4.1 Virial expansion of imperfect gas

Transcription

1 4 CHAPTER 4. INTERACTING FLUID SYSTEMS 4.1 Virial expansion of imperfect gas A collection of identical classical particles interacting via a certain potential U may be described by the following Hamiltonian H = i p i m + U({r i}), (4.1.1) where m is the particle mass and r i the position vector of the i-th particle. It is often assumed that U may be decomposed into binary interactions as U = i,j φ(r ij ), (4.1.) where i, j denotes the pair (disregarding the order) of particles i and j, r ij = r i r j, and φ is the potential of the binary interaction. If we discuss a spherically symmetric interaction, then φ ij φ(r ij ) = φ( r ij ) with a slight abuse of the function symbol. How realistic is the binary interaction description? If the molecule is not simple and if the phase is dense, it is known that three-body interactions are very important. However, it is also known that we could devise an effective binary interaction 3 that incorporates approximately the many-body effects. In this case the binary potential parameters are fitting parameters to reproduce macroscopic observables (thermodynamic and scattering data). 4 Here, three-body interaction does not imply simultaneous binary interactions among three particle, but a genuine three-body interaction in the sense that even the intereaction between two particles is modified by the presence of the third particle. 3 Later we will discuss the potential of mean force. Do not confuse this effective potential and the effective binary interaction being discussed here. In the present case, the effectiveness implies to write truly many-body interactions (approximately) in terms of binary interactions. The potential of mean force we will discuss later is the potential of the effective force between a particular pair of particles surrounded by many other interacting particles. 4 How good is the effective two-body interaction potential? The two-body correlation may be fitted with an effective binary interaction, but generally, we cannot fit the three-body correlation function. See, for example, an experimental work: C. Russ, M. Brunner, C. Bechinger, and H. H. von Grünberg, Three-body forces at work: Three-body potentials derived from triplet correlations in colloidal suspensions, Europhys. Lett. 69, 468 (005). How good is the ab initio quantum mechanically obtained binary potential? For noble gases if we take into account the tripledipole interaction, fairly good results for gas-liquid coexistence curve seems obtained. See A. E. Nasrabad and U. K. Deiters, Prediction of thermodynamic properties of krypton by Monte Carlo simulation using ab initio interaction potentials, J. Chem. Phys. 119, 947 (003).

2 4.1. VIRIAL EXPANSION OF IMPERFECT GAS 5 We already know from (..7) that the classical canonical partition function Z can be written as a product of the classical ideal gas partition function Z ideal and the configurational partition function Q, where Q = e β P i,j φ ij, (4.1.3) V is the system volume, and V is the average (..10) over the configuration space. If statistical mechanics is correct, all the phases (for which quantum effects may be ignored), gas phase, liquid phase, solid phase, etc., must be in Q. Isotope effects Isotope effects show up in spin-statistics relation and in mass difference. The spinstatistics relation affects only low temperature properties of diatomic molecules. Therefore, the effect due to mass difference is the only remaining isotope effect in most cases. There are two effects, modifying the de Broglie wave length and modifying the interaction potential (through changing the effective mass of electrons). The ionic potential is of the order of 10 ev, so H-D mass difference could affect it and modify the interaction potential to the extent that cannot be totally ignored relative to k B T. In any case, however, unless the H-D effect is involved, isotope effects are usually very small for equilibrium properties around room temperature. For low molecular weight compounds, the isotope effect is at most 1 or K for phase transition temperatures. 5 However, the replacement of H with D in polymers can have a large effect. For example, the phase separation temperatures could change by 10 K. Polystyrene and D-polystyrene melts cannot mix. Thus, isotope effect may be amplified by many-body effects. Incidentally, heavy water disrupts the spindle to inhibit cell division, so seeds cannot germinate with heavy water; also it can cause male infertility. The difference in de Broglie thermal wavelength can change the structure of condensed phases. For example, even if the interaction potential is the same, lighter isotopes tend to be delocalized. 6 In classical statistical mechanics, configurational and kinetic parts are separated, so unless there is a mass effect on the interaction potential, no isotope effect can be explained. Therefore, without modifying the potential to take account of the delocalization effect, classical statistical mechanics cannot explain the effect of de Broglie wavelength difference. In terms of the configurational partition function Q, we can write (cf. V A/ V = n A/ n = P V ) P V Nk B T = 1 n W, (4.1.4) n 5 For example, even for water the effect is small: the triple point: H O is 0.01 C, D O is 3.8 C; boiling point (at 1atm): H O is 100 C, D O is C. 6 A. Cunsolo, D. Colognesi, M. Sampoli, R. Senesi, and R. Verbeni, Signatures of quantum behavior in the microscopic dynamics of liquid hydrogen and deuterium, J. Chem. Phys. 13, , (005). The paper observes that ordinary hydrogen delocalizes more than heavy hydrogen in the liquid phase. V

3 6 CHAPTER 4. INTERACTING FLUID SYSTEMS where W = (1/N) log Q and n = N/V (the number density). For gas phases, our main task is to obtain the virial expansion of the equation of state: 7 P V Nk B T = 1 k k + 1 β kn k, (4.1.5) k= = 1 + B(T )n + C(T )n + D(T )n 3 +, (4.1.6) where B, C, D, are called virial coefficients. The systematic calculation of virial coefficients is not very simple, but certain general useful ideas worth remembering are used: Mayer s f, cumulant expansion, 1-PI diagrams, etc. 8 φ Fig Sketch of Mayer s f. It is short-ranged and usually between 1 and 1. 0 r f 0 r _1 Expanding W or Q in terms of the number density is roughly equivalent to expanding it in terms of interactions. Unfortunately, the binary interaction potential has a hard core due to the Pauli exclusion principle, so the binary interaction potential φ is not bounded. Therefore, we need something smaller to facilitate our expansion. Mayer introduced Mayer s f-function: f(r) = e βφ(r) 1. (4.1.7) In terms of f we can write Q = i<j e f ij L V. (4.1.8) 7 This form was introduced by Kammerlingh-Onnes in Except for somewhat simplified notations, the exposition here is based on R. Abe, Statistical Mechanics (University of Tokyo Press) [in Japanese].

4 4.1. VIRIAL EXPANSION OF IMPERFECT GAS 7 where e x 9 L = 1 + x is the linearized exponential function. Elementary calculation of the second virial coefficientq Before jumping into a systematic calculation, let us compute the second virial coefficient in an elementary fashion. Q = = 1 V N 1 V N = ( ) N V d 3 r 1 d 3 r d 3 r N (1 + f ij ), (4.1.9) i<j d 3 r 1 d 3 r d 3 r N 1 + f ij +, (4.1.10) i<j d 3 r 1 d 3 r f 1 + = ( ) N d 3 ρf(ρ) + (4.1.11) V At the last step the integration variables have been switched from r 1, r to r 1, ρ = r r 1. The integration over r 1 gives V. Therefore, W = 1 n d 3 ρf(ρ). (4.1.1) Comparing this with (4.1.6), we obtain B(T ) = 1 d 3 ρf(ρ) = 1 d 3 ρ (1 e βφ(ρ) ). (4.1.13) If the binary potential is a hard core (of diameter σ) + short-ranged attractive tail,r { 1 for ρ < σ, f(ρ) = (4.1.14) βφ(ρ) for ρ σ, so we get where we have introduced B(T ) = 3 πσ3 a = 1 σ a k B T, (4.1.15) φ(ρ)4πρ dρ, (4.1.16) which is finite, since we have assumed a short-ranged attraction. We need log Q instead of Q itself. A general technique to study the logarithm of the generating function is the cumulant expansion. Let X be a stochastic variable with all the moments well defiend. e θx is the (moment) generating function: e θx = 1 + n=1 θ n n! Xn, (4.1.17) 9 The reason to introduce e L is to use cumulant expansion explained below as conveniently as possible later.

5 8 CHAPTER 4. INTERACTING FLUID SYSTEMS where θ is a real number. Cumulants are introduced as r log e θx = n=1 θ n n! Xn C = e θx 1 C. (4.1.18) The second equality is for the mnemonics sake. X n C is called the n-th order cumulant. To compute cumulants, we must relate them to the ordinary moments. It is easy to extend the definition of cumulants to multivariate cases. The following two properties are worth remembering: (i) A necessary and sufficient condition for X to be Gaussian is X n C = 0 for all n 3. (ii) If X and Y are independent stochastic variables, then X n Y m C = 0 for any positive integers n and m. (i) is obvious from an explicit calculation. (ii) is obvious from e αx+βy = e αx e βy. (ii) is crucial in our present context. We must have an explicit formulas for multivariate cases, but this extension becomes almost trivial with the aid of a clever notation (Hadamard s notation): 10 Let A and B be D-dimensional vectors. Then, we write A B = D i=1 A B i i. (4.1.19) For (nonnegative) integer D-vector N = (N 1,, N D ) N! = D N i!. (4.1.0) i=1 With the aid of these notations, the multivariate Taylor expansion just looks like the one-variable case: f(x + x 0 ) = 1 M! f (M ) (x 0 ), (4.1.1) M where x is a D-vector and f (M ) (x) = 10 The notation is standard in partial differential equation theory. ( ) M f(x) (4.1.) x

6 4.1. VIRIAL EXPANSION OF IMPERFECT GAS 9 and the summation is over all the nonnegative integer D-vector M: M N D. The multinomial theorem reads (x x D ) n = K n! K! xk, (4.1.3) where the summation is over all nonnegative component vector K such that K K D = K 1 = n. Here, 1 = (1,, 1) (there are D 1 s). The multivariate cumulants are defined as (see (4.1.18)) Actually, what we need later is only log e θ X = e θ X 1. (4.1.4) C X M C = X M +, (4.1.5) where has all the terms with extra insertion of (i.e., the terms decomposed into the product of moments). Cumulant in terms of moments Using log(1 + x) = n n 1 xn ( 1) n, (4.1.6) we expand the LHS of (4.1.4) as log 1 + θ N N! N 0 X N = ( 1) n 1 n=1 n N 0 θ N N! X N n. (4.1.7) To expand the RHS we use (4.1.3). Therefore, the RHS of (4.1.7) becomes ( 1) n 1 n=1 n K 1=n [ n! θ N K! N! X N ] K. (4.1.8) Here, [ ] K may be a slight abuse of the notation, but now each component of K denotes how many θ N terms appear. This double summation over n and K can be rewritten as [ (K 1 1)!( 1) K 1 1 θ N X N ] K. (4.1.9) K! N! K

7 30 CHAPTER 4. INTERACTING FLUID SYSTEMS This should be compared with (4.1.4). Hence, we obtain X M = M! C P i KiN i=m ( ) K i 1 i!( 1) P 1 i Ki i K i! X N i N i! i Ki. (4.1.30) Here, the summation is over all possible decomposition of M into nonzero nonnegative integer vectors N i with multiplicity K i. We formally apply (4.1.4) to (4.1.8): log Q = log i<j e f ij L V = M 0 1 f M. (4.1.31) M! C The cumulants can be expressed in terms of the moments f N V, but thanks to the definition of e L, no moment containing the same f more than once shows up (i.e., the components of N are 0 or 1). Let us itemize several key ideas helpful to express cumulants in terms of moments. (1) There are many cumulants or moments, so a diagrammatic expression of f M is advantageous. In these diagrams, particles correspond to the vertices and f ij is denoted by a line connecting two vertices corresponding to particles i and j (See Fig for illustrations) Fig The left diagram corresponds to f 1 f 3 f 14 f 4, and the right one to f 34 f 46 f 67 f 73. There are many diagrams in these days, but the theory we are discussing here is the first systematic use of diagrammatics. () If a diagram consists of two disconnected parts, then the particle positions can be changed freely independently, so the corresponding cumulants vanish. The same is true for diagrams that can be decomposed into disjoint pieces by removing one particle (one vertex). This is because the average of f M is the average over the relative coordinates between particles; if a diagram has a hinge around which two parts can be rotated independently, the interparticle vectors belonging to these two parts are statistically independent. Therefore, only the 1-PI diagrams (= one-particle irreducible diagram; a diagram that does not decompose into disjoint parts by removing one vertex) matter.

8 4.1. VIRIAL EXPANSION OF IMPERFECT GAS 31 Fig PI diagrams and a non 1-PI diagram 1-PI diagrams NOT 1-PI (3) If M is a vector whose components are 0 or 1 and gives a 1-PI diagram, then we may identify f M C and f M V (as already announced) in the large volume limit. This may be understood as follows. Suppose the diagram corresponding to f M has m particles. Then, (let us call these m particles 1,, m) f M V = 1 dr V m 1 dr m f M. (4.1.3) The integral is of the order of V, because we may place the cluster at any position in the space. Therefore, the moment is of order 1/V m 1. Consider a decomposition M 1 + M = M. M 1 * * M Fig A decomposition of a 1-PI diagram into two subdiagrams results in at least two common vertices (with ) shared by the subdiagrams. In this example, the edge between these shared vertices belong to M 1. The original diagram is 1-PI, so the subdiagrams corresponding to M 1 (containing m 1 vertices) and M (containing m vertices) share at least two vertices. Therefore, m 1 + m m and f M 1 V f M V = f M V O[V 1 ]. (4.1.33) (4) The above consideration implies that we may ignore in the large volume limit all the cumulants f M C with M having some component(s) larger than 1. Now, (4.1.31) reads (notice that M! = 1) log Q = f M V M 0. (4.1.34)

9 3 CHAPTER 4. INTERACTING FLUID SYSTEMS The summation is over all the choices of 1-PI diagrams with no multiple edges connecting two vertices directly. If f M corresponds to a k vertex 1-PI diagram, there are ( N ) k N k ways to choose k particles. Because f M = O[1/V k 1 ], the overall contribution of such diagrams to the summation is proportional to N k /V k 1 Nn k 1. Since log Q must be extensive, this is just the right contribution from such diagrams: ( ) N log Q = f M, (4.1.35) k V,k k= D(k) where D(k) is the 1-PI silhouettes (see Fig ) with k vertices: D(k) f M = 1 V,k V k 1 d(independent relative particle coordinates) f M. (4.1.36) The sum here is over all the topologically distinguishable assignments of the particles 1,, k to the vertices of the silhouette D(k). For example, for k = 4 there are three distinct silhouettes as shown in Fig (b), and for one of them there are ways to assign 4 particles as exhibited in (a). If the silhouettes are identical, then the integrated values are identical, so we have only to count the ways for such assignments. For the example (a) there are 6 ways. 6 (a) = For example, (b) = etc., should not be double counted Fig (a) Silhouette implies the diagrams that are identical if particles are indistinguishable. (b) For k = 4 there are three different kinds of silhouettes (topologically different silhouettes), each of which have several ways (as denoted with a prefactor) of assigning 4 particles.

10 4.1. VIRIAL EXPANSION OF IMPERFECT GAS 33 Finally, we obtain the virial expansion of the free energy: 1 N log Q = n k k + 1 β k, (4.1.37) k=1 where β k = 1 d(independent relative particle coordinates)(all 1-PI k + 1 particle silhouettes ). k! (4.1.38) Silhouette implies that the number of ways counted as in Fig has been taken into account (see (4.1.41); implying that, for example, the numerical coefficients as 3 and 6 in this formula should be included). This integral is called the k-irreducible cluster integral. For example, β 1 = f 1 dr 1, (4.1.39) β = 1 f 1 f 3 f 31 dr 1 dr 13. (4.1.40) β 3 consists of three different silhouettes: β 3 = 1 (3f 1 f 3 f 34 f f 1 f 3 f 34 f 41 f 13 + f 1 f 3 f 34 f 41 f 13 f 4 ) dr 1 dr 13 dr 14. 3! (4.1.41) Diagrammatic expressions are as follows: [ ] From (4.1.37) and (4.1.4) we finally obtain the virial expansion of equation of state: P V Nk B T = 1 k k + 1 β kn k, (4.1.4) k=1 = 1 + B(T )n + C(T )n + D(T )n 3 +. (4.1.43)

11 34 CHAPTER 4. INTERACTING FLUID SYSTEMS Notice that when we compute the virial coefficients, V, N is taken with n being kept constant. The series expansion and this limit are not commutative. Therefore, from the nature of the series (4.1.43) we cannot conclude anything about the existence or absence of phase transition. 11 It is known that this series has a finite convergence radius Van der Waals equation of state Van der Waals 13 proposed the following equation of state (van der Walls equation of state: P = Nk BT V Nb an V, (4..1) where a and b are materials constants. Here, P, N, T, V have the usual meaning in the equation of state of gases. His key ideas are: (1) The existence of the real excluded volume due to the molecular core should reduce the actual volume from V to V Nb; this would modify the ideal gas law to P HC (V Nb) = Nk B T. Here, subscript HC implies hard core. () The attractive binary interaction reduces the actual pressure from P HC to P = P HC a/(v/n), because the wall-colliding particles are actually pulled back by their fellow particles in the bulk. The most noteworthy feature of the equation is that liquid and gas phases are described by a single equation. Maxwell was fascinated by the equation, and gave the liquid-gas coexistence condition (Maxwell s rule). q 11 However, this does not mean that we cannot obtain the critical temperature from the coefficients. See T. Kihara and J. Okutani, Chem. Phys. Lett., 8, 63 (1971). See Problem Virial expansion converges r Theorem [Lebowitz and Penrose] The radius of convergence of the virial expansion (4.1.4) is at least 0.89 (e βb + 1) 1 C(T ) 1, where B is the lower bound of φ. See D. Ruelle, Statistical Mechanics (World Scientific, 1999) p van der Waals biography See the following page for his scientific biography: