4.3 Applications

X<sup>1</sup> ¼ Xq ¼ ∇

X<sup>2</sup> ¼ Xv ¼ ∇

X<sup>3</sup> ¼ Xs ¼ ∇

can be approximated as

Non-Equilibrium Particle Dynamics

which is equivalent to

<sup>μ</sup> <sup>¼</sup> <sup>∂</sup><sup>G</sup>

<sup>d</sup> <sup>G</sup> Nw � �

where the following mathematical identity was used

<sup>∇</sup> <sup>μ</sup><sup>k</sup>

Jq Jv Js

2 6 4 <sup>T</sup> <sup>¼</sup> <sup>μ</sup>k<sup>∇</sup>

2 6 4

where Onsager's reciprocal relationship, Lij ¼ Lji, is employed.

ture, pressure, and molar solute fraction:

coupled to their driving forces, such as

94

<sup>∂</sup><sup>N</sup> <sup>¼</sup> <sup>G</sup>

tesimal change of Gibbs free energy is, in particular, written as

Nw þ Ns

∂S ∂E � �

∂S ∂V � �

∂S ∂Ns � �

where subscripts q, v, and s of X indicates heat, volume of solvent, and solute, respectively. In Eq. (92), entropy S is differentiated by energy E, keeping V, and Ns invariant, which are applied to Eqs. (93) and (94). Eq. (94) indicates that the driving force is a negative gradient of the chemical potential divided by the ambient temperature. Within the isothermal-isobaric ensemble, Gibbs free energy is defined as

where Hð Þ ¼ E þ PV is enthalpy. If the solute concentration is diluted (i.e., Nw ≫ Ns), it is referred to as a weak solution. As such, the overall chemical potential

> <sup>≃</sup> <sup>G</sup> Nw

where H and S represent molar enthalpy and entropy, respectively. An infini-

where V is a molar volume of the system, μ<sup>s</sup> is the solute chemical potential, and c ¼ Ns=Nw is the molar fraction of solute molecules. The gradient of the solvent chemical potential was rewritten as a linear combination of gradients of tempera-

> 1 T � �

In general, fluxes of heat, solvent volume, and solute molecules are intrinsically

Lqq Lqv Lqs Lqv Lvv Lvs Lqs Lvs Lss

þ 1

> 3 7 5

2 6 4

Xq Xv Xs

3 7

V,Ns

E,Ns

E,V

¼ ∇

¼ ∇

<sup>¼</sup> <sup>∇</sup> � <sup>μ</sup><sup>s</sup> T � �

1 T � �

P T � �

G ¼ H � TS (95)

dG ¼ �SdT þ VdP þ μsdNs (97)

≃ dμ<sup>w</sup> ¼ �SdT þ VdP þ μsdc (98)

∇μ<sup>w</sup> ¼ �S∇T þ V∇P þ μs∇c (99)

¼ μ<sup>w</sup> ¼ H � ST (96)

<sup>T</sup> <sup>∇</sup>μ<sup>k</sup> ð Þ (100)

<sup>5</sup> (101)

(92)

(93)

(94)
