Advanced Topics in Equilibrium Statistical Mechanics

Transcription

1 Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson 2. Classical Fluids A. Coarse-graining and the classical limit For concreteness, let s now focus on a fluid phase of a simple monatomic substance, e.g. Argon. We can write generally that Q(N,V,T) = ν e βeν where the sum is over all quantum states ν that the electrons and nuclei of the atoms can be in. This is a hard sum to do, since it requires solving a many-body quantum mechanics problem to find the states. Instead, we use our physical intuition that the light electrons are moving much faster than the heavy nuclei, so that for a given nuclear configuration, the quantum states of the electrons are nearly at local equilibrium (Born-Oppenheimer approximation).

2 This suggests the rewriting Q= ν e βeν R e βer,i i }{{} R e βẽr = e βẽr where R is a sum over nuclear states (i.e. position and momentum treat with classical mechanics) and i e βer,i is a sum over electronic states (use quantum mechanics). The second line is a definition of Ẽ R, the effective energy (actually free energy, since it is β dependent) of just the nuclear states with the electronic states averaged out. This is an example of the notion of coarse-graining in statistical mechanics small objects (electrons, nuclei) fundamental interactions coarsegrain larger objects (atoms) effective interactions We have now made a lot of progress in getting rid of the quantum mechanics in our problem, assuming that we can evaluate the effective interactions between atoms that enter ẼR. Indeed, this is what ab initio quantum chemistry tries to do. There are now a large number of user-friendly software packages for computing the effective interactions between atoms and molecules that are very useful in deducing classical descriptions of fluids. B. Classical phase space averages What now are the classical states R that we are supposed to sum over? In classical mechanics, the state of a particle is determined by specifying its position r =(x, y, z) and momentum p =(p x,p y,p z ). (We can then figure out its future state by integrating Newton s equation F = ma = ṗ.) Thus, for an N-atom gas, R (r,...,r }{{ N } ; p },...,p {{ N } ) and R is a 6N-dimensional phase space. We expect that Q = R e βẽr C dr... dr N dp... dp N e βẽr where C is a constant prefactor. We adopt the shorthands: (r,...,r N ; p,...,p N )=(r N, p N ) 2

3 Note that we use a slightly different notation than that of Chandler. dr...dr N dp...dp N = dr N dp N Also, Ẽ R = Ẽ(rN, p N ) H(r N, p N ) where H is known as the Hamiltonian. The Hamiltonian (don t confuse with enthalpy!) is the sum of the effective kinetic energy of the atoms (nuclei) and the effective potential energy associated with their mutual interactions. We can separate the two as H(r N, p N )=K(p N )+U(r N ) Finally, what is C? There are various ways of deriving it, the simplest of which is the work out Q quantum mechanically for the special case of U =0 (non-interacting boson or fermion gas) then take the classical limit (h 0, β 0). For each momentum degree of freedom this yields a factor of /h (Planck s constant) and an overall factor of /N! to correct overcounting of configurations corresponding to different label permutations of indistinguishable particles. (See section of Chandler for details.) Thus, the canonical partition function in the classical limit is: Q(N,V,T) = N!h 3N dr N dp N e βh(rn,p N ) Notice that the integrals over particle positions are confined the the volume V of the fluid, while the momentum integrals are unrestricted in the 3N-dimensional momentum space. Expressions such as this are easily generalized to multi-component fluid systems. For example, the canonical partition function of a binary mixture of A and B atoms is Q = dr NA dr NB dp NA dp NB e βh(rn A,...,p N B ) N A!N B!h 3NA h 3NB Returning to the one-component system, it is now convenient to define a phase space distribution function f(r N, p N )= e βh(rn,p N ) dr N dp N e βh which is normalized such that dr N dp N f =. Ensemble averages in the canonical ensemble are thus defined by P (r N, p N ) dr N dp N f(r N, p N )P (r N, p N ) 3

4 for any property P (r N, p N ) of interest. For example, the average energy is E= H(r N, p N ) = dr ) N dp N fh = Q Q β as before! V,N The expression A(N,V,T) = k B T ln Q(N,V,T) remains the fundamental thermodynamic connection for this ensemble. This is about as far as we can go without specifying the form of the Hamiltonian H. Evaluating any thermodynamic property or Q seems to involve doing 6N integrals, where N 0 23! Indeed, equilibrium statistical mechanics is really all about the evaluation of high dimensional integrals. Life gets a bit easier if we think physically about the form of the effective Hamiltonian that is obtained by removing the electronic degrees of freedom. We expect K(p N )= N i= p 2 i 2m, p2 i = p 2 ix + p 2 iy + p 2 iz for the kinetic energy where m is the effective mass of the atom (m m nucleus ). Then, noting that H = N i= p 2 i 2m + U(rN ), the phase space distribution function f(r N, p N ) thus factors as { N } f(r N, p N )= φ(p i ) P (r N ) i= where and P (r N )= φ(p i )= e βu(rn ) drn e βu(rn ) e βp2 i /2m dpi e βp2 i /2m, prob. of observing system at configuration space point r N Maxwell-Boltzmann momentum distribution and where again, p i p i. The Maxwell-Boltzmann distribution gives the probability density of observing a particle with (3-d) vector momentum p in an equilibrium fluid. It is very important to note that all particles (atoms) have their momenta distributed independently, regardless of the fluid density (e.g. liquid, gas, or solid) or form of U(r N ). 4

5 Gaussian integrals, such as those appearing in the MB distribution are very important in statistical mechanics. Here we need ( ) /2 dx e 2π 2 ax2 =,a>0 a The denominator of the MB distribution is thus ( dp x dp y dp z e β(p2 x +p2 y +p2 z 2πm )/2m = β ) 3/2 Note that all three components of a particle s momentum p are themselves independently distributed ( equipartitioned ): Thus, e.g. p 2 x = dpp 2 xφ(p) = φ(p)= 3 α= g(p α) g(p x )= e βp2 x /2m (2πm/β) /2 dp x g(p x )=mk B T This implies that the mean-squared velocity component of any atom in the fluid is given by v 2 x = k B T/m and that p 2 = p 2 x + p 2 y + p 2 z =3mk B T We can now draw some conclusions about a classical fluid at equilibrium: At the same T, increasing m implies smaller average RMS velocities v 2 /2 The kinetic energy is equally partitioned among the three translational modes at equilibrium K = N p2 2m = N 2m 3 mk BT = 3 2 Nk BT x, y, z Notice that when we calculate with the MB distribution, sometimes it is handy to mix spherical polar and cartesian coordinates. For example, the average speed v = v of a molecule is v = m p = m dp p φ(p) = m dp p e βp2 /2m (2πm/β) 3/2 Since the integrand depends only on p = p, 2π π dp = dp p 2 dφ dθ sin θ 0 0 } 0 {{ } 4π 5

6 or v = 4π m(2πm/β) 3/2 0 dp p 3 e βp2 /2m =[8k B T/(πm)] /2 } {{ } 2m 2 /β 2 Finally, we can use our Gaussian integral formulae to simplify Q: Q = N!h 3N dp N e βk(pn ) dr N e βu(rn ) }{{} (2πmk BT ) /2 3 N Q = N!λ 3N T dr N e βu(rn ) where λ T h/ 2πmk B T is the so-called thermal wavelength. It is thus common to write: Q(N,V,T) = Q c (N,V,T) N!λ 3N T where Q c (N,V,T) dr N e βu(rn ) is the configurational partition function. Evidently, Q c is where the remaining work has to be done in evaluating the 3N coordinate integrals. Lecture 3 Recap: Classical Limit, canonical partition function Q = ν e βeν Q= N!λ 3N T Q c (N,V,T), Q c = dr N e βu(rn ) λ T = h/ 2πmk B T thermal wavelength P ν f(r N, p N )= P (r N ) N i= α= g(p x )= e βp2 x /2m (2πm/β) /2 3 g(p iα ),P(r N )= e βu(rn ) Q c Maxwell-Boltzmann Distribution What is the significance of λ T? Recall from quantum mechanics that the DeBroglie wavelength of a particle with momentum p, λ = h/p, is the wavelength associated with the wave mechanics picture of a quantum particle. In quantum mechanics we represent localized particles by wave packets: 6

7 where the width of the wave packet is denoted by x. This is the scale of the uncertainty in position of the quantum particle at some instant in time. The Fourier transform of such a wave function has a broad peak centered at k =2π/λ with width k p/ h x, which immediately gives the Heisenberg uncertainty principle: x p h. This principle relates the characteristic scales of position and momentum uncertainty of a quantum particle. In an equilibrium system, we could thus estimate x by computing p p 2 /2 MB k B Tm.It follows that x h/ length scale over which k B Tm λ T atomic positions are smeared With this interpretation of λ T, a reasonable way to assess the importance of quantum effects is to compare λ T to the characteristic size (or spacing in a liquid or solid) of the atoms or molecules that we wish to describe classically. Thus, quantum effects should be negligible when λ T σ, where σ is an atomic or molecular diameter. For example, argon at its triple point temperature, T = 84K, has λ T 0.30Å, whereas σ 3.5Å. Thus argon should be quite accurately described by by classical mechanics at this temperature. We will now finish this section by extending the classical limit to the grand canonical ensemble: In the grand canonical ensemble, the phase space distribution also depends on N, which fluctuates. Thus, in the classical limit: f(r N, p N ; N) = Q G (µ, V, T ) where the grand partition function can be written: Q G (µ, V, T )= N=0 N=0 e Nβµ enβµ βh(r N!h3N Q(N,V,T) }{{} N!λ 3N Q c(n,v,t) T z N N! Q c(n,v,t) N,p N ) where z eβµ is referred to as an activity. Note that this is consistent with λ 3 T the thermodynamic sense of the word. Averages follow naturally P (r N, p N ; N) = dr N dp N f(r N, p N ; N)P (r N, p N ; N) N=0 7

8 and thermodynamic properties follow (as before) from: pv = k B T ln Q G E = β ln Q ) G + µ µ,v β µ ln Q ) G β,v ) ) N = β µ ln Q G C. Intermolecular Potentials β,v = ln QG ln z The above results, while restricted to systems that obey classical mechanics, are exact. However, any calculations based on these formulae require an explicit form for the effective potential energy U(r N ) of interaction among atoms or molecules. It is often the case that the biggest limitation on theoretical calculations for fluids is obtaining an accurate representation of U(r N ). Ab initio quantum chemical methods are advancing rapidly, but many systems (hydrogen-bonding fluids, molten metals and salts) remain challenging for the purpose of parameterizing U(r N ). Most calculations on liquids and gases are based on the notion of pairadditive potentials, namely β,v N N U(r N ) 2 u(r i, r j ) = i< i j j u(r i, r j ) N(N ) 2 pairs pairs In the case of nearly spherical atoms like argon, or molecules like methane, the pair potential u depends only on the distance r ij r i r j between a pair of atoms. Thus, u(r i, r j )=u(r ij ). This would seem restrictive, but non-spherical molecules can be treated in an interaction-site model by superposing spherically symmetric potentials at different atomic sites in a molecule. For example, in the case of the nitrogen molecule we can express the potential energy as a sum of spherically symmetric site-site interactions: 8

9 N N U i< j 2 α 2 β u αβ ( r iα r jβ ), where u αβ (r) isa2 2 matrix of site-site potentials. What are reasonable forms for the pair potential u(r ij )? At short distances, neutral molecules repel strongly due to electronic overlap. The simplest model is thus {,r<d Hard-sphere u(r) = 0, r > d fluid where d is a molecular diameter. (Note that it is customary to define the zero of potential energy for isolated molecules separated by a large distance.) A problem with the hard sphere model is that while it has a fluid-solid transition, there are no attractive forces necessary to induce gas-liquid transitions. A slightly more realistic model, which describes fluids with both types of phase transitions, is {,r<d squareu(r) = ɛ, d < r < γd well 0, r>γd fluid Mathematically, however, we expect u(r) to be a continuous, smooth function. Neutral molecules at large separations have dipole-induced dipole attractions that vary as r 6. A robust form that fits experiments on argon, methane and other simple quasi-spherical molecules is [ (σ ) 2 ( σ ) ] 6 Lennard-Jones u(r) =4ɛ 6-2 potential r r 9

10 Finally, we note that effective ion-ion pseudo potentials for liquid metals, e.g. sodium or potassium, look like The pair approximation for U(r N ) is also often seriously in question in such systems! 0