and its total energy is purely kinetic. Yet, in a dense packed state the exclusion forces impose strong correlations between molecular positions. In the low temperatures quantum regime, these correlations are manifested by certain complex boundary conditions on the 3N dimensional wave function (rN ) of the N free particles. Finding the form of or even its energy eigenvalues is an insoluble many body problem. Hence, to enable explicit calculations, we make use of a cell model type approximation, assuming that each particle moves independently within a cell with 9xed boundaries. We consider hard spheres of diameter and take as a measure of the linear size of a cell the nearest neighbor distance, a, for a chosen lattice L con9guration, at a given v = −1 . a is related to v by v = cL a3 , where√cL is the volume (in units of a3 ) of a unit cell of L. For L = FCC; HCP, cL = 1= 2. At close packing its volume per particle is vc = cL 3 . The linear motion of a particle is restricted practically to an interval of length lf ≈ a − , the “free length” of a particle, which is distributed according to some probability density pf (l) depending on the spatial arrangement of the particles. In a disordered phase the l’s are distributed
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continuously over the range 0 ¡ l ¡ ∞. In an ordered phase all particles have the same l = lf and pf (l) = (l − lf ). Ignoring minor re9nements relating to the shape of the cell, we assume that each particle moves independently along each of the three Cartesian directions. This amounts to representing the particle cell by a cube. By assumption, decomposes into a product of one-particle wave functions, each being a product of three one-coordinate wave functions. Thus, N (rN ) = (9) i (ri ); i (r) = ’i (x)’ i (y)’i (z) i=1
with ˝ 2 d 2 ’i = )ix ’i (10) 2m d x2 being the wave equation for translational motion within an interval − l2i ¡ x ¡ l2i . The solutions ’i to (10), satisfying the boundary conditions ’(±li =2) = 0, are even and odd standing waves: 1; 3; : : : cos nx ; n= ; (11) ’i = ’(x; n; li ) = A 2; 4; : : : sin li 2 h )ix = )x (n; li ) = n2 : (11a) 8ml2i (The case n = 0, with ’(x) ≡ 0, is excluded). We then have per degree of freedom h2 : (12) )ix = )x (n; li ) = )1i n2 ; where )1i = 8ml2i The ground state of particle i is )i = 3)1i . The pairs (n; li ) specify “internal states” of particle i : li is the linear size of the cell available to the free motion of the particle and n is the quantum number of its energy level. The particles are distributed over all possible states according to two distribution laws: the probability density pf (l) and, 2 given l = li , the Boltzmann distribution law for energy levels, pn (th ; li ) ˙ e−th )1i n , where th is the reciprocal temperature of a heat bath. The energy /uctuations of the system,
2 2 )i )j − )i ; (13) (-E) = −
ij
i
can now be evaluated following closely the procedure applied in Section 5 of Ref. [10]. Note that the angular brackets in (13) refer to averaging over both the internal states of each particle and over the con9gurations of the N particles. Introducing the local energy density )(r) = )i (r − ri ) ; (14) i
and performing a mean 9eld type partial averaging over the internal states of particles, all particles being identical, we write )(r) J = )J (r − ri ) = )(r) J ; (15) i
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where )J =
377
)i =N
(15a)
i
is the average energy per particle, and (r) is the local particle density. Here )i denotes an average over all the states (n) of particle i (see Eqs. (21) and (22) below). Spatial averaging of (15) produces )(r) J = ). J Rewriting (13) in terms of the partially averaged densities (15), we must separate out from the sums the terms with i = j. The latter produces additional contributions to the energy /uctuations. Thus, with -)(r) J = )(r) J − )(r) J = -(r))J ; we have
-)(1)J )(2) J d1 d2 = )J2
-(1)-(2) d1 d2 = N )J2
(16)
[G(r) − ] d 3 r ; (17)
where G(r) = G(r; t = 0) = (r) + g(r) is the van Hove correlation function at t = 0, and we obtain for the total energy /uctuations, the following generalization of (17) in Ref. [10] (-E)2 = -)(1)-)(2) d1 d2 = )J2 -(1)-(2) d1 d2 + N (-))2 (18) = N )J2 [G(r) − ] d 3 r + N (-))2 ; where (-))2 =
1 2 [)i − )i 2 ] : N i
Using the structure factor (scattering function) expression F(k) = 1 + [g(r) − 1]eik·r d 3 r ;
(19)
(20)
we can rewrite (18) as (-E)2 =N = )J2 F(0) + (-))2 :
(18 )
The 9rst term on the rhs of (18 ) represents energy /uctuations due to structural, i.e., local density /uctuations, whereas the second term represents the particles’ internal energy /uctuations. An equation similar to (18 ) was obtained in Ref. [10] for the case of cohesive systems. Both equations are based on an approximation (i.e., (15)) which relates structural energy /uctuations to density /uctuations and neglects the e4ect of higher order correlations. In both the cohesive and repulsive cases, the energy is de9ned relative to a zero energy reference state of high dilution. Given the representation (6) of g(r), the scattering function F(k) given by (20) can be evaluated analytically, thus giving F(0) with high accuracy (see Fig. 2).
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Fig. 2. Static scattering function F(k; L; D): solid line, D = 0:006, Eq. (A.5); dashed line, same D, FFT of (6); dotted line, D = 0:014, Eq. (A.5); inset: same, 0 6 k 6 1:
Averaging over a particle’s energy quantum states can be performed via the ith particle partition function, 3 ∞ 2 qi = q(th ; li ) = e−th )1i n ; (21) n=1
from which we obtain the ith particle average energy, @ log qi ; )i = )(th ; li ) = − @th
(22)
and its /uctuations (-)i )2 = )i2 − )i 2 =
@2 log qi : 2 @th
(23)
By (21) and (23), the internal energy /uctuations vanish in the Tth → 0 limit, leaving only the structural contribution to (-E)2 . Moreover, even when Tth = 0 but small, assuming the particle mass to be suQciently large so that the thermal wavelength 0th ≡
−1 2 th h =2m is smaller than a typical free length li , we have classically )i = (3=2)th , −2 2 (-)i ) = (3=2)th . Hence at low Tth (high th ) the thermal energy contributes little to the total energy /uctuations. The summations in (15a) and (19) amount to averaging over all the free lengths li . In the th → ∞ limit we then have from (21) and (22), with (12), )J = 3)1i =N = 3) (=l)2 ; ) ≡ h2 =(8m2 ) ; (24) i
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which is the average energy per particle at zero thermal (bath) temperature. Averaging over free lengths requires an estimate of the distribution pf (l), which we wish to express in terms of structure data, speci9cally in terms of the radial distribution g(r). For the ith particle r = li + is a typical distance between the particle and its nearest neighbors. Information on its spatial distribution can be deduced from the shape of the main (9rst) peak in g(r). Thus, we can use the following estimate: rm1 rm1 2 2 2 r g(r) dr r 2 g(r) dr ; (25) (=l) = l where l = r − and rm1 is the position of the 9rst minimum of g(r). Since for a hard spheres system g(r) = gHS (r) possesses densely distributed singularities [25] at all r ¿ , and gHS () = 0, the analytic parameterized expression (6) cannot strictly represent gHS (r) although it can approximate it to high accuracy. Moreover, since g(r) = g(r; L; D) does not vanish at r = , substituting it into (25) requires replacing the denominator l = r − in the integrand by a function l (r) which is nonzero for all 0 ¡ r ¡∞ and satis9es l (r) ∼ r − for r, and l (r) → 0 for r → 0. We choose for l (r) a function proportional to the function W (r) in (6), the latter determining the rate of decrease of g(r) to zero with decreasing r (see the appendix), and perform the numerical integration in (25) replacing the lower integration limit by zero. Thus we obtain (=l)2 and by (24) the energy E as functions of D (and L) over a whole range of D values (see Fig. 3). We have chosen for proportionality factor in l (r) a value such that in the D = 0 limit the exact value (=l)2 = [=(a − )]2 = [=l (a)]2 is obtained. We choose a density = rc corresponding to a random close packed system [26,27], with a density √ fraction rc =c = vc =vrc = 0:859, i.e., a packing fraction 1rc = 0:636, where vc = 3 = 2 is the volume per particle of the ordered close packed system.
3. Evaluating the entropy of a strongly repulsive glassy system from structure data We can follow now the scheme presented in Ref. [10] to evaluate the entropy of a strongly repulsive system with a given structure by integrating (1). Taking into account the boundary conditions (asymptotic behavior) lim S(E) = 0;
E→EL
lim (E) = ∞;
E→EL
a formal integration of (1) gives E E∞ dE S(E) = d EJ : (-E)2 EL EJ
lim (E) = 0;
E→E∞
lim 2 (-E)2 (E) = 0 ;
E→EL
(26)
(27)
In (26) the 9rst two conditions specify the properties of the singular point of S(E) at the ordered (ground) state L, with energy EL , which follow from the Third Law. The third condition is the assumption of an in9nite temperature at the highest (possibly
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2 Fig. 3. Energy )=(3) J ) ≡ (=l) as function of D.
in9nite) energy, E∞ . The last condition follows from a general asymptotic behavior of heat capacities (de9nable only through the entropy in the case of glass) C=
@E @E = − 2 = 2 (-E)2 ; @T @
(28)
in the T → 0 limit. By adopting the representation g(r) = g(r; L; D), the energy E and the energy /uctuations (-E)2 , given by (18), become functions of D (and L), and the integrals in (27) can be transformed into integrals in terms of the variable D. Thus we obtain for the entropy per particle (see Ref. [10], Eq. (35 )): D∞ J D J J dD + J D) J dD ; s(D) = u(D) (29) 3(D) u(D)3( J J D D D 0 where we have put )J = )(D) J = E=N; u(D) = )(D) J − )(0), J and de9ne the functions 3 and by (-E)2 =N = D (D);
3(D) = u (D)= (D) :
(30)
Constructing rational approximants for )(D) J and (D) from their respective numerical values over an interval Da ¡ D ¡ D (omitting the small values D ¡ Da where numerical integration becomes diQcult) and using these approximants as analytic continuations into the entire range 0 ¡ D 6 D, the integrals in (29) can be performed
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Fig. 4. Scattering function at k = 0; F(0; D)=D as function of D.
analytically and evaluated to give the desired s(E) = s(D). By (30) is implied that (-E)2 = 0 in the limit of the ordered state, D = 0. The same applies to the low angle static structure factor, for which we put F(0) ≡ F(0; D) = D&(D)
(31)
in conformity with (18). However, &(0) = 0; (0) = 0 and limD→0 u(D)=D is 9nite. In Figs. 4 and 5 are given F(0; D) and s(D), respectively for the random close packing density fraction rc =c = vc =vrc = 0:859 (=a = 0:95056 for L = FCC). In both repulsive and cohesive /uid systems structure characteristics are determined mainly by the repulsive forces [28,29]. Thus one could choose to represent g(r) of a typical disordered strongly repulsive system by g(r; L = FCC; D = 0:006) exhibiting a second and third peak structure similar to that of the model cohesive system studied in Ref. [10]. As in the latter case, it was not possible to complete the integration of (1) without using the boundary condition = @S=@E ≈ 0 for some “maximum disorder” limit point D∞ . It was found that for 0:01 6 D∞ 6 0:025; S(D) changed only within about 10%. From Fig. 5 one 9nds the estimate S(D = 0:006) ≈ 25kB , which is about 5 times larger than the value S = 4:84kB for the cohesive system. We note that this estimate, which follows from characterization of the state of the HS system by the value D = 6 × 10−2 , falls within the range 3 × 10−2 ¡ D ¡ 8 × 10−2 , where S(D)
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Fig. 5. Entropy s(D) as function of D; D∞ = 0:015.
rises steeply from 2kB to 40kB , implying a pronounced change in structure within this range. From the foregoing results the following conclusions can be made: Considering (24) and (18 ), it follows from (27) that Planck’s constant and the particle mass, appearing only in the combination ) = h2 =(8m2 ), are completely eliminated from the entropy. Thus, quantum e4ects enter only through the dependence of the wave function (11) and its energy eigenvalues (11a) on the boundary conditions imposed by the intermolecular exclusion forces and structure of the hard sphere system. Another striking feature of the repulsive system, contrary to the cohesive system [10], is the decrease of the energy function E(D) with increasing D (see Fig. 3). This is a direct consequence of the asymmetry of the 9rst peak of g(r), which gives more weight to larger r values in the evaluation of (=l)2 in (24). Since S(D) is increasing with D, it follows that the structural temperature T = (@S=@E)−1 is negative and the ordered state S = 0, i.e., D = 0, has the highest energy.
4. Energy and pressure equations of state at Tth = 0 Contrary to the entropy, the energy and the pressure are proportional to ) . Near close packing at some volume vc which depends on the particular set of constraints c in
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the glass sample, they also depend strongly on the available free length. An estimate of the dense packed energy and pressure can be obtained from (24), by replacing (=l)2 by (=lf )2 , where lf = a − is an e4ective free length. Putting v = cL a3 ; vc = cL 3 , one obtains 3 2 d(a=) @)J )J = 3) ; p= − = 6) : (32) @v lf dv lf Near close packing, dv=v = 3(da=a) and v − vc ≈ dv; lf ≈ da. Hence da lf ≈ ; ≈ v − vc dv 3vc and
)J = 27)
vc v − vc
3vc ≈ lf v − vc
2
;
pvc ≈ 54)
(33)
vc v − vc
3 :
(34)
Eqs. (32) and (34) follow directly from the introduction of quantum states, the resulting dependence on v−vc being in contrast to the dense packed limit of classical cell theory, vc : (35) )J ∼ (3=2)kB T; pvc = 3kB T v − vc 5. Conclusion The outlined method for evaluating the entropy via (1) makes use of a special choice of parameterization of the structure (g(r)) based on an ordered state L as a reference state. This parameterization provides a path in a virtual structure space {L; D} connecting the observed state of a system to a state L of zero entropy. Thus integration of (1) can be performed along this path. Yet the actual implementation of the method has required a distinction between cohesive systems, namely systems whose vapor pressure is zero in the zero temperature limit and whose energy is thus predominantly potential energy, versus repulsive systems, which are maintained under a positive external pressure, hence possess predominantly kinetic energy in this limit. An extreme case of repulsive systems is a system of particles with purely exclusion interactions, where the energy is only kinetic. Using the particular HS cell model for this extreme case, it was possible to get an estimate of the structural entropy. The model illustrates how the low lying quantum states determine the constants of integration of (1). The semiclassical formula for the entropy, based on the above formulated cell model, S = NkB [3 log(lf =0th ) + 32 ], gives only the contribution of thermal motion to the entropy, not including the contribution of structural disorder. Even as such it is applicable only as long as the free length lf 0th and fails for larger thermal wave lengths 0th ¿ lf , giving even a negative entropy for 0th 1:65lf . The classical kinetic energy of a HS system is independent of the structure, and only through quantum mechanics does it become related to structure, making it possible to apply (1) to the evaluation of the entropy. At very low ambient temperature, the quantum mechanical
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expression for the entropy becomes independent of Planck’s constant ˝ and the particle mass m as a result of the wave function becoming dependent mainly on the geometry of boundary conditions. The characteristic wavelength of becomes then the particle “free length”, 0 ≈ lf , which is related only to particle diameter and interparticle distance, while ˝ and m enter only through 0th , which disappears from the expression for in the Tth → 0 limit. Considering the decrease of the entropy function with increasing energy, states of negative temperature are known to exist in paramagnetic systems (see Ref. [11, pt. I, #73]) where nuclear spins (magnetic moments) which are weakly coupled to the lattice degrees of freedom, are subject to a sudden reversal of a magnetic 9eld and relax only slowly to an equilibrium state. Similarly we can assume that in the present glass model of repulsively interacting particles possessing only zero point kinetic energy, such a state cannot be realized for a long time and eventually some structural repulsive energy is transferred to excite particle motion, accompanied by increasing structural disorder. Noting that strongly repulsive systems can be realized only at high external pressure, the implied transition from ordered to amorphous state for the dense HS system at zero ambient temperature can serve as a model for the observed phenomenon of high pressure induced amorphization, in particular at low temperatures, as e.g. the transition of ice at T = 77 K and p ∼ 10 kbar from crystalline to amorphous ice [30], or the crystalline to amorphous transition of silica [31] at T = 300 K and p = 25–35 kbar. Acknowledgements The present article resulted thanks to a remark by Prof. Benny Gerber who noted the inconsistency of applying the scheme for evaluating the entropy of a cohesive system to a hard sphere system. Appendix. Analytic representation of the radial distribution and the static scattering function The expression (6) holds strictly only for r = 0, but can be interpreted for all r as the static van Hove correlation function G(r) ≡ G(r; t = 0) = (r) + g(r), which includes also self-correlations of particle densities, if W (r = 0)Daa2 , where a is the nearest neighbor distance in L. Applying the Poisson sum transformation formula to (6), we obtain an equivalent representation as a sum over all points b5 of the reciprocal lattice to L: 2 1 2 G(r) = C5 e−Wb5 eib5 ·r 0 ; C5 = eib5 ·a : (6 ) n 5
Expressions similar to (6) (but not identical to it) have been repeatedly used [32,33] to reproduce g(r) data. Expressions similar to (6 ) are used as well [34,35], mainly to
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obtain good 9ts to scattering data in the low k range (see below). For the numerical evaluation of g(r), the following specialization of W (r) can be made. Because of exclusion e4ects, we expect g(r) = 0 to hold to a good approximation within an entire domain 0 6 r 6 , where is a measure of molecular diameter. This is achieved by choosing for W (r) a function fast decreasing to zero for r 6 . For practical reasons of convergence at small r we choose for W the particular form r W = W0 + Dr0 ! ; (A.1) r0 where !(x) = !1 (x) = log cosh x;
(A.2)
with r0 = r = =log 2. Another form used was [21]
!(x) = !2 (x) = 1 + x2 − 1 ;
(A.3)
with r0 = . Both choices of !(x) have the asymptotic form !(x) = 12 x2 + O(x4 );
x→0;
but for x→∞ !1 (x) = x − log 2 + O(e
−2x
);
1 : !2 (x) = x − 1 + O x
Both forms of !(x) have the advantage over an earlier choice [36] in that they ensure the requirement of a positive scattering function [37,38], F(k) ¿ 0, for all k, but (A.2) has been found preferable since it produces a better 9t to g(r) at the steep rise on the small r side of the 9rst peak. We also choose l (r) = r !1 (r=r ) to represent the “free length” function discussed in Section 2. Writing (20) in the form F(k) = (G(r) − )eik·r d 3 r ; (A.4) and substituting (6 ) into (A.4), using (A:1) with either (A.2) or (A.3), we can obtain F(k) in closed form. With (A.2) we obtain 2 F(k) = r03 C5 e−W0 b5 [K(05 ; u; u5 )] ; (A.5) 5
with K(05 ; u; u5 ) = 4
I (05 ; u − u5 ) − I (05 ; u + u5 ) 2uu5
(A.6)
and the dimensionless parameters 05 = Db25 r0 ; u5 = b5 r0 ; u = kr0 , where [39] ∞ 0 u 0 u −0!1 (x) 0−2 ; I (0; u) ≡ e cos ux d x = 2 B +i ; −i 2 2 2 2 0 where B(x; y) = <(x)<(y)=<(x + y) is the Beta function. The series in (A.5) converges for all k values, being especially useful for evaluating F(k) at small k, and gives
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accurate results for k up to about the position of the second peak of F(k). However, beyond it F(k) deviates increasingly from the correct value and goes to zero when k → ∞ (see Fig. 2). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
W.G. Hoover, F.H. Ree, J. Chem. Phys. 47 (1967) 4873. W.G. Hoover, F.H. Ree, J. Chem. Phys. 49 (1968) 3609. W. Kauzmann, Chem. Rev. 43 (1948) 219–256. J. JVackle, Philos. Mag. B 44 (1981) 533–545. J. JVackle, Physica B 127 (1984) 79–86. F.H. Stillinger, Science 267 (1995) 1935. F.H. Stillinger, J. Chem. Phys. 88 (1988) 7818–7825. S. Baer, J. Non-Cryst. Solids 250 –252 (1999) 586–590. D. Kivelson, H. Reiss, J. Phys. Chem. B 103 (1999) 8337–8343. S. Baer, B. Ilan, J. Phys.: Condens. Matter 9 (1997) 7785–7798. L.D. Landau, E.M. Lifshitz, Statistical Physics, 3rd Edition, Pergamon Press, Oxford, 1980. R. Brout, Phys. Rev. 115 (1959) 824. A.I. Khinchin, Mathematical Foundations of Statistical Mechanics, Dover, New York, 1949. R.D. Levine, M. Tribus, The Maximum Entropy Formalism, MIT Press, Cambridge, MA, 1979. W.T. Grandy, Phys. Rep. 62 (1980) 175–266. B.J. Alder, W.G. Hoover, D.A. Young, J. Chem. Phys. 49 (1968) 3688–3696. T. Morita, J. Math. Phys. 5 (1964) 1401. J.S. Rowlinson, Liquids and Liquid Mixtures, 2nd Edition, Butterworths, London, 1969. P. Scho9eld, Proc. Phys. Soc. (London) 88 (1966) 149. W.B. Brown, Philos. Trans. A 250 (1957) 221. S. Baer, L. Gutman, M. Silbert, J. Non-Cryst. Solids 192–193 (1995) 106. S. Baer, J. Phys.: Condens. Matter 2 (1990) 9675. P.W. Anderson, Concepts in Solids, Benjamin, New York, 1964, p. 3. B. Ilan, S. Baer, Phys. Rev. B 54 (1996) 14881–14883. F.H. Stillinger, J. Comput. Phys. 7 (1971) 367. J.L. Finney, J. Phys. Coll. C2 36 (Suppl. 4) (1975) C2-1. K. Gotoh, J.L. Finney, Nature 252 (1974) 202. N.W. Ashcroft, J. Lekner, Phys. Rev. 145 (1966) 404. L. Verlet, Phys. Rev. 165 (1968) 201–214. O. Mishima, L.D. Calvert, E. Whalley, Nature 310 (1984) 393–395. R.J. Hemley, A.P. Jephcoat, H.K. Mao, L.C. Ming, M.H. Manghnani, Nature 334 (1988) 52–54. Y.I. Frenkel, Kinetic Theory of Liquids, Oxford University Press, Oxford, 1946, Dover, New York, 1955 (Chapter III.4). A.H. Narten, H.A. Levy, Science 165 (1969) 447. T. Iijima, K. Nishikawa, Chem. Phys. Lett. 115 (1985) 522. K. Nishikawa, T. Iijima, Bull. Chem. Soc. Japan 59 (1986) 117. S. Baer, Physica A 91 (1978) 603. S. Baer, M. Silbert, Phys. Chem. Liq. 16 (1986) 121. S. Baer, J. Non-Cryst. Solids 106 (1988) 92. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1965, #3.985.
www.elsevier.com/locate/physa
Structure–entropy relationship in repulsive glassy systems Shalom Baer ∗ Department of Physical Chemistry, The Hebrew University, 91904 Jerusalem, Israel Received 5 March 2001
Abstract The entropy of glass can be evaluated from the experimental structure data and given laws of intermolecular forces. The method is based on the functional relation @2 S=@E 2 = − (-E)2 −1 , which connects the entropy function S = S(E) to structure via the energy E and the spatial energy /uctuations (-E)2 . This method, previously applied to a model cohesive system, is extended to strong repulsive systems. In cohesive systems at low thermal temperature, E is mainly potential energy which can be determined from pair potentials and molecular pair distributions. In contrast, in strong repulsive systems, characteristic of systems subject to high external pressure, E is mainly kinetic and its dependence on structure can be derived only by quantum mechanics which relates the strong repulsive forces to an e4ective volume available for molecular motion. This dependence has a form peculiar to the wave nature of the particles, and is illustrated by a cell model treatment of a disordered dense packed hard spheres system. In the low thermal temperature limit, it leads to an entropy independent of Planck’s constant and of the particle mass. To integrate the above equation we use a model of the radial distribution g(r) in the form of an analytic function, g(r) = g(r; L; D), where L is a set of parameters specifying a lattice characterizing the dominant local con9gurations of atoms and D is a “structural di4usion” parameter providing a measure of the degree of spatial decay of coherence between local structures in the amorphous system and the degree of structural disorder. The model provides a representation of structure by a point in the low dimensional parameter space {L; D}. Integration is performed along a path connecting the ordered state (L; 0) to (L; D). Whereas S = S(D) increases with D, for strongly repulsive systems E = E(D) decreases with D, leading to an ordered state with highest energy. This implies a transition from an ordered to a more stable amorphous phase, in accordance with the observed phenomenon of high pressure induced amorphization, a transic 2001 tion under high pressure and low temperatures from a crystalline to an amorphous state. Elsevier Science B.V. All rights reserved. PACS: 05.70: Ce; 61.43:−j; 05.10:−a Keywords: Entropy; Glass structure; High pressure ∗
Tel.: +972-2-658-5269; fax: +972-2-651-3742. E-mail address: [email protected] (S. Baer).
c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 2 7 3 - 4
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1. Introduction The problem of 9nding the entropy of a system arises whenever it is required to determine the relative stability of di4erent states of the system. This problem enters for instance in the determination of the tie line between /uid and solid branches of the pressure obtained in separate molecular dynamics runs [1,2], or in the interpretation of heat capacity data of a supercooled liquid and glass compared to that of a crystal [3]. In the former case, it is solved by imposing constraints on the system which enable to follow the entropy (free energy) change along a path from the solid state to a state of dilute gas constrained to single occupancy of cells of volume V=N , whose entropy is known. However, in the case of glass the application of calorimetry has led to erroneous conclusions on its entropy [3–9], since glass has a frozen-in structure which is not in internal equilibrium, hence the contribution of the disordered structure to the entropy cannot be probed by heat exchange with an external bath. In a previous paper [10] a scheme was presented for obtaining the entropy from structure data, provided the molecular interaction potentials are known. It is based on the following general relation between the entropy as a function of energy, S = S(E), and the spatial energy dispersion (-E)2 : @2 S 1 =− : (-E)2 @E 2
(1)
This relation holds for any macroscopic system irrespective of whether it is in internal equilibrium or not. It follows from the property of extensivity (additivity) of a macroscopic system, i.e., the system can be viewed as composed of many subsystems, themselves macroscopic [11,12], and the energies of the subsystems add up to the total energy of the system. Moreover, the subsystems constitute an ensemble of systems in di4erent microstates, and this ensemble uniquely speci9es the macrostate of the physical system. Hence by the Central Limit Theorem of probability theory [13], the energy distribution in this ensemble is asymptotically a sharply peaked Gaussian around the average energy: J 2 (E − E) : (2) p(E) ∼ C exp − 2(-E)2 Using the information theoretic de9nition for the entropy [14,15] S = kB log (E);
(E) = number density of microstates ;
(3)
this distribution can be obtained by maximizing (3), subject to the constraint of a J producing a Boltzmann distribution, i.e., a canonical given average energy, E = E, ensemble representation of the system, p(E) ˙ (E)e−E :
(4)
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373
Comparing (4) with (2), one can put now for the entropy, replacing EJ by E, S = kB log (E) ;
(5a)
@S : (5b) @E As de9ned in (5b), = (E) = (kB T )−1 is a reciprocal canonical temperature uniquely determined by the ensemble specifying the macrostate of the system at a given energy E, and is derivable directly from the entropy. However, only when the system can be brought into complete thermal equilibrium with an external heat bath at a temperature Tth , can one have an equality T = Tth . When the thermal motion of atoms contributes negligibly to the disorder in the system (see below) one can refer to S and T as structural entropy and temperature, respectively. Given the energy, the system is understood to be constrained by additional parameters. For instance, keeping the volume V 9xed in a homogeneous system in complete internal equilibrium, its entropy function is S = S(E; V ). The change of entropy when such a parameter is varied can be followed if the parameter can be coupled to a suitable force which performs measurable work [9]. More sophisticated constraints can be realized in computer experiments, such as single cell occupancy [1,2,16]. However, other constraints, only implied by their indirect e4ect [17], must be e4ective in the case of glasses and amorphous solids, keeping the systems practically unchanged in time. Denoting the particular set of constraints by c, we have strictly (E) ≡ (E; c), (S ≡ S(E; c)), and c is kept constant in the di4erentiations in (1) and (5). Since both E and (-E)2 are known functionals of molecular distributions and interaction potentials [18–20], one can integrate (1) to obtain the entropy if the structure of the system can be characterized by a small set of parameters and a path can be found in this parameter space, connecting the given state to a reference state of known entropy. Such a path is indeed provided by a modeling of molecular distributions by parametrized analytic functions, based on a general model for structure in condensed systems [21]. It turns out that experimental radial distribution data, g(r), can be represented accurately by an analytic expression g(r) = g(r; L; D) depending on a small set of parameters (L; D). L is an underlying “local lattice” (or set of di4erent lattices) representing dominant spatial con9gurations of neighboring atoms, and D is a “structural di4usion” parameter (or set of parameters) representing the decay with distance of the coherence (matching) between atomic con9gurations at di4erent localities. In its simplest version, the model leads to the expression n 1 (r − a − a + a )2 1 g(r) = g(r; L; D) = exp − ; n (4W )3=2 4W 0 =
; =1
(6) where the -sum extends over the positions a of all unit cells of L, the -sums run over all (n) points in a unit cell, and is the particle density in L. The brackets 0 denote averaging over all the orientations of L, or of r: W = W (r) has the asymptotic forms W ∼ D(r − ) (see the appendix). Since the methodology leading to g(r; L; D)
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also leads to speci9c expressions for higher order distributions [22] in terms of L and D, one can view the set (L; D), having been chosen optimally to represent g(r) data, as uniquely specifying the structure (macrostate) of the system, and providing the desired parameter space determining both E and (-E)2 . Now, when D = 0, the right-hand side (rhs) of (6) becomes practically a sum of -functions representing g(r) of a perfect lattice structure L. This is a completely ordered system [23] whose entropy must be S(EL ) = 0, where EL is the lattice energy. Hence, integrating (1) along the path 0 6 D 6 D;
L 9xed ;
(7)
one obtains the entropy S(E) = S(L; D) of the system with the given structure (L; D). Applying the above outlined method, the entropy of a model liquid and solid amorphous metal was evaluated [10] using a given radial distribution g(r; L; D) with L = FCC and a typical liquid D as structure data. The interaction between the ions was assumed to be a Coulomb potential, including the interaction with a background of opposite charges, and a very short range strong repulsive pair potential was added according to a scheme devised by Born [24]. The energy was expressed as a function of L and D by 1 E = E(L; D) = N u(r)g(r; L; D) d 3 r ; (8) 2 where u(r) is an e4ective pair potential. Because of the cohesive forces, (8) is practically the total energy at liquid and lower temperatures. The kinetic energy of thermal motion of the atoms is then negligible compared to the interaction energy and the pressure required to keep the system in a condensed state (which is the vapor pressure of the system) is practically small and vanishes in the limit Tth → 0. On the other hand, in a repulsive system, where the forces between the particles are purely repulsive, a positive pressure is required to counteract the kinetic pressure of the atoms and to con9ne the system to a 9nite volume V , down to the Tth → 0 limit. Thus, the foregoing scheme for evaluating the entropy from structure data requires modi9cations taking into account the positive kinetic energy in the Tth → 0 limit. This is illustrated in the following by the extreme case of a hard sphere system, where the interaction energy (8) vanishes. We will make use of a quantum cell version, related to the given g(r) data, to evaluate E and (-E)2 and will apply it to obtain the entropy of a strongly repulsive system as outlined above. The g(r) data we choose are typical of an amorphous solid (see Fig. 1), whose atoms are thermally practically unexcited, i.e., the solid is at Tth = 0, but at a nonzero canonical temperature, T ¿ 0. For glass, S(Tth = 0) ¿ 0, and the energy E(Tth = 0) is not the energy of an ordered solid. 2. Quantum cell model and the structure of a glassy repulsive system In a repulsive system, where the forces between the particles are purely repulsive, the pressure required to con9ne the system to a 9nite volume V is positive and does not vanish in the Tth → 0 limit. Contrary to the classical ideal gas whose energy
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Fig. 1. Radial distribution g(r) = g(r; L; D): solid line, D = 0:006; dashed line, D = 0:010; dotted line, D = 0:014; (L = FCC; r0 = 1:3714 (see the appendix)).
E = (3=2)NkB T vanishes at T = 0, quantum mechanics requires a positive, volume and structure dependent “zero point” energy which maintains a positive internal pressure. These properties of the repulsive system are illustrated by a model of a purely repulsive system with only exclusion forces between particles, e.g. a system of hard spheres (HS). Such a system has zero potential energy and its total energy is purely kinetic. Yet, in a dense packed state the exclusion forces impose strong correlations between molecular positions. In the low temperatures quantum regime, these correlations are manifested by certain complex boundary conditions on the 3N dimensional wave function (rN ) of the N free particles. Finding the form of or even its energy eigenvalues is an insoluble many body problem. Hence, to enable explicit calculations, we make use of a cell model type approximation, assuming that each particle moves independently within a cell with 9xed boundaries. We consider hard spheres of diameter and take as a measure of the linear size of a cell the nearest neighbor distance, a, for a chosen lattice L con9guration, at a given v = −1 . a is related to v by v = cL a3 , where√cL is the volume (in units of a3 ) of a unit cell of L. For L = FCC; HCP, cL = 1= 2. At close packing its volume per particle is vc = cL 3 . The linear motion of a particle is restricted practically to an interval of length lf ≈ a − , the “free length” of a particle, which is distributed according to some probability density pf (l) depending on the spatial arrangement of the particles. In a disordered phase the l’s are distributed
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continuously over the range 0 ¡ l ¡ ∞. In an ordered phase all particles have the same l = lf and pf (l) = (l − lf ). Ignoring minor re9nements relating to the shape of the cell, we assume that each particle moves independently along each of the three Cartesian directions. This amounts to representing the particle cell by a cube. By assumption, decomposes into a product of one-particle wave functions, each being a product of three one-coordinate wave functions. Thus, N (rN ) = (9) i (ri ); i (r) = ’i (x)’ i (y)’i (z) i=1
with ˝ 2 d 2 ’i = )ix ’i (10) 2m d x2 being the wave equation for translational motion within an interval − l2i ¡ x ¡ l2i . The solutions ’i to (10), satisfying the boundary conditions ’(±li =2) = 0, are even and odd standing waves: 1; 3; : : : cos nx ; n= ; (11) ’i = ’(x; n; li ) = A 2; 4; : : : sin li 2 h )ix = )x (n; li ) = n2 : (11a) 8ml2i (The case n = 0, with ’(x) ≡ 0, is excluded). We then have per degree of freedom h2 : (12) )ix = )x (n; li ) = )1i n2 ; where )1i = 8ml2i The ground state of particle i is )i = 3)1i . The pairs (n; li ) specify “internal states” of particle i : li is the linear size of the cell available to the free motion of the particle and n is the quantum number of its energy level. The particles are distributed over all possible states according to two distribution laws: the probability density pf (l) and, 2 given l = li , the Boltzmann distribution law for energy levels, pn (th ; li ) ˙ e−th )1i n , where th is the reciprocal temperature of a heat bath. The energy /uctuations of the system,
2 2 )i )j − )i ; (13) (-E) = −
ij
i
can now be evaluated following closely the procedure applied in Section 5 of Ref. [10]. Note that the angular brackets in (13) refer to averaging over both the internal states of each particle and over the con9gurations of the N particles. Introducing the local energy density )(r) = )i (r − ri ) ; (14) i
and performing a mean 9eld type partial averaging over the internal states of particles, all particles being identical, we write )(r) J = )J (r − ri ) = )(r) J ; (15) i
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where )J =
377
)i =N
(15a)
i
is the average energy per particle, and (r) is the local particle density. Here )i denotes an average over all the states (n) of particle i (see Eqs. (21) and (22) below). Spatial averaging of (15) produces )(r) J = ). J Rewriting (13) in terms of the partially averaged densities (15), we must separate out from the sums the terms with i = j. The latter produces additional contributions to the energy /uctuations. Thus, with -)(r) J = )(r) J − )(r) J = -(r))J ; we have
-)(1)J )(2) J d1 d2 = )J2
-(1)-(2) d1 d2 = N )J2
(16)
[G(r) − ] d 3 r ; (17)
where G(r) = G(r; t = 0) = (r) + g(r) is the van Hove correlation function at t = 0, and we obtain for the total energy /uctuations, the following generalization of (17) in Ref. [10] (-E)2 = -)(1)-)(2) d1 d2 = )J2 -(1)-(2) d1 d2 + N (-))2 (18) = N )J2 [G(r) − ] d 3 r + N (-))2 ; where (-))2 =
1 2 [)i − )i 2 ] : N i
Using the structure factor (scattering function) expression F(k) = 1 + [g(r) − 1]eik·r d 3 r ;
(19)
(20)
we can rewrite (18) as (-E)2 =N = )J2 F(0) + (-))2 :
(18 )
The 9rst term on the rhs of (18 ) represents energy /uctuations due to structural, i.e., local density /uctuations, whereas the second term represents the particles’ internal energy /uctuations. An equation similar to (18 ) was obtained in Ref. [10] for the case of cohesive systems. Both equations are based on an approximation (i.e., (15)) which relates structural energy /uctuations to density /uctuations and neglects the e4ect of higher order correlations. In both the cohesive and repulsive cases, the energy is de9ned relative to a zero energy reference state of high dilution. Given the representation (6) of g(r), the scattering function F(k) given by (20) can be evaluated analytically, thus giving F(0) with high accuracy (see Fig. 2).
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Fig. 2. Static scattering function F(k; L; D): solid line, D = 0:006, Eq. (A.5); dashed line, same D, FFT of (6); dotted line, D = 0:014, Eq. (A.5); inset: same, 0 6 k 6 1:
Averaging over a particle’s energy quantum states can be performed via the ith particle partition function, 3 ∞ 2 qi = q(th ; li ) = e−th )1i n ; (21) n=1
from which we obtain the ith particle average energy, @ log qi ; )i = )(th ; li ) = − @th
(22)
and its /uctuations (-)i )2 = )i2 − )i 2 =
@2 log qi : 2 @th
(23)
By (21) and (23), the internal energy /uctuations vanish in the Tth → 0 limit, leaving only the structural contribution to (-E)2 . Moreover, even when Tth = 0 but small, assuming the particle mass to be suQciently large so that the thermal wavelength 0th ≡
−1 2 th h =2m is smaller than a typical free length li , we have classically )i = (3=2)th , −2 2 (-)i ) = (3=2)th . Hence at low Tth (high th ) the thermal energy contributes little to the total energy /uctuations. The summations in (15a) and (19) amount to averaging over all the free lengths li . In the th → ∞ limit we then have from (21) and (22), with (12), )J = 3)1i =N = 3) (=l)2 ; ) ≡ h2 =(8m2 ) ; (24) i
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379
which is the average energy per particle at zero thermal (bath) temperature. Averaging over free lengths requires an estimate of the distribution pf (l), which we wish to express in terms of structure data, speci9cally in terms of the radial distribution g(r). For the ith particle r = li + is a typical distance between the particle and its nearest neighbors. Information on its spatial distribution can be deduced from the shape of the main (9rst) peak in g(r). Thus, we can use the following estimate: rm1 rm1 2 2 2 r g(r) dr r 2 g(r) dr ; (25) (=l) = l where l = r − and rm1 is the position of the 9rst minimum of g(r). Since for a hard spheres system g(r) = gHS (r) possesses densely distributed singularities [25] at all r ¿ , and gHS () = 0, the analytic parameterized expression (6) cannot strictly represent gHS (r) although it can approximate it to high accuracy. Moreover, since g(r) = g(r; L; D) does not vanish at r = , substituting it into (25) requires replacing the denominator l = r − in the integrand by a function l (r) which is nonzero for all 0 ¡ r ¡∞ and satis9es l (r) ∼ r − for r, and l (r) → 0 for r → 0. We choose for l (r) a function proportional to the function W (r) in (6), the latter determining the rate of decrease of g(r) to zero with decreasing r (see the appendix), and perform the numerical integration in (25) replacing the lower integration limit by zero. Thus we obtain (=l)2 and by (24) the energy E as functions of D (and L) over a whole range of D values (see Fig. 3). We have chosen for proportionality factor in l (r) a value such that in the D = 0 limit the exact value (=l)2 = [=(a − )]2 = [=l (a)]2 is obtained. We choose a density = rc corresponding to a random close packed system [26,27], with a density √ fraction rc =c = vc =vrc = 0:859, i.e., a packing fraction 1rc = 0:636, where vc = 3 = 2 is the volume per particle of the ordered close packed system.
3. Evaluating the entropy of a strongly repulsive glassy system from structure data We can follow now the scheme presented in Ref. [10] to evaluate the entropy of a strongly repulsive system with a given structure by integrating (1). Taking into account the boundary conditions (asymptotic behavior) lim S(E) = 0;
E→EL
lim (E) = ∞;
E→EL
a formal integration of (1) gives E E∞ dE S(E) = d EJ : (-E)2 EL EJ
lim (E) = 0;
E→E∞
lim 2 (-E)2 (E) = 0 ;
E→EL
(26)
(27)
In (26) the 9rst two conditions specify the properties of the singular point of S(E) at the ordered (ground) state L, with energy EL , which follow from the Third Law. The third condition is the assumption of an in9nite temperature at the highest (possibly
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2 Fig. 3. Energy )=(3) J ) ≡ (=l) as function of D.
in9nite) energy, E∞ . The last condition follows from a general asymptotic behavior of heat capacities (de9nable only through the entropy in the case of glass) C=
@E @E = − 2 = 2 (-E)2 ; @T @
(28)
in the T → 0 limit. By adopting the representation g(r) = g(r; L; D), the energy E and the energy /uctuations (-E)2 , given by (18), become functions of D (and L), and the integrals in (27) can be transformed into integrals in terms of the variable D. Thus we obtain for the entropy per particle (see Ref. [10], Eq. (35 )): D∞ J D J J dD + J D) J dD ; s(D) = u(D) (29) 3(D) u(D)3( J J D D D 0 where we have put )J = )(D) J = E=N; u(D) = )(D) J − )(0), J and de9ne the functions 3 and by (-E)2 =N = D (D);
3(D) = u (D)= (D) :
(30)
Constructing rational approximants for )(D) J and (D) from their respective numerical values over an interval Da ¡ D ¡ D (omitting the small values D ¡ Da where numerical integration becomes diQcult) and using these approximants as analytic continuations into the entire range 0 ¡ D 6 D, the integrals in (29) can be performed
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381
Fig. 4. Scattering function at k = 0; F(0; D)=D as function of D.
analytically and evaluated to give the desired s(E) = s(D). By (30) is implied that (-E)2 = 0 in the limit of the ordered state, D = 0. The same applies to the low angle static structure factor, for which we put F(0) ≡ F(0; D) = D&(D)
(31)
in conformity with (18). However, &(0) = 0; (0) = 0 and limD→0 u(D)=D is 9nite. In Figs. 4 and 5 are given F(0; D) and s(D), respectively for the random close packing density fraction rc =c = vc =vrc = 0:859 (=a = 0:95056 for L = FCC). In both repulsive and cohesive /uid systems structure characteristics are determined mainly by the repulsive forces [28,29]. Thus one could choose to represent g(r) of a typical disordered strongly repulsive system by g(r; L = FCC; D = 0:006) exhibiting a second and third peak structure similar to that of the model cohesive system studied in Ref. [10]. As in the latter case, it was not possible to complete the integration of (1) without using the boundary condition = @S=@E ≈ 0 for some “maximum disorder” limit point D∞ . It was found that for 0:01 6 D∞ 6 0:025; S(D) changed only within about 10%. From Fig. 5 one 9nds the estimate S(D = 0:006) ≈ 25kB , which is about 5 times larger than the value S = 4:84kB for the cohesive system. We note that this estimate, which follows from characterization of the state of the HS system by the value D = 6 × 10−2 , falls within the range 3 × 10−2 ¡ D ¡ 8 × 10−2 , where S(D)
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Fig. 5. Entropy s(D) as function of D; D∞ = 0:015.
rises steeply from 2kB to 40kB , implying a pronounced change in structure within this range. From the foregoing results the following conclusions can be made: Considering (24) and (18 ), it follows from (27) that Planck’s constant and the particle mass, appearing only in the combination ) = h2 =(8m2 ), are completely eliminated from the entropy. Thus, quantum e4ects enter only through the dependence of the wave function (11) and its energy eigenvalues (11a) on the boundary conditions imposed by the intermolecular exclusion forces and structure of the hard sphere system. Another striking feature of the repulsive system, contrary to the cohesive system [10], is the decrease of the energy function E(D) with increasing D (see Fig. 3). This is a direct consequence of the asymmetry of the 9rst peak of g(r), which gives more weight to larger r values in the evaluation of (=l)2 in (24). Since S(D) is increasing with D, it follows that the structural temperature T = (@S=@E)−1 is negative and the ordered state S = 0, i.e., D = 0, has the highest energy.
4. Energy and pressure equations of state at Tth = 0 Contrary to the entropy, the energy and the pressure are proportional to ) . Near close packing at some volume vc which depends on the particular set of constraints c in
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the glass sample, they also depend strongly on the available free length. An estimate of the dense packed energy and pressure can be obtained from (24), by replacing (=l)2 by (=lf )2 , where lf = a − is an e4ective free length. Putting v = cL a3 ; vc = cL 3 , one obtains 3 2 d(a=) @)J )J = 3) ; p= − = 6) : (32) @v lf dv lf Near close packing, dv=v = 3(da=a) and v − vc ≈ dv; lf ≈ da. Hence da lf ≈ ; ≈ v − vc dv 3vc and
)J = 27)
vc v − vc
3vc ≈ lf v − vc
2
;
pvc ≈ 54)
(33)
vc v − vc
3 :
(34)
Eqs. (32) and (34) follow directly from the introduction of quantum states, the resulting dependence on v−vc being in contrast to the dense packed limit of classical cell theory, vc : (35) )J ∼ (3=2)kB T; pvc = 3kB T v − vc 5. Conclusion The outlined method for evaluating the entropy via (1) makes use of a special choice of parameterization of the structure (g(r)) based on an ordered state L as a reference state. This parameterization provides a path in a virtual structure space {L; D} connecting the observed state of a system to a state L of zero entropy. Thus integration of (1) can be performed along this path. Yet the actual implementation of the method has required a distinction between cohesive systems, namely systems whose vapor pressure is zero in the zero temperature limit and whose energy is thus predominantly potential energy, versus repulsive systems, which are maintained under a positive external pressure, hence possess predominantly kinetic energy in this limit. An extreme case of repulsive systems is a system of particles with purely exclusion interactions, where the energy is only kinetic. Using the particular HS cell model for this extreme case, it was possible to get an estimate of the structural entropy. The model illustrates how the low lying quantum states determine the constants of integration of (1). The semiclassical formula for the entropy, based on the above formulated cell model, S = NkB [3 log(lf =0th ) + 32 ], gives only the contribution of thermal motion to the entropy, not including the contribution of structural disorder. Even as such it is applicable only as long as the free length lf 0th and fails for larger thermal wave lengths 0th ¿ lf , giving even a negative entropy for 0th 1:65lf . The classical kinetic energy of a HS system is independent of the structure, and only through quantum mechanics does it become related to structure, making it possible to apply (1) to the evaluation of the entropy. At very low ambient temperature, the quantum mechanical
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expression for the entropy becomes independent of Planck’s constant ˝ and the particle mass m as a result of the wave function becoming dependent mainly on the geometry of boundary conditions. The characteristic wavelength of becomes then the particle “free length”, 0 ≈ lf , which is related only to particle diameter and interparticle distance, while ˝ and m enter only through 0th , which disappears from the expression for in the Tth → 0 limit. Considering the decrease of the entropy function with increasing energy, states of negative temperature are known to exist in paramagnetic systems (see Ref. [11, pt. I, #73]) where nuclear spins (magnetic moments) which are weakly coupled to the lattice degrees of freedom, are subject to a sudden reversal of a magnetic 9eld and relax only slowly to an equilibrium state. Similarly we can assume that in the present glass model of repulsively interacting particles possessing only zero point kinetic energy, such a state cannot be realized for a long time and eventually some structural repulsive energy is transferred to excite particle motion, accompanied by increasing structural disorder. Noting that strongly repulsive systems can be realized only at high external pressure, the implied transition from ordered to amorphous state for the dense HS system at zero ambient temperature can serve as a model for the observed phenomenon of high pressure induced amorphization, in particular at low temperatures, as e.g. the transition of ice at T = 77 K and p ∼ 10 kbar from crystalline to amorphous ice [30], or the crystalline to amorphous transition of silica [31] at T = 300 K and p = 25–35 kbar. Acknowledgements The present article resulted thanks to a remark by Prof. Benny Gerber who noted the inconsistency of applying the scheme for evaluating the entropy of a cohesive system to a hard sphere system. Appendix. Analytic representation of the radial distribution and the static scattering function The expression (6) holds strictly only for r = 0, but can be interpreted for all r as the static van Hove correlation function G(r) ≡ G(r; t = 0) = (r) + g(r), which includes also self-correlations of particle densities, if W (r = 0)Daa2 , where a is the nearest neighbor distance in L. Applying the Poisson sum transformation formula to (6), we obtain an equivalent representation as a sum over all points b5 of the reciprocal lattice to L: 2 1 2 G(r) = C5 e−Wb5 eib5 ·r 0 ; C5 = eib5 ·a : (6 ) n 5
Expressions similar to (6) (but not identical to it) have been repeatedly used [32,33] to reproduce g(r) data. Expressions similar to (6 ) are used as well [34,35], mainly to
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obtain good 9ts to scattering data in the low k range (see below). For the numerical evaluation of g(r), the following specialization of W (r) can be made. Because of exclusion e4ects, we expect g(r) = 0 to hold to a good approximation within an entire domain 0 6 r 6 , where is a measure of molecular diameter. This is achieved by choosing for W (r) a function fast decreasing to zero for r 6 . For practical reasons of convergence at small r we choose for W the particular form r W = W0 + Dr0 ! ; (A.1) r0 where !(x) = !1 (x) = log cosh x;
(A.2)
with r0 = r = =log 2. Another form used was [21]
!(x) = !2 (x) = 1 + x2 − 1 ;
(A.3)
with r0 = . Both choices of !(x) have the asymptotic form !(x) = 12 x2 + O(x4 );
x→0;
but for x→∞ !1 (x) = x − log 2 + O(e
−2x
);
1 : !2 (x) = x − 1 + O x
Both forms of !(x) have the advantage over an earlier choice [36] in that they ensure the requirement of a positive scattering function [37,38], F(k) ¿ 0, for all k, but (A.2) has been found preferable since it produces a better 9t to g(r) at the steep rise on the small r side of the 9rst peak. We also choose l (r) = r !1 (r=r ) to represent the “free length” function discussed in Section 2. Writing (20) in the form F(k) = (G(r) − )eik·r d 3 r ; (A.4) and substituting (6 ) into (A.4), using (A:1) with either (A.2) or (A.3), we can obtain F(k) in closed form. With (A.2) we obtain 2 F(k) = r03 C5 e−W0 b5 [K(05 ; u; u5 )] ; (A.5) 5
with K(05 ; u; u5 ) = 4
I (05 ; u − u5 ) − I (05 ; u + u5 ) 2uu5
(A.6)
and the dimensionless parameters 05 = Db25 r0 ; u5 = b5 r0 ; u = kr0 , where [39] ∞ 0 u 0 u −0!1 (x) 0−2 ; I (0; u) ≡ e cos ux d x = 2 B +i ; −i 2 2 2 2 0 where B(x; y) = <(x)<(y)=<(x + y) is the Beta function. The series in (A.5) converges for all k values, being especially useful for evaluating F(k) at small k, and gives
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accurate results for k up to about the position of the second peak of F(k). However, beyond it F(k) deviates increasingly from the correct value and goes to zero when k → ∞ (see Fig. 2). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
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