Thermodynamics Interaction Studies Solids, Liquids and Gases Part 10 potx

Macroscopic thermodynamic results19, 20 showed that ZnII adsorbed on manganite was largely irreversible adsorption and desorption isotherms corresponding to the forward and backward reac

Advances in Interfacial Adsorption Thermodynamics:

Fig 8 Comparison between calculated and measured isotherms under different Cp

conditions in Cd–goethite system Lines are calculated from the Cp effect isotherm equation

0.435

1.778 C eq

   Points are adsorption data from Figure 1b

According to MEA theory, for the ideal reversible adsorption reactions, changes in Cp have

no influence on the reversibility of MEA states, and it should have no Cp effect in such systems when experimental artifacts are excluded.11, 18 For partially irreversible adsorption reactions, changes in Cp may significantly affect the irreversibility and the microscopic MEA structures, and a Cp effect should fundamentally exist in irreversible adsorption systems 11, 17Therefore, the MEA theory provided a rational explanation for the phenomena of Cp effect and non-Cp effect from the fundamental thermodynamic principle

4 Microscopic measurement of metastable-equilibrium adsorption state

It should be noted that, when the Cp effect isotherm equations are used in the modeling of practical adsorption processes, they may be totally empirical and does not imply particular physical mechanism The macroscopic adsorption behavior is fundamentally controlled by the microscopic reaction mechanism of adsorbed molecules on solid surfaces Therefore, the direct Measurement on the microstructures at solid-water interfaces is crucial to verifying the MEA principle

Macroscopic thermodynamic results19, 20 showed that Zn(II) adsorbed on manganite was largely irreversible (adsorption and desorption isotherms corresponding to the forward and backward reactions did not coincide, see Figure 9), but the adsorption of Zn (II) on δ-MnO2 was highly reversible (there was no apparent hysteresis between the adsorption and desorption isotherms, see Figure 10) This contrast adsorption behavior between the two forms of manganese oxides could be explained from the different microscopic structures

Advances in Interfacial Adsorption Thermodynamics:

Fig 11 Corner-sharing linkage (a) and interlayer structures of Zn(II) adsorbed on δ-MnO2 (b) (a) RZn–O = 2.07 Å, RMn–O = 1.92 Å, RZn–Mn = 3.52 Å (b) Squares were vacant sites,

illustration diagram adapted from Wadsley,27 Post and Appleman,28 and Manceau et al 25

Fig 12 Two types of linkage between adsorbed Zn(II) (octahedron and tetrahedron) and MnO6 octahedra on the γ-MnOOH surfaces (a) Double-corner linkage mode; (b) edge-linkage mode

Extended X-ray absorption fine structure (EXAFS) analysis showed that Zn(II) was adsorbed onto δ-MnO2 in a mode of corner-sharing linkage, which corresponded to only one Zn–Mn distance of 3.52 Å (Figure 11) However, there were two linkage modes for adsorbed Zn(II)

on manganite surface as inner-sphere complexes, edge-sharing linkage and corner-sharing linkage, which corresponded to two Zn–Mn distances of 3.07 and 3.52 Å (Figure 12) The

532

edge-sharing linkage was a stronger adsorption mode than that of the corner-sharing linkage, which would make it more difficult for the edge linkage to be desorbed from the solid surfaces than the corner linkage.20 So adsorption of Zn(II) onto manganite was more irreversible than that on δ-MnO2 This implied that the adsorption reversibility was influenced by the proportion of different bonding modes between adsorbate and adsorbent

in nature

Due to the contrast adsorption linkage mode, Zn(II) adsorbed on δ-MnO2 and manganite can

be in very different metastable-equilibrium adsorption (MEA) states, which result in the different macroscopic adsorption–desorption behavior For example, the extents of inconstancy of the equilibrium adsorption constant and the particle concentration effect are very different in the two systems Adsorption of metals on δ-MnO2 and manganite may therefore be used as a pair of model systems for comparative studies of metastable-equilibrium adsorption

5 Temperature dependence of metastable-equilibrium adsorption

Since temperature (T) is expected to affect both adsorption thermodynamics and kinetics,

the adsorption–desorption behavior may be T-dependent The adsorption irreversibility of

Zn(II) on anatase at various temperatures was studied using a combination of macroscopic thermodynamic methods and microscopic spectral measurement

Adsorption isotherm results29 showed that, when the temperature increased from 5 to 40 °C, the Zn(II) adsorption capacity increased by 130% (Figure 13) The desorption isotherms significantly deviate from the corresponding adsorption isotherms, indicating that the adsorption of zinc onto anatase was not fully reversible The thermodynamic index of irreversibility (TII) proposed by Sander et al.30 was used to quantify the adsorption irreversibility The TII was defined as the ratio of the observed free energy loss to the maximum possible free energy loss due to adsorption hysteresis, which was given by

qD ); Ceq is the solution concentration of hypothetical reversible desorption state γ (Ceq , eq

q ) CeqS and CeqD are determined based on the experimental adsorption and desorption isotherms, and are easily obtained from the adsorption branch where the solid-phase concentration is equal to qeqD

Based on the definition, the TII value lies in the range of 0 to 1, with 1 indicating the maximum irreversibility The TII value (0.63, 0.34, 0.20) decreased by a factor of >3 when the temperature increased from 5 to 40 °C This result indicated that the adsorption of Zn(II) on the TiO2 surfaces became more reversible with increasing temperature.29

EXAFS spectra results showed that the hydrated Zn(II) was adsorbed on anatase through edge-sharing linkage mode (strong adsorption) and corner-sharing linkage mode (weak adsorption), which corresponded to two average Zn–Ti atomic distances of 3.25±0.02 and 3.69±0.03 Å, respectively.29 According to the DFT results (Figure 14),13 EXAFS measured the

Advances in Interfacial Adsorption Thermodynamics:

Fig 13 Adsorption and desorption isotherms of Zn(II) on anatase at various temperatures Symbols, experimental data; solid lines, model-fitted adsorption isotherms; dashed lines, model-fitted desorption isotherms S5, S20, and S40 indicate where desorption was initiated and samples selected for subsequent EXAFS analysis Data given as mean of duplicates and errors refer to the difference between the duplicated samples

corner-sharing linkage mode at the Zn-Ti distance of 3.69 Å may be a mixture of coordinated bidentate binuclear (BB, 3.48 Å) and 6-coordinated monodentate mononuclear (MM, 4.01 Å) MEA states DFT calculated energies showed that the MM complex was an energetically unstable MEA state compared with the BB (-8.58 kcal/mol) and BM (edge-sharing bidentate mononuclear, -15.15 kcal/mol) adsorption modes,13 indicating that the

4-MM linkage mode would be a minor MEA state, compared to the BB and BM MEA state In the X-ray absorption near-edge structure analysis (XANES), the calculated XANES of BB and BM complexes reproduced all absorption characteristics (absorption edge, post-edge absorption oscillation and shape resonances) from the experimental XANES spectra (Figure 15).13 Therefore, the overall spectral and computational evidence indicated that the corner-sharing BB and edge-sharing BM complexation mode coexisted in the adsorption of Zn(II)

on anatase

As the temperature increased from 5 to 40 °C, the number of strong adsorption sites (edge linkage) remained relatively constant while the number of the weak adsorption sites (corner linkage) increased by 31%.29 These results indicate that the net gain in adsorption capacity and the decreased adsorption irreversibility at elevated temperatures were due to the increase in available weak adsorption sites or the decrease in the ratio of edge linkage to corner linkage Both the macroscopic adsorption/desorption equilibrium data and the molecular level evidence indicated a strong temperature dependence for the metastable-equilibrium adsorption of Zn(II) on anatase

534

Fig 14 Calculated Zn(II)–TiO2 surface complexes using density functional theory: (a)

dissolved Zn(II) with six outer-sphere water molecules; (b) monodentate mononuclear (MM); (c) bidentate binuclear (BB); (d) bidentate mononuclear (BM) Purple, red, big gray, small gray circles denote Zn, O, Ti, H atoms, respectively Distances are shown in

angstroms

Advances in Interfacial Adsorption Thermodynamics:

9660 9680 9700 9720 9740 0.6

1.2 1.8

4-coord BB exp pH=6.3 exp pH=6.8

Photon Energy (eV)

Fig 15 Calculated XANES spectra of 4-oxygen coordinated BB and 5-oxygen coordinated

BM complex and experimental XANES spectra

6 pH dependence of metastable-equilibrium adsorption

According to MEA theory, both adsorbent/particle concentration (i.e., Cp) and adsorbate concentration could fundamentally affect equilibrium adsorption constants or isotherms when a change in the concentration of reactants (adsorbent or adsorbate) alters the reaction irreversibility or the MEA states of the apparent equilibrium On the other hand, a general theory should be able to predict and interpret more phenomena To test new phenomenon predicted by MEA theory can not only cross-confirm the theory itself but also provide new insights/applications in broadly related fields The influence of adsorbate concentration on adsorption isotherms and equilibrium constants at different pH conditions was therefore studied in As(V)-anatase system using macroscopic thermodynamics and microscopic spectral and computational methods.14, 31, 32

The thermodynamic results14 showed that, when the total mass of arsenate was added to the TiO2 suspension by multiple batches, the adsorption isotherms declined as the multi-batch increased, and the extent of the decline decreased gradually as pH decreased from 7.0 to 5.5 (Figure 16) This result provided a direct evidence for the influence of adsorption kinetics (1-batch/multi-batch) on adsorption isotherm and equilibrium constant, and indicated that the influence varied with pH

According to MEA theory, for a given batch adsorption reaction under the same thermodynamic conditions, when the reaction is conducted through different kinetic pathways (1-batch/multi-batch), different MEA states (rather than a unique ideal equilibrium state) could be reached when the reaction reaches an apparent equilibrium (within the experimental time such as days).14 Equilibrium constants or adsorption isotherms, which are defined by adsorption density, are inevitably affected by the reactant concentration when they alter the final MEA states.11, 12

536

Fig 16 Adsorption isotherms of As (V) on TiO2 in 0.01mol/L NaNO3 solution at 25 °C under different pH TiO2 particle concentration is 1g/L 1-batch stands for a series of total arsenate being added to TiO2 suspension in one time, and 3-batch stands for the total

arsenate being added averagely to TiO2 suspension in 3 times every 4 hours EXAFS samples were marked by ellipse, in which the initial total As (V) concentration is 0.80 mmol/L

Advances in Interfacial Adsorption Thermodynamics:

Fig 17 DFT calculated structure of inner-sphere and H-bond adsorption products of arsenate

on TiO2: (a) monodentate mononuclear arsenate H-bonded to a H2O surface functional group occupying the adjacent surface site (MM1); (b) monodentate mononuclear arsenate H-bonded

to a -OH surface functional group occupying the adjacent surface site (MM2); (c) bidentate binuclear (BB) complex; (d) H-bonded complex Red, big gray, small gray, purple circles denote O, Ti, H, As atoms, respectively Distances are shown in angstroms

The EXAFS coordination number of CN1 and CN2 represented statistically the average number of nearest Ti atoms around the As atom corresponding to a specific interatomic distance We used the coordination number ratio of CN1/CN2 to describe the relative proportion of BB mode to MM mode in adsorption samples The CN1/CN2 was 1.6 and 2.2 for 1-batch and 3-batch adsorption samples at pH 7.0, respectively (Table 1),14 indicating that 3-batch adsorption samples contained more BB adsorbed arsenate than that of 1-batch adsorption samples This result was cross-confirmed by measuring the spectral shift of X-ray absorption near edge structure (XANES) and Fourier transform infrared spectroscopy (FTIR)

538

DFT calculation showed that the theoretical XANES transition energy of BB complex was 0.62eV higher than that of MM complex Therefore, the blue-shift of As (V) K-absorption edge observed from 1-batch to 3-batch adsorption samples suggested a structural evolution from MM to BB adsorption as the multi-batch increased (Figure 18).31

Fig 18 The first derivative K-edge XANES spectra of As (V) adsorption on anatase

The DFT calculated frequency analysis showed that the As-OTi asymmetric stretching vibration (υas) of MM and BB complexes located at 855 and 835 cm-1, respectively On the basis of this theoretical analysis, the FTIR measured red-shift of As-OTi υas vibration from 1-batch sample (849 cm-1) to 3-batch sample (835 cm-1) suggested that the ratio of BB/MM in 3-batch sample was higher than that in 1-batch sample (Figure 19).32

The good agreement of EXAFS results of CN1/CN2 with XANES and FTIR analysis also validated the reliability of the CN ratio used as an index to approximate the proportion change of surface complexation modes BB complex occupies two active sites on adsorbent surface whereas MM occupies only one For monolayer chemiadsorption, a unit surface area

of a given adsorbent can contain more arsenate molecules adsorbed in MM mode than that

in BB mode Therefore, the increase of the proportion of BB complex from 1-batch to 3-batch addition mode was shown as the decrease of adsorption density in 3-batch isotherm (Figure 16)

Table 1 showed that the relative proportion of BB and MM complex was rarely affected by

pH change from 5.5 to 7.0, indicating that the pH dependence for the influence of adsorption kinetics (1-batch/multi-batch) on adsorption isotherm was not due to inner-sphere chemiadsorption.14 The influence of pH on adsorption was simulated by DFT theory through changing the number of H+ in model clusters Calculation of adsorption energy showed that the thermodynamic favorability of inner-sphere and outer-sphere adsorption was directly related to pH (Table 2).14 As pH decreased, the thermodynamic favorability of inner-sphere and outer-sphere arsenate adsorption on Ti-(hydr)oxides increased This DFT result explained why the adsorption densities of arsenate (Figure 16) and equilibrium adsorption constant (Table 2) increased with the decrease of pH

Advances in Interfacial Adsorption Thermodynamics:

Theoretical equilibrium adsorption constant (K) of calculated surface complexes (BB, MM

and H-bonded complexes in this adsorption system) that constructed real equilibrium adsorption constant were significantly different in the order of magnitude under the same thermodynamic conditions (Table 2) The theoretical K were in the order of BB (6.80×1042)

>MM (3.13×1039) >H-bonded complex (3.91×1035) under low pH condition, and in the order

of MM (1.54×10-5) > BB (8.72×10-38) >H-bonded complex (5.01×10-45) under high pH condition Therefore, even under the same thermodynamic conditions, the real equilibrium adsorption constant would vary with the change of the proportion of different surface complexes in real equilibrium adsorption

DFT results (Table 2) showed that H-bond adsorption became thermodynamically favorable (-203.1 kJ/mol) as pH decreased H-boned adsorption is an outer-sphere electrostatic attraction essentially (see Figure 17d), so it was hardly influenced by reactant concentration (multi-batch addition mode).14 Therefore, as the proportion of outer-sphere adsorption complex increased under low pH condition, the influence of adsorption kinetics (1-batch/multi-batch) on adsorption isotherm would weaken (Figure 16)

Both the macroscopic adsorption data and the microscopic spectral and computational results indicated that the real equilibrium adsorption state of As(V) on anatase surfaces is generally a mixture of various outer-sphere and inner-sphere metastable-equilibrium states The coexistence and interaction of outer-sphere and inner-sphere adsorptions caused the extreme complicacy of real adsorption reaction at solid-liquid interface, which was not taken into account in traditional thermodynamic adsorption theories for describing the macroscopic relationship between equilibrium concentrations in solution and on solid surfaces The reasoning behind the adsorbent and adsorbate concentration effects is that the conventional adsorption thermodynamic methods such as adsorption isotherms, which are

540

defined by the macroscopic parameter of adsorption density (mol/m2), can be inevitably ambiguous, because the chemical potential of mixed microscopic MEA states cannot be unambiguously described by the macroscopic parameter of adsorption density Failure in recognizing this theoretical gap has greatly hindered our understanding on many adsorption related issues especially in applied science and technology fields where the use

of surface concentration (mol/m2) is common or inevitable

Bidentate binuclear complexes

0 H2AsO4- ( H2O)12+ [Ti2(OH)4(H2O)6]4+ →

[Ti2(OH)4(H2O)4AsO2(OH)2]3+(H2O)2+ 12H2O -244.5 6.80×1042

1 [Ti2(OH)4(H2O)4AsO2(OH)2]H2AsO4- ( H2O)12+ [Ti2(OH)5(H2O)5]3+(H2O)2 + OH3+ → -( H2O)11 13.1 5.15×10-3

2 H2AsO4- ( H2O)12+ [Ti2(OH)6(H2O)4]2+ →

[Ti2(OH)4(H2O)4AsO2(OH)2]3+(H2O)2 + 2OH-(H2O)10 211.5 8.72×10-38

Monodentate mononuclear complexes

0 [Ti2(OH)4(H2O)5AsO2(OH)2]H2AsO4- ( H2O)12+ [Ti2(OH)4(H2O)6]3+ H2O + 12H2O 4+→ -225.4 3.13×1039

1-1 H2AsO4- ( H2O)12+ [Ti2(OH)5(H2O)5]3+→

[Ti2(OH)4(H2O)5AsO2(OH)2]3+ H2O + OH-( H2O)11 32.1 2.37×10-61-2 [Ti2(OH)5(H2O)4AsO2(OH)2]H2AsO4- ( H2O)12+ [Ti2(OH)5(H2O)5]2+ H2O + 12H2O 3+ → -135.6 5.72×1023

2 H2AsO4- ( H2O)12+ [Ti2(OH)6(H2O)4]2+→

[Ti2(OH)5(H2O)4AsO2(OH)2]2+ H2O + OH-( H2O)11 27.5 1.54×10-5

H-bond complexes

0 [Ti2(OH)4(H2O)6AsO2(OH)2]H2AsO4- ( H2O)12+ [Ti2(OH)4(H2O)6]3+ + 12H2O 4+ → -203.1 3.91×1035

1 H2AsO4- ( H2O)12+ [Ti2(OH)5(H2O)5]3+ →

[Ti2(OH)4(H2O)6AsO2(OH)2]3+ + OH-( H2O)11 54.4 2.96×10-10

2 [Ti2(OH)4(H2O)6AsO2(OH)2]H2AsO4- ( H2O)12+ [Ti2(OH)6(H2O)4]3+ + 2OH2+ -(H2O)10 → 252.9 5.01×10-45Table 2 Calculated ΔGads (kJ/mol) and equilibrium adsorption constant K at 25 °C of

arsenate on various protonated Ti-(hydr)oxide surfaces

Metastable-equilibrium adsorption (MEA) theory pointed out that adsorbate would exist on solid surfaces in different forms (i.e MEA states) and recognized the influence of adsorption reaction kinetics and reactant concentrations on the final MEA states (various outer-sphere and inner-sphere complexes) that construct real adsorption equilibrium state Therefore, traditional thermodynamic adsorption theories need to be further developed by taking metastable-equilibrium adsorption into account in order to accurately describe real equilibrium properties of surface adsorption

Advances in Interfacial Adsorption Thermodynamics:

7 Acknowledgment

The study was supported by NNSF of China (20073060, 20777090, 20921063) and the Hundred Talent Program of the Chinese Academy of Science We thank BSRF (Beijing), SSRF (Shanghai), and KEK (Japan) for supplying synchrotron beam time

8 References

[1] Atkins , P W.; Paula, J d., Physical Chemistry, 8th edition Oxford University Press:

Oxford, 2006

[2] Sverjensky, D A., Nature 1993, 364 (6440), 776-780

[3] O'Connor, D J.; Connolly, J P., Water Res 1980, 14 (10), 1517-1523

[4] Voice, T C.; Weber, W J., Environ Sci Technol 1985, 19 (9), 789-796

[5] Honeyman, B D.; Santschi, P H., Environ Sci Technol 1988, 22 (8), 862-871

[6] Benoit, G., Geochim Cosmochim Acta 1995, 59 (13), 2677-2687

[7] Benoit, G.; Rozan, T F., Geochim Cosmochim Acta 1999, 63 (1), 113-127

[8] Cheng, T.; Barnett, M O.; Roden, E E.; Zhuang, J L., Environ Sci Technol 2006, 40,

3243-3247

[9] McKinley, J P.; Jenne, E A., Environ Sci Technol 1991, 25 (12), 2082-2087

[10] Higgo, J J W.; Rees, L V C., Environ Sci Technol 1986, 20 (5), 483-490

[11] Pan, G.; Liss, P S., J Colloid Interface Sci 1998, 201 (1), 77-85

[12] Pan, G.; Liss, P S., J Colloid Interface Sci 1998, 201 (1), 71-76

[13] He, G Z.; Pan, G.; Zhang, M Y.; Waychunas, G A., Environ Sci Technol 2011, 45 (5),

1873-1879

[14] He, G Z.; Zhang, M Y.; Pan, G., J Phys Chem C 2009, 113, 21679-21686

[15] Nyffeler, U P.; Li, Y H.; Santschi, P H., Geochim Cosmochim Acta 1984, 48 (7),

1513-1522

[16] Dzombak, D A.; Morel, F M M., J Colloid Interface Sci 1986, 112 (2), 588-598

[17] Pan, G.; Liss, P S.; Krom, M D., Colloids Surf., A 1999, 151 (1-2), 127-133

[18] Pan, G., Acta Scientiae Circumstantia 2003, 23 (2), 156-173(in Chinese)

[19] Li, X L.; Pan, G.; Qin, Y W.; Hu, T D.; Wu, Z Y.; Xie, Y N., J Colloid Interface Sci 2004,

271 (1), 35-40

[20] Pan, G.; Qin, Y W.; Li, X L.; Hu, T D.; Wu, Z Y.; Xie, Y N., J Colloid Interface Sci 2004,

271 (1), 28-34

[21] Bochatay, L.; Persson, P., J Colloid Interface Sci 2000, 229 (2), 593-599

[22] Bochatay, L.; Persson, P.; Sjoberg, S., J Colloid Interface Sci 2000, 229 (2), 584-592

[23] Drits, V A.; Silvester, E.; Gorshkov, A I.; Manceau, A., Am Mineral 1997, 82 (9-10),

946-961

[24] Post, J E.; Veblen, D R., Am Mineral 1990, 75 (5-6), 477-489

[25] Manceau, A.; Lanson, B.; Drits, V A., Geochim Cosmochim Acta 2002, 66 (15), 2639-2663

[26] Silvester, E.; Manceau, A.; Drits, V A., Am Mineral 1997, 82 (9-10), 962-978

[27] Wadsley, A D., Acta Crystallographica 1955, 8 (3), 165-172

[28] Post, J E.; Appleman, D E., Am Mineral 1988, 73 (11-12), 1401-1404

[29] Li, W.; Pan, G.; Zhang, M Y.; Zhao, D Y.; Yang, Y H.; Chen, H.; He, G Z., J Colloid

Interface Sci 2008, 319 (2), 385-391

[30] Sander, M.; Lu, Y.; Pignatello, J J A thermodynamically based method to quantify true

sorption hysteresis; Am Soc Agronom: 2005; pp 1063-1072

542

[31] He, G Z.; Pan, G.; Zhang, M Y.; Wu, Z Y., J Phys Chem C 2009, 113 (39), 17076-17081

[32] Zhang, M Y.; He, G Z.; Pan, G., J Colloid Interface Sci 2009, 338 (1), 284-286

In practice, for example to calculate the lattice stability, the construction of the phase diagram

is to find the phase equilibria based on the comparison of the Gibbs free energies amongthe possible phases Hence, the most important factor is the accuracy and precesion of thegiven Gibbs free energy values, which are usually acquired by the experimental assessments.Once the required thermodynamic data are obtained, the phase diagram constructionbecomes rather straightforward with modern computation techniques, so called CALPHAD(CALculation of PHAse Diagrams) (Spencer, 2007) Hence, the required information forconstructing a phase diagram is the reliable Gibbs free energy information The Gibbs free

energy G is defined by

where E is the internal energy, P is the pressure, V is the volume of the system, T is the temperature and S is the entropy. The state which provides the minimum of the

free energy under given external conditions at constant P and T is the equilibrium state.

However, there is a critical issue to apply the conventional CALPHAD method in generalmaterials design Most thermodynamic information is relied on the experimental assessments,which do not available occasionally to be obtained, but necessary For example, the directthermodynamic information of silicon solubility in cementite had not been available for longtime (Ghosh & Olson, 2002; Kozeschnik & Bhadeshia, 2008), because the extremely low siliconsolubility which requires the information at very high temperature over the melting point

of cementite The direct thermodynamic information was available recently by an ab initio method (Jang et al., 2009) However, the current technology of ab initio approaches is usually

limited to zero temperature, due to the theoretical foundation; the density functional theory(Hohenberg & Kohn, 1964) guarrentees the unique total energy of the ground states only Theexample demonstrates the necessity of a systematic assessment method from first principles

In order to obtain the Gibbs free energy from first principles, it is convenient to use the

equilibrium statistical mechanics for grand canonical ensemble by introducing the grand

21

where β is the inverse temperature (kBT) −1 with the Boltzmann’s constant kB, μ i is the

chemical potential of the ith component, N i is the number of atoms The sum of ζ runs

over all accessible microstates of the system; the microstates include the electronic, magnetic,

vibrational and configurational degrees of freedom The corresponding grand potentialΩ isfound by

Towards the Authentic Ab Intio Thermodynamics 3

In principle, we can calculate any macroscopic thermodynamic states if we have the completeknowledge of the (grand) partition function, which is abled to be constructed from firstprinciples However, it is impractical to calculates the partition function of a given systembecause the number of all accessible microstates, indexed byζ, is enormously large.

Struggles have been devoted to calculate the summation of all accessible states The number ofall accessible states is evaluated by the constitutents of the system and the types of interactionamong the constituents The general procedure in statistical mechanics is nothing more thanthe calculation of the probability of a specific number of dice with the enormous number

of repititions of the dice tosses The fundamental principles of statistical mechanics of a

mechanical system of the degrees of freedom s is well summarized by Landau & Lifshitz (1980) The state of a mechanical system is described a point of the phase space represented

by the generalized coordinates q i and the corresponding generalized momenta p i, where the

index i runs from 1 to s The time evolution of the system is represented by the trajectory in

the phase space Let us consider a closed large mechanical system and a part of the entire

system, called subsystem, which is also large enough, and is interacting with the rest part of

the closed system An exact solution for the behavior of the subsystem can be obtained only

by solving the mechanical problem for the entire closed system

Let us assume that the subsystem is in the small phase volumeΔpΔq for short intervals The probability w for the subsystem stays in the ΔpΔq during the short interval Δt is

dw=ρ(p1, p2, , p s , q1, q2, , q s)dpdq, (13)whereρ is a function of all coordinates and momenta in writing for brevity ρ(p, q) Thisfunction ρ represents the density of the probability distribution in phase space, called (statistical) distribution function Obviously, the distribution function is normalized as



One should note that the statistical distribution of a given subsystem does not depend onthe initial state of any other subsystems of the entire system, due to the entirely outweighedeffects of the initial state over a sufficiently long time

A physical quantity f = f(p, q) depending on the states of the subsystem of the solvedentire system is able to be evaluated, in the sense of the statistical average, by the distributionfunction as

In addition, the Liouville’s theorem

2 Phenomenological Landau theory

A ferromagnet in which the magnetization is the order parameter is served for illustrativepurpose Landau & Lifshitz (1980) suggested a phenomenological description of phase

transitions by introducing a concept of order parameter. Suppose that the interactionHamiltonian of the magnetic system to be

∑

i,j

where Si is a localized Heisenberg-type spin at an atomic site i and J ij is the interaction

parameter between the spins Siand Sj

In the ferromagnet, the total magnetization M is defined as the thermodynamic average of the

Let us consider a situation that an external magnetic field H is applied to the system Landau’s

idea1is to introduce a function,L (m, H, T), known as the Landau function, which describes

the “thermodynamics” of the system as function of m, H, and T The minimum of Lindicatesthe system phase at the given variable values To see more details, let us expand the Ladau

function with respect to the order parameter m:

L ( m, H, T) =∑4

n

where we assumed that both the magnetization m and the external magnetic field H are

aligned in a specific direction, say ˆz When the system undergoes a first-order phase transition,

the Landau function should have the properties

∂ L

∂m

Towards the Authentic Ab Intio Thermodynamics 5

for the minima points A and B For the case of the second-order phase transition, it is required

and then the Ladau function near the critical point is

L ( m, h, t) = c1hm+d2tm2+c3hm3+b4m4, d2>0, b4>0 (25)Enforcing the inversion symmetry,L ( m, H, T) = L (−m, − H, T), the Landau function will be

Let us consider the second-order phase transition with H=0 For T > TC, the minimum of

L is at m=0 For T=TC, the Landau function has zero curvature at m=0, where the point

is still the global minimum For T < TC, the Landau function Eq (26) has two degenerate

minima at m s=m s(T), which is explicitly

as 1/t for t → 0 both for the regions of t > 0 and t <0

For the first-order phase transition, we need to consider Eq (25) with c1=0 and changing thecoefficient symbols to yield

For H=0, the equilibrium value of m is obtained as

where c=3C/4b The nonzero solution is valid only for t < t ∗ , by defining t ∗ ≡ bc2/a Let

T c is the temperature where the coefficient of the term quadratic in m vanishes Suppose t1

is the temperature where the value ofLat the secondary minimum is equal to the value at

m=0 Since t ∗ is positive, this occurs at a temperature greater than T c For t < t ∗, a secondary

minimum and maximum have developed, in addition to the minimum at m=0 For t < t1,the secondary minimum is now the global minimum, and the value of the order parameterwhich minimizesL jumps discontinuously from m=0 to a non-zero value This is a first-order

transition Note that at the first-order transition, m(t1)is not arbitrarily small as t → t −1 In

other words, the Landau theory is not valid Hence, the first-order phase transition is arosen

by introducing the cubic term in m.

Since the Landau theory is fully phenomenological, there is no strong limit in selecting orderparameter and the corresponding conjugate field For example, the magnetization is the orderparameter of a ferromagnet with the external magnetic field as the conjugate coupling field,the polarization is the order parameter of a ferroelectric with the external electric field asthe conjugate coupling field, and the electron pair amplitude is the order parameter of asuperconductor with the electron pair source as the conjugate coupling field When a systemundergoes a phase transition, the Landau theory is usually utilized to understand the phasetransition

The Landau theory is motivated by the observation that we could replace the interactionHamiltonian Eq (18)

by∑ij S i J ij  S j  If we can replace S i S j by S i  S j , it is also possible to replace  S i S j by S i  S j 

on average if we assume the translational invariance The fractional error implicit in thisreplacement can be evaluated by

ε ij=

S i S j  −  S i  S j  

where all quantities are measured for T < TCunder the Landau theory The numerator is just

a correlation function C and the interaction range r

i −rj ∼ R will allow us to rewrite ε



where f is a function of the correlation length ξ For T TC, the correlation lengthξ ∼ R, and the order parameter m is saturated at the low temperature value The error is roughly

Towards the Authentic Ab Intio Thermodynamics 7

where a is the lattice constant and d is the dimensionality of the interaction In Eq (36),

theory is self-consistent

On the other hand, the correlation length grows toward infinity near the critical point; R

ξ for t → 0 A simple arithematics yields m ∼ | t | β , where a critial exponent β is 1

2 for aferromagnet This result leaves us the error

ε R ∼ 1

| t |2β

 a R

d

which tends to infinity as t → 0 Hence, the Landau theory based on the mean-fieldapproximation has error which diverges as the system approaches to the critical point.Mathematically, the Landau theory expands the Landau function in terms of the order

parameter The landau expansion itself is mathematically non-sense near the critical point

for dimensions less than four Therefore, the Landau theory is not a good tool to investigatesignificantly the phase transitions of the system

3 Matters as noninteracting gases

Materials are basically made of atoms; an atom is composed of a nucleus and the surrounding

electrons However, it is convenient to distinguish two types of electrons; the valence electrons are responsible for chemical reactions and the core electrons are tightly bound around the nucleus to form an ion for screening the strongly divergent Coulomb potential from the

nucleus It is customary to call valence electrons as electrons

The decomposition into electrons and ions provides us at least two advantages in treatingmaterials with first-principles First of all, the motions of electrons can be decoupledadiabatically from the those of ions, since electrons reach their equilibrium almostimmediately by their light mass compared to those factors of ions The decoupling ofthe motions of electrons from those of ions is accomplished by the Born-Oppenheimeradiabatic approximation (Born & Oppenheimer, 1927), which decouples the motions ofelectrons approximately begin independent adiabatically from those of ions In practice, themotions of electrons are computed under the external potential influenced by the ions at

their static equilibrium positions, before the motions of ions are computed under the external

potential influenced by the electronic distribution Hence, the fundamental information forthermodynamics of a material is its electronic structures Secondly, the decoupled electrons ofspin half are identical particles following the Fermi-Dirac statistics (Dirac, 1926; Fermi, 1926).Hence, the statistical distribution function of electrons is a closed fixed form This featurereduces the burdens of calculation of the distribution function of electrons

3.1 Electronic subsystem as Fermi gas

The consequence of the decoupling electrons from ions allows us to treat the distributionfunctions of distinguishable atoms, for example, an iron atom is distinguished from a carbonatom, can be treated as the source of external potential to the electronic subsystem Modelling

of electronic subsystem was suggested firstly by Drude (1900), before the birth of quantummechanics He assumes that a metal is composed of electrons wandering on the positivehomogeneous ionic background The interaction between electrons are cancelled to allow us

549

Towards the Authentic Ab Intio Thermodynamics

for treating the electrons as a noninteracting gas Albeit the Drude model oversimplifies thereal situation, it contains many useful features of the fundamental properties of the electronicsubsystem (Aschcroft & Mermin, 1976; Fetter & Walecka, 2003; Giuliani & Vignale, 2005).

As microstates is indexed as i of the electron subsystem, the Fermi-Dirac distribution function

is written in terms of occupation number of the state i,

n0i = 1

where i is the energy of the electronic microstate i and μ is the chemical potential of the

electron gas At zero-temperature, the Fermi-Dirac distribution function becomes

where p is the single-particle momentum, k is the corresponding wave vector, the grand

potential in Eq (3) is calculated in a continuum limit2as

PV= 23

If we have knowledge of the single-particle energy dispersion relation, the wavenumber integral is also

replaced by an integral over energy as gV(2π)−3

d 3kF( k) → g∞

−∞d D ( )F( ), whereD ( )is the density of states.

Towards the Authentic Ab Intio Thermodynamics 9

and the chemical potential from the relation N= (∂(PV)/∂μ) TVas

3.2 Elementary excitation as massive boson gas

For the case of ions, the treatment is rather complex One can immediately raise the sametreatment of the homogeneous noninteracting ionic gas model as we did for the electronicsubsystem Ignoring the nuclear spins, any kinds of ions are composed of fully occupiedelectronic shells to yield the effective zero spin; ions are massive bosons It seems, if the systemhas single elemental atoms, that the ionic subsystem can be treated as an indistinguishablehomogeneous noninteracting bosonic gas, following the Bose-Einstein statistics (Bose, 1926;Einstein, 1924; 1925) However, the ionic subsystem is hardly treated as a boson gas.Real materials are not elemental ones, but they are composed of many different kinds ofelements; it is possible to distinguish the atoms They are partially distinguishable eachother, so that a combinatorial analysis is required for calculating thermodynamic properties(Ruban & Abrikosov, 2008; Turchi et al., 2007) It is obvious that the ions in a materialare approximately distributed in the space isotropically and homogeneously Such phases

are usually called fluids As temperature goes down, the material in our interests usually

crystalizes where the homogeneous and isotropic symmetries are broken spontaneously andindividual atoms all occupy nearly fixed positions

In quantum field theoretical language, there is a massless boson, called Goldstone boson,

if the Lagrangian of the system possesses a continuous symmetry group under which thethe ground or vacuum state is not invariant (Goldstone, 1961; Goldstone et al., 1962) Forexample, phonons are emerged by the violation of translational and rotational symmetry ofthe solid crystal; a longitudinal phonon is emerged by the violation of the gauge invariance

in liquid helium; spin waves, or magnons, are emerged by the violation of spin rotationsymmetry (Anderson, 1963) These quasi-particles, or elementary excitations, have known inmany-body theory for solids (Madelung, 1978; Pines, 1962; 1999) One has to note two facts:(i) the elementary exciations are not necessarily to be a Goldstone boson and (ii) they are not

551

Towards the Authentic Ab Intio Thermodynamics

necessarily limited to the ionic subsystem, but also electronic one If the elementary excitationsare fermionic, thermodynamics are basically calculable as we did for the non-interactingelectrons gas model, in the beginning of this section If the elementary excitations are(Goldstone) bosonic, such as phonons or magnons, a thermodynamics calculation requiresspecial care In order to illustrative purpose, let us see the thermodynamic information of asystem of homogeneous noninteracting massive bosons.

The Bose-Einstein distribution function gives the mean occupation number in the ith state as

n0i = 1

Since the chemical potential of a bosonic system vanishes at a certain temperature T0, a specialcare is necessary during the thermodynamic property calculations (Cornell & Wieman, 2002;Einstein, 1925; Fetter & Walecka, 2003) The grand potential of an ideal massive boson gas,where the energy spectrum is also calculated as in Eq (41), is

with the consideration of the fact ≥0

In the classical limit T → ∞, or β → 0, for fixed N, we have

Towards the Authentic Ab Intio Thermodynamics 11

The classical chemical potentialμ cis now calculated as

βμ c=ln

⎡

⎣ N gV



ζ3 2

2

whereΓ and ζ are Gamma function and zeta function, respectively For μ=0 andβ > β0, the

integral in Eq (53) is less than N/V because these conditions increase the denominator of the

integrand relative to its value atβ0

The breakdown of the theory was noticed by Einstein (1925) and was traced origin ofthe breakdown was the converting the conversion of the summation to the integral of theoccupation number counting in Eq (53) The total number of the ideal massive Bose gas iscounted, using the Bose-Einstein distribution function Eq (49), by

of the integrand in Eq (53) The number density

of the Bose particles with energies >0 is computed by Eq (53) to be

3

1

β

52



ζ

52

The constant-volume heat capacity forβ > β0becomes

C V= 52

⎡

⎣Γ



5 2



ζ5 2



Γ3 2



ζ3 2

2√

2

4π2Γ

52



ζ

52



m3g

It is interesting to find that the pressure approaches to zero as temperature goes to zero,

i.e β → ∞ In other words, the Bose gas exerts no force on the walls of the container at

T = 0, because all of the particles condensate in the zero-momentum state The pressure

is independent of the number density N/V, depending only on temperature β The two

different summations above and below the temperatureβ0lead us that the heat capacity as afunction of temperature has discontinuity in its slope (Landau & Lifshitz, 1980) as



ζ3 2

This implies that a homogeneous massive ideal Bose gas system exhibits a phase transition at

β0without interaction This phenomenon is known as the Bose-Einstein condensation A good

review for the realization of the Bose-Einstein condensation is provided by Cornell & Wieman(2002)

3.3 Elementary excitations as massless boson gas

As we stated previously, some elementary excitations emerged by the spontaneous symmetry

breaking are massless bosons (Goldstone, 1961; Goldstone et al., 1962) as well as gauge bosons, which are elemental particles, e.g. photons, arosen from the fundamental interactions,

electromagnetic fields for photons, with gauge degrees of freedom Whether a boson is a

Goldstone boson or a gauge one, the procedure described above is not appliable to themassless character of the boson, because its energy spectrum is not in the form of Eq (41).One has to remind that a masselss boson does not carry mass, but it carries momentum andenergy The energy spectrum of a massless boson is given by

and the frequency ω is obtained by the corresponding momentum p = ¯hk through a

dispersion relation

The number of bosons N in the massless boson gas is a variable, and not a given constants

as in an ordinary gas Therefore, N itself must be determined from the thermal equilibrium

condition, the (Helmholtz) free energy minimum(∂F/∂N)T,V =0 Since(∂F/∂N)T,V =μ,

this gives

Towards the Authentic Ab Intio Thermodynamics 13

In these conditions, the mean occupation number is following the Planck distribution function(Planck, 1901)

n0k= 1

which was originally suggested for describing the distribution function problem ofblack-body radiations Note that the Planck distribution function is a special case of theBose-Einstein distribution function with zero chemical potential

Considering the relation Eq (4) and the condition Eq (69), the grand potential of a masslessBose gas subsystem becomes the same as the Helmholtz free energy The details of thethermodynamic properties are depending on the dispersion relation of the bosons Photons

are quantized radiations based on the fact of the linearity of electrodynamics,3so that photons

do not interact with one another The photon dispersion relation is linear,

which propotional to the fourth power of the temperature; Boltzmann’s law The constant

volume heat capacity of the radiation is

The procedure for the photonic subsystem is quite useful in computing thermodynamicsfor many kinds of elementary exciations, which are usually massless (Goldstone) bosons

in a condensed matter system It is predicted that any crystal must be completelyordered, and the atoms of each kind must occupy entirely definite positions, in a state

of complete thermodynamic equilibrium (Andreev & Lifshitz, 1969; Leggett, 1970) It iswell known that the ions vibrate even in the zero temperature with several vibrationmodes (Aschcroft & Mermin, 1976; Callaway, 1974; Jones & March, 1973a; Kittel, 2005;Landau & Lifshitz, 1980; Madelung, 1978; Pines, 1999) The energy spectrum of a phonon

in a mode j and a wavevector q contains the zero vibration term

iq=



n jq+12



where n jq is the occupation number of the single-particle modes of j and q The corresponding

partition function is then written as

asD ( ω) = Nδ(ω− ω0), where delta function is centered atω0 This model is, in turn, useful

Towards the Authentic Ab Intio Thermodynamics 15

to treat high temperature phonon thermodynamics The thermal energy of the noninteractingphonon system is

of the heat capacity as

whereΘDis known as the Debye temperature

4 Matters as interacting liquids

The discussions in Sec 3 has succeed to describe the thermodynamic properties of materials

in many aspects, and hence such descriptions were treated in many textbooks However,the oversimplified model fails the many important features on the material properties

One of the important origin of such failures is due to the ignorance of the electromagnetic

interaction among the constituent particles; electrons and ions, which carry electric charges.However, the inclusion of interactions among the particles is enormously difficult to treat

To date the quantum field theory (QFT) is known as the standard method in dealing withthe interacting particles There are many good textbooks on the QFT (Berestetskii et al.,1994; Bjorken & Drell, 1965; Doniach & Sondheimer, 1982; Fetter & Walecka, 2003; Fradkin,1991; Gross, 1999; Itzykson & Zuber, 1980; Mahan, 2000; Negele & Orland, 1988; Parisi, 1988;Peskin & Schroeder, 1995; Zinn-Justin, 1997) in treating the interacting particles systematically

in various aspects In this article, the idea of the treatments will be reviewed briefly, instead

of dealing with the full details

The idea of noninteracting particles inspires an idea to deal with the electronic subsystem

as a sum of independent particles under a given potential field (Hartree, 1928), with theconsideration of the effect of Pauli exclusion principle (Fock, 1930), which it is known asthe exchange effect This idea, known as the Hartree-Fock method, was mathematicallyformulated by introducing the Slater determinant (Slater, 1951) for the many-body electronicwave function The individual wave function of an electron can be obtained by solving eitherSchrödinger equation (Schrödinger, 1926a;b;c;d) for the nonrelativistic cases or Dirac equation(Dirac, 1928a;b) for the relativistic ones.4

Since an electron carries a fundamental electric charge e in its motion, it is necessary to deal

with electromagnetic waves or their quanta photons Immediate necessity was arosen inorder to deal with both electrons and photons in a single quantum theoretical framework

in consideration of the Einstein’s special theory of relativity (Einstein, 1905) Jordan & Pauli(1928) and Heisenberg & Pauli (1929) suggested that a new formalism to treat both the

4 The immediate relativistic version of the Schrödinger equation was derived by Gordon (1926) and Klein (1927), known as the Klein-Gordon equation The Klein-Gordon equation is valid for the Bose-Einstein particles, while the Dirac equation is valid for the Fermi-Dirac particles.

557

Towards the Authentic Ab Intio Thermodynamics

electrons and the radiations as quantized objects in such a way of a canonical transformation

to the normal modes of their fields; this method is called the second quantization or the field quantization The canonical transformation technique to the normal mode is a well

established classical method for continuous media (Goldstein, 1980) The idea treats boththe electrons and radiations as continuous fields and quantized them for their own normalmodes (Heisenberg & Pauli, 1929; Jordan & Pauli, 1928) The practically available solutionswas suggested for the nonrelativistic case by Bethe (1947) followed by the fully relativisticcase by Dyson (1949a;b); Feynman (1949a;b); Schwinger (1948; 1949a;b); Tomonaga (1946),

so called the renormalization for cancelling the unavoidable divergencies appeared in the

quantum field theory This kind of theory on the electrons and radiations is called as thequantum electrodynamics (QED), which is known to be the most precise theory ever achieved(Peskin & Schroeder, 1995) with the error between the theory and experiment to be less than

a part per billion (ppb) (Gabrielse et al., 2006; 2007; Odom et al., 2006)

4.1 The concepts of quantum field theory

Feynman (1949b) visulized the underlying concept of the quantum field theory byreinterpreting the nonrelativistic Schrödinger equation with the Green’s function concept

As in classical mechanics, a Hamiltonian operator ˆH contains all the mechanical interactions

of the system The necessary physical information of the system is contained in the wavefunctionψ The Schrödinger equation

i¯h ∂ψ

describes the change in the wave functionψ in an infinitesimally time interval Δt as due to the operation if an operator is e −i H ¯hˆΔt This description is equivalent to the description that the

wave functionψ(x2, t2)at x2and t2is evolved one from the wave functionψ(x1, t1)at x1and

t1through the equation

where we abbreviated 1 for x1, t1 and 2 for x2, t2 and define K(2, 1) =0 for t2 < t1 It is

straightforward to show that K can be defined by that solution of

whereδ(2, 1) =δ(t2− t1)δ(x2− x1)δ(y2− y1)δ(z2− z1)and the subscript 2 on ˆH2means

that the operator acts on the variables of 2 of K(2, 1) The kernel K is now called as the Green’s

function and it is the total amplitude for arrival at x2, t2starting from x1, t1 The transitionamplitude for finding a particle in stateχ(2), if it was inψ(1), is



χ ∗(2)K(2, 1)ψ(1)d3x1d3x2 (93)