Skip to main content
SpringerLink
Log in
Book cover

pp 1–73Cite as

Density Functional Theory for Liquid Structure and Thermodynamics

  • Chapter
  • First Online:

Part of the book series: Structure and Bonding

Abstract

Density functional theory (DFT) provides a powerful computational tool for study of the structure and thermodynamic properties of both bulk and inhomogeneous fluids. On the one hand, DFT is able to describe the microscopic structure and meso/macroscopic properties on the basis of intermolecular forces; and on the other hand, it connects seamlessly with conventional phenomenological equations for modeling macroscopic phenomena. The DFT-based methods are generic yet versatile they are naturally applicable to systems with multiple length scales that may fail alternative computational methods. This chapter presents a tutorial overview of the basic concepts of DFT for classical systems, the mathematical relations linking the microstructure and correlation functions to measurable thermodynamic quantities, and connections of DFT to conventional liquid-state theories. While for pedagogy the discussion is limited to one-component simple fluids, similar ideas and concepts are applicable to mixtures and polymeric systems of practical concern. This chapter also covers a few theoretical approaches to formulate the thermodynamic functional. Some illustrative examples are given on applications to liquid structure, interfacial properties, and surface and colloidal forces.

This is a preview of subscription content, log in via an institution.

Notes

  1. 1.

    For a simple liquid, the capillary length \(L_{\rm c} = \sqrt{\gamma/\Delta \rho g}\) is on the order of a few millimeters but the thermal length \(L_{\rm T} = \sqrt{k_{\rm B} T/\gamma}\) is only a few Angstroms. The gravitational force and capillary wave is insignificant if L c >> L T.

  2. 2.

    For van der Waals attraction, ΓA (z) ∼ 1/z 6 and following the definition ψ(z) ∼ 1/z 4, which is an even function of z.

  3. 3.

    For a molecule of n segments, the number of external degrees of freedom is 3nc F. For argon, n = 1 and c F = 1. For molecules when n > 1, c F < 1 because of chain connectivity. For a long paraffin, c F ≈ 1/3.

  4. 4.

    A Legendre transform switches independent variables in the fundamental equation of thermodynamics. For example, F = E − TS is a Legendre transform that changes the fundamental equation of thermodynamics from E = E(S, V, N) to F = F(T,V,N). Here the thermodynamic variable is a function.

  5. 5.

    This notation does not mean that a pair correlation function depends only on separation distance r. Instead, g(r) is a function of the system and depends on temperature, density, and intermolecular interactions.

  6. 6.

    For a function that depends only on the radial distance r, the Fourier transform is defined as \(\hat {f}(k)=\int {\rm d}\vec {r}{\rm e}^{-i\vec {k} \cdot \vec {r}} f(r)=(4\pi /k)\int_{0}^{\infty} {r\sin (kr)f(r){\rm d}r}\).

  7. 7.

    In some literature, the OZ equation defines of the second-order direct correlation function. While such definition is legitimate mathematically, it applies only to the second-order and to some extent it makes the physical meaning of the direct correlation less transparent.

  8. 8.

    A cavity correlation function describes the correlation between two “cavities” in the fluid. A cavity is a real particle that has no interaction with another cavity particle. Because the potential of mean force (βW(r) = −ln[g(r)]) is the overall interaction between two molecules in the system, it follows that \(-\ln [y(\vec{r} )]\) includes all indirect interactions, or effects of all other molecules on the overall interaction.

  9. 9.

    The concept of virial was first introduced by Clausius; in Latin it means force. The virial equation is not the same as the virial expansion.

  10. 10.

    A second-order tensor is a linear mapping of a vector onto another vector. Here, mapping means an arbitrary arithmetic calculation. For example, the gradient of a vector is a second-order tensor, which specifies the change in this vector with respect to position.

  11. 11.

    We put a negative sign here because, according to the conventional notation, a pressure is defined as the force applied toward a surface per unit area (compression push). In Fig. 12, the force is in the outward direction of the surface (pull, tension).

  12. 12.

    Pressure tensor is also used to describe the mechanical properties of materials, including strain, stress, and elasticity. Stress is a concept that is equivalent to pressure tensor but defined in a different perspective, \(\hat {\tau }=-\hat {P}\). The diagonal elements of a stress tensor are called normal stress. While a positive normal pressure means a force pushing on a surface, a positive normal stress stands for a pulling force out of there surface, or a tension. The nondiagonal elements of a stress tensor are called shear stress.

  13. 13.

    Here the force due to the external field is not included.

Abbreviations

A :

Surface area

a :

Energy parameter in van der Waals' equation of state

B :

MSA parameter

b :

Volume parameter in van der Waals' equation of state

b(r):

Bridge function

C :

Number of carbon atoms per molecule

c :

Parameter in van der Waals' square gradient theory

C F :

Semi-empirical parameter introduced by Prigogine

c(r):

Direct correlation function

E :

Energy/ground-state energy

F :

Helmholtz energy

\(\vec {F}\) :

Force

F [ρ(r)] :

Intrinsic Helmholtz energy functional

\(f(\vec {r})\) :

Local Helmholtz energy

g(r):

Radial distribution function

g :

Gravitational constant

H :

Interfacial thickness/pore width

h(r):

Total correlation function

h :

Elevation

\(\tilde {I}\) :

Unit tensor

K :

Kinetic energy

k B :

Boltzmann constant

L c :

Capillary length

L T :

Thermal length

M n :

Molecular weight

\(\vec {M}\) :

Momentum transfer

m :

Molecular mass

N :

Number of molecules

n α :

Weighted densities

\(\vec {n}\) :

Normal vector

P :

Pressure

\(\tilde {P}\) :

Pressure tensor

\(\vec {p}\) :

Molecular momentum

p ν :

Probability density

Q :

Surface charge density

q i :

Ionic charge

R :

Radius

r :

Center-to-center distance

\(\vec {r}\) :

Position vector

S :

Entropy

\({\mathbb S}(\vec {k} )\) :

Static structure factor

\(s(\vec {r} )\) :

Local entropy

T :

Temperature, K

U :

Internal energy

\(u(\vec {r} )\) :

Local internal energy

V :

Volume

v :

Cell volume

W :

Reversible work

W(r):

Potential of mean force

\(w(\vec {r} )\) :

Weight function

\(\chi _0^{({\rm A})}\) :

Local fraction of nonbonded associating site A

y(r):

Cavity correlation function

z :

Distance

BMCSL:

Boublik–Mansoori–Carnahan–Starling–Leland

DFT:

Density functional theory

FMSA:

First-order mean-spherical approximation

FMT:

Fundamental measure theory

HNC:

Hypernetted chain

LDA:

Local-density approximation

LJ:

Lennard-Jones

MC:

Monte Carlo

MHNC:

Modified hypernetted chain

MSA:

Mean-spherical approximation

OZ:

Ornstein–Zernike

PB:

Poisson–Boltzmann

PDT:

Potential distribution theorem

PY:

Percus–Yevick

RHNC:

Reference hypernetted chain

SAFT:

Statistical associating fluid theory

TDDFT:

Time-dependent density functional theory

WDA:

Weighted-density approximation

β:

1/k B T

Γ:

Intermolecular potential

γ:

Surface tension

δ (r):

Dirac delta function

ε:

Interaction energy

εD :

Dielectric constant

ζ:

Zeta potential

η:

Packing density

θ(r):

Heaviside step function

κ:

Inverse screening length

κT :

Isothermal compressibility

Λ:

Thermal wavelength

λ:

Coupling parameter

μ:

Chemical potential

Ξ:

Grand partition function

\(\rho (\vec{r} )\) :

Density profile

σ:

Diameter

\(\phi (\vec{r} )\) :

One-body potential function

\(\chi (\vec{r}^{\prime} ,\vec{r}^{\prime \prime})\) :

Density–density correlation function

ψ(z):

Function used in the square-gradient theory

Ψ(1,2,…,N):

Wave function

Ω:

Grand potential

ν:

Microstate

\(\nu (\vec{r})\) :

One-body external potential

ς:

Inhomogeneous factor

*:

Reduced quantities

id:

Ideal gas

ex:

Excess

ext:

External

0:

Bulk/reference system

α:

Index of weight functions = 0, 1, 2, 3, V1 and V2

A:

Intermolecular attraction

ass:

Association

c:

critical point

el:

Electric

hs:

Hard sphere

L:

Liquid

LJ:

Lennard-Jones

R:

Intermolecular repulsion

V:

Vapor

References

  1. Rowlinson JS (1979) J Stat Phys 20:197

    Article  ADS  MathSciNet  Google Scholar 

  2. Waals JD van der (1893) Verh K Akad Wet Amsterdam 1:8

    Google Scholar 

  3. Mezger M, Reichert H, Schoder S, Okasinski J, Schroder H, Dosch H, Palms D, Ralston J, Honkimaki V (2006) Proc Natl Acad Sci USA 103:18401

    Article  PubMed  ADS  CAS  Google Scholar 

  4. Frenkel D (2006) Science 314:768

    Article  PubMed  CAS  Google Scholar 

  5. Hansen JP, McDonald IR (2006) Theory of simple liquids, 3rd edn. Elsevier, London

    Google Scholar 

  6. Evans R (1992) Density functionals in the theory of nonuniform fluids. In: Henderson (ed) D Fundamentals of inhomogeneous fluids. Dekker, New York, p 85

    Google Scholar 

  7. Davis HT (1996) Statistical mechanics of phases, interfaces, and thin films, VCH, New York

    Google Scholar 

  8. Wu JZ, Li ZD (2007) Annu Rev Phys Chem 58:85

    Article  PubMed  CAS  ADS  Google Scholar 

  9. Wu JZ (2006) AIChE J 52:1169; Hu Y, Liu HL, Wang WC (2002), Fluid phase equilibria 194(97).

    Article  CAS  Google Scholar 

  10. Gibbs JW (1876) Trans Conn Acad 3:108–343

    Google Scholar 

  11. Rowlinson JS, Widom B (1982) Molecular theory of capillarity, Clarendon Press, Oxford

    Google Scholar 

  12. Cahn JW, Hilliard JE (1958) J Chem Phys 28:258

    Article  ADS  CAS  Google Scholar 

  13. Landau LD, Lifschitz L (1935) Phys Z Sowjetunion 8:153

    MATH  Google Scholar 

  14. Mitsui T, Furuichi J (1953) Phys Rev 90:193

    Article  ADS  CAS  Google Scholar 

  15. Bonn D, Ross D (2001) Rep Prog Phys 64:1085

    Article  ADS  CAS  Google Scholar 

  16. Mello EVL, de Caixeiro ES (2006) J Phys Chem Solids 67:165

    Article  ADS  CAS  Google Scholar 

  17. Gameiro M, Mischaikow K, Wanner T (2005) Acta Material 53:693

    Article  CAS  Google Scholar 

  18. Carnahan NF, Starling KE (1969) J Chem Phys 51:635

    Article  ADS  CAS  Google Scholar 

  19. Fu D, Wu J (2005) Ind Eng Chem Res 44:1120

    Article  CAS  Google Scholar 

  20. Cornelisse PMW, Peters CJ, Arons JD (1997) J Chem Phys 106:9820

    Article  ADS  CAS  Google Scholar 

  21. Trokhymchuk A, Alejandre J (1999) J Chem Phys 111:8510

    Article  ADS  CAS  Google Scholar 

  22. Dominik A, Tripathi S, Chapman WG (2006) Ind Eng Chem Res 45:6785

    Article  CAS  Google Scholar 

  23. Dee GT, Sauer BB (1998) Adv Phys 47:161

    Article  ADS  CAS  Google Scholar 

  24. Geerlings P, Proft F, De Langenaeker W (2003) Chem Rev 103:1793

    Article  PubMed  CAS  Google Scholar 

  25. Thomas LH (1927) Proc Cambridge Philos Soc 23:542

    Article  MATH  CAS  Google Scholar 

  26. Fermi E (1928) Z Phys 48:73

    Article  ADS  CAS  Google Scholar 

  27. Hohenberg P, Kohn W (1964) Phys Rev 136:B864

    Article  ADS  MathSciNet  Google Scholar 

  28. Kohn W, Sham LJ (1965) Phys Rev 140:A1133

    Article  ADS  MathSciNet  Google Scholar 

  29. Parr RG, Yang W (1989) Density-functional theory of atoms and molecules. Oxford University Press, New York

    Google Scholar 

  30. Ebner C, Saam WF, Stroud D (1976) Phys Rev A 14:2264

    Article  ADS  Google Scholar 

  31. Evans R (1979) Adv Phy 28:143

    Article  ADS  CAS  Google Scholar 

  32. Mermin ND (1965) Phys Rev 127:A1441

    Article  MathSciNet  Google Scholar 

  33. Narten AH, Levy D (1971) J Chem Phys 55:2263

    Article  ADS  CAS  Google Scholar 

  34. Okhulkov AV, Demianets YN, Gorbaty YE (1994) J Chem Phys 100:1578

    Article  ADS  CAS  Google Scholar 

  35. Percus JK, Yevick GJ (1958) Phys Rev 110:1

    Article  MATH  ADS  CAS  MathSciNet  Google Scholar 

  36. Lee LL (1988) Molecular thermodynamics of nonideal fluids. Butterworths, Boston

    Google Scholar 

  37. Wertheim M (1963) Phys Rev Lett 10:321

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. Baxter RJ (1968) J Chem Phys 49:2770

    Article  ADS  CAS  Google Scholar 

  39. Blum L (1975) Mol Phys 30:1529

    Article  ADS  CAS  Google Scholar 

  40. Malijevsky A, Labik S, Smith WR (1991) Mol Phys 72:193

    Article  ADS  CAS  Google Scholar 

  41. Caccamo C (1996) Phys Rep 274:1

    Article  ADS  CAS  Google Scholar 

  42. Rosenfeld Y, Ashcroft NW (1979) Phys Rev A (Gen Phys) 20:1208

    Article  ADS  CAS  Google Scholar 

  43. Li ZD, Wu JZ (2005) Macromol Symp 219: 51

    Article  CAS  Google Scholar 

  44. Irving JH, Kirkwood JG (1950) J Chem Phys 18:817

    Article  ADS  CAS  MathSciNet  Google Scholar 

  45. Yu YX, Wu JZ (2002) J Chem Phys 117:10156

    Article  ADS  CAS  Google Scholar 

  46. Henderson D, Blum L, Lebowitz JL (1979) J Electroanal Chem 102:315

    Article  CAS  Google Scholar 

  47. Roth R, Evans R, Dietrich S (2000) Phys Rev E 62:5360

    Article  ADS  CAS  Google Scholar 

  48. Biben T, Bladon P, Frenkel D (1996) J Phys Condens Matter 8:10799

    Article  ADS  CAS  Google Scholar 

  49. Roth R, Gotzelmann B, Dietrich S (1999) Phys Rev Lett 83:448

    Article  ADS  CAS  Google Scholar 

  50. Cuesta JA, Martinez-Raton Y, Tarazona P (2002) J Phys Condens Matter 14:11965

    Article  ADS  CAS  Google Scholar 

  51. Rosenfeld Y (1989) Phys Rev Lett 63:980

    Article  PubMed  ADS  CAS  Google Scholar 

  52. Tarazona P (2002) Physica A 306:243

    Article  MATH  ADS  MathSciNet  Google Scholar 

  53. Rosenfeld Y (2002) J Phys Condens Matter 14:9141

    Article  ADS  CAS  Google Scholar 

  54. Boublik T (1970) J Chem Phys 53:471

    Article  ADS  CAS  Google Scholar 

  55. Mansoori GA, Carnahan NF, Starling KE, Leland TW, Jr. (1971) J Chem Phys 54:1523

    Article  ADS  CAS  Google Scholar 

  56. Roth R, Evans R, Lang A, Kahl G (2002) J Phys Condens Matter 14:12063

    Article  ADS  CAS  Google Scholar 

  57. Groot RD, Eerden JP, van der Faber NM (1987) J Chem Phys 87:2263

    Article  ADS  CAS  Google Scholar 

  58. Lafuente L, Cuesta JA (2003) J Chem Phys 119:10832

    Article  ADS  CAS  Google Scholar 

  59. Schmidt M (1999) Phys Rev E 60:R6291

    Article  ADS  CAS  Google Scholar 

  60. Winkelmann J (2001) J Phys Condens Matter 13:4739

    Article  ADS  CAS  MathSciNet  Google Scholar 

  61. Tang YP, Wu JZ (2003) J Chem Phys 119:7388

    Article  ADS  CAS  Google Scholar 

  62. Tang YP, Wu JZ (2004) Phys Rev E 70:011201

    Article  ADS  CAS  Google Scholar 

  63. Tang YP (2003) J Chem Phys 118:4140

    Article  ADS  CAS  Google Scholar 

  64. Chapman WG, Gubbins KE, Jackson G, Radosz M (1989) Fluid Phase Equil 52:31

    Article  CAS  Google Scholar 

  65. Muller EA, Gubbins KE (2001) Ind Eng Chem Res 40:2193

    Article  CAS  Google Scholar 

  66. Segura CJ, Chapman WG, Shukla KP (1997) Mol Phys 90:759

    Article  ADS  CAS  Google Scholar 

  67. Segura CJ, Vakarin EV, Chapman WG, Holovko MF (1998) J Chem Phys 108:4837

    Article  ADS  CAS  Google Scholar 

  68. Segura CJ, Zhang J, Chapman WG (2001) Mol Phys 99:1

    Article  ADS  CAS  Google Scholar 

  69. Yu YX, Wu JZ (2002) J Chem Phys 116:7094

    Article  ADS  CAS  Google Scholar 

  70. Paricaud P, Galindo A, Jackson G (2002) Fluid Phase Equil 194:87

    Article  Google Scholar 

  71. Tripathi S, Chapman WG (2003) J Chem Phys 118:7993

    Article  ADS  CAS  Google Scholar 

  72. Pizio O, Patrykiejew A, Sokolowski S (2000) J Chem Phys 113:10761

    Article  ADS  CAS  Google Scholar 

  73. Malo BM, Huerta A, Pizio O, Sokolowski S (2000) J Phys Chem B 104:7756

    Article  CAS  Google Scholar 

  74. Huerta A, Pizio O, Bryk P, Sokolowski S (2000) Mol Phys 98:1859

    Article  ADS  CAS  Google Scholar 

  75. Yu YX, Wu JZ, Gao GH (2004) Chin J Chem Eng 12:688

    CAS  Google Scholar 

  76. Chow RS, Takamura K (1988) J Colloid Interface Sci 125:226

    Article  CAS  Google Scholar 

Download references

Acknowledgements

The author is grateful to Prof. John Prausnitz for reading many sections of this chapter and to Prof. Lloyd Lee for helpful comments. The research is sponsored by the US Department of Energy (DE-FG02-06ER46296) and uses the computational resources of the National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the US Department of Energy under contract no. DE-AC03-76SF00098.

Author information

Authors and Affiliations

  1. Department of Chemical and Environmental Engineering, University of California, Riverside, CA, 92521-0425, USA

    J.Z. Wu

Authors
  1. J.Z. Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J.Z. Wu .

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag London

About this chapter

Cite this chapter

Wu, J. (2008). Density Functional Theory for Liquid Structure and Thermodynamics. In: Structure and Bonding. Springer, Berlin, Heidelberg. https://doi.org/10.1007/430_2008_3

Download citation

  • DOI: https://doi.org/10.1007/430_2008_3

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

Publish with us

Policies and ethics

Search

Navigation