Two attractive sites A and B

To enable the formation of chain-associate configurations, Wertheim introduces hard spheres with two attractive sites Γ ={A, B}. He simplifies the treatment by prohibiting interactions between attractive sites of the same type, i.e. u_AA(r_mAnA) = u_BB(r_mBnB) = 0. The intro-duction of the singly-bonded condition again eliminates graphs containing bond-connected networks with three or more sites. Wertheim further excludes doubly-bonded hard spheres by restricting the range of attractive interactions a_AB such that, for a given attractive site geometry, two sites on particle m can never simultaneously bond to two sites located on particle n because of prohibitive hard-sphere overlap.

With these restrictions in mind, one obtains

c⁽⁰⁾[{ρ}]−c⁽⁰⁾_R [ρ] = B^{A B}A +

A B

A B

B A

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+ ...

+

A B

A B

B A

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+ ...

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+ ...

(1.114)

where the single-chain approximation has already been incorporated. Contrary to the case of one attractive site, two attractive sites allow chain-associate s-mers with arbitrary values of s (greater or equal unity). In the first row of eq. (1.114), we grouped graphs with similar configurations as previously regarded in eq. (1.95), i.e. one dimer surrounded by solely repulsively interacting monomers. With each following row, the number of chain-members increases by one, i.e. the second row contains 3-mer graphs, the third row represents 4-mer graphs, etc... The graph sum, eq. (1.114), actually also contains graphs with ring-like, attractively interacting hard spheres. Ring-structures are not eliminated after applying the single-chain approximation. The first member in the class of ring-graphs consists of three hard spheres. Wertheim limits consideration to chain-associate networks and neglects graphs of ring-associates. That is an approximation, because the molecular model does not prohibit ring-associates, with s^ring ≥ s^ring,min, where s^ring,min depends on the molecular model. An exception is the case where two sites are located at an angle of 180^◦ and with vanishing interaction range aAB. Thens^ring,min becomes infinite and ring-structures are irrelevant.

Each chain-associate structure in eq. (1.114) contains two field points as chain ends with each having one unbonded and one occupied attractive site. These field points are eitherσ_A-field points with bonded attractive siteB orσ_B-field points with bonded attractive site A. Both attractive sites of field points in between the two chain ends of a chain-associate structure are bonded to neighbouring chain members. These field points are σ₀ -field points and appear only in (s ≥ 3)-mers. Each (s ≥ 3)-mer contains (s−2) σ₀-field points. Differentiating eq. (1.114) with respect to σB(r) leads, according to eq. (1.77), to

cA(r) as

cA(r) = B^{A B}A +

A B

A B

B A

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+ ...

+

A B

A B

B A

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+ ...

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+ ...

(1.115)

Functionc_B(r) is analogously derived by functionally differentiating eq. (1.114) with respect to σ_A(r) and reads

c_B(r) = ^AB ^BA +

A B

A B

B A

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+ ...

+

A B

A B

B A

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+ ...

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+ ...

(1.116)

We will later need function c_AB(r) being the functional derivative of eq. (1.114) with respect to σ₀(r) as

c_AB(r) =

A B

A B

B A

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+ ...

+ 2

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+ 2

B A

A B

B A A B

+ 2

B A

A B

B A A B

+ ...

(1.117) As in the case of one attractive site, one can formulate a relation between c⁽⁰⁾[{ρ}]−c⁽⁰⁾_R [ρ], eq. (1.114), and the functions cA(r), eq. (1.115), and cB(r), eq. (1.116), as

c⁽⁰⁾[{ρ}]−c⁽⁰⁾_R [ρ] = 1 2

Z

(σ_A(r)c_B(r) +σ_B(r)c_A(r)) dr (1.118)

We emphasize that this relation is only valid in the single-chain approximation scheme. The integration in eq. (1.118) requires a factorization by 1/2, since we integrate over both c_A(r) and c_B(r) to derive c⁽⁰⁾[{ρ}]−c⁽⁰⁾_R [ρ]. With the relations in eq. (1.82), we get

c⁽⁰⁾[{ρ}]−c⁽⁰⁾_R [ρ] = Z

σ_A(r)σ_B(r) ρ₀(r) − 1

2(σA(r) +σB(r))

dr (1.119)

The difference of the intrinsic Helmholtz energy, eq. (1.87), and of the pressure, eq. (1.88), then leads with Q(r) = (σ_A(r)σ_B(r))/ρ₀(r)−σ_A(r)−σ_B(r) (in the case of Γ = {A, B} in eq. (1.86)) to

β(A[{ρ}]− AR[ρ]) = Z

ρ(r)

lnρ0(r)

ρ(r) + 1− 1 2

σA(r)

ρ(r) +σB(r) ρ(r)

dr (1.120)

β(p−p_R)V = Z

ρ²(r)

∂ln(ρ₀(r)/ρ(r))

∂ρ(r) −1 2

∂(σ_A(r)/ρ(r))

∂ρ(r) +∂(σ_B(r)/ρ(r))

∂ρ(r)

dr (1.121) To obtain a self-consistent, analytical relation between the multiple densities σ_α(r), similar to eq. (1.102), we focus on the difference between c⁽⁰⁾[{ρ}]−c⁽⁰⁾_R [ρ] for one attractive site, eq. (1.94), and same quantity for two attractive sites, eq. (1.114) by considering the singly-bonded condition and the single-chain approximation: For one attractive site, only (s≤ 2)-mers can occur, thus leading to a single row in our graphical representation of c⁽⁰⁾[{ρ}]− c⁽⁰⁾_R [ρ]. For two attractive sites, however, we additionally get the 3-mer subtotal as the 2^nd row in c⁽⁰⁾[{ρ}] −c⁽⁰⁾_R [ρ], as well as an infinite number of additional rows, for the higher s-mers. It is obvious that capturing all rows (and thus all s-mers) in analytical expressions is not possible, because for each row the order s of required particle-distribution functions g^(s)(r⁰,r⁰⁰, ...,r^s) increases. Wertheim thus regardss-mers only up to a valuen. Depending on the numbern of rows the theory is capturing, it is referred to as TPTn, i.e. Thermodynamic Perturbation Theory of n-th order. We note, the expressions in eq. (1.120) and (1.121) are order-independent. Order-dependence is introduced in the self-consistent density relations.

In what follows it is useful to introduce the density ratios ν_A(r) = ρ(r)

σ_A(r), ν_B(r) = ρ(r)

σ_B(r) (1.122)

τ_A(r) = σ_A(r)

ρ₀(r), τ_B(r) = σ_B(r)

ρ₀(r) (1.123)

for a compact notation.

In homogeneous systems, one directly realizes σ_A = σ_B ≡ σ_G, which leads to τ_A = τ_B ≡ τ and ν_A = ν_B ≡ ν, because only singly-bonded attractive sites are allowed and interactions between attractive sites of the same type are prohibited. The number of bonded

attractive sites A in the system therefore equals the number of bonded attractive sites B.

The same holds for the number of unbonded attractive sites A andB. Wertheim introduces the subscript G for either siteA or B. In this case, eq. (1.120) and (1.121) simplify to

β(A[{ρ}]− A_R[ρ])

N =−lnντ + 1− 1

ν (1.124)

β(p−p_R)

ρ =ρ

1 ν

∂ν

∂ρ 1

ν −1

− 1 τ

∂τ

∂ρ

(1.125) TPT1: first order theory

In TPT1, the first row of c_A(r), eq. (1.115), and of c_B(r), eq. (1.116), is considered, i.e.

the dimer graph subtotal. Factoring the only remaining attractive f_AB-bond and the corre-sponding σ_A-field point and σ_B-field point of the dimer-associate, respectively, we formulate c_A(r) and c_B(r) in terms of the two-particle distribution function g_R⁽²⁾(r,r⁰) as

c_A(r) = Z

f_AB(r,r⁰)g_R⁽²⁾(r,r⁰)σ_A(r⁰)dr⁰ (1.126) c_B(r) =

Z

f_BA(r,r⁰)g_R⁽²⁾(r,r⁰)σ_B(r⁰)dr⁰ (1.127) Further, function c_AB(r), eq. (1.117), consists of irreducible graphs with a dashed root point with both attractive sites AandB bonded. We find such dashedroot points only in (s≥ 2)-mers. Because the first-order theory, TPT1, limits consideration to chain-associate dimers (s≤2), we have

cAB(r) = 0 (1.128)

Substituting the c_ψ(r)-functions given in eqs. (1.126), (1.127), and (1.128) according to eq. (1.82) and using the density ratios in eq. (1.122) and (1.123), one obtains

τ_A(r)−1 = Z

f_AB(r,r⁰)g⁽²⁾_R (r,r⁰)σ_A(r⁰)dr⁰ (1.129) τ_B(r)−1 =

Z

f_BA(r,r⁰)g⁽²⁾_R (r,r⁰)σ_B(r⁰)dr⁰ (1.130) ρ(r)

ρ₀(r)−τ_A(r)τ_B(r) = 0 (1.131)

For a known density profile ρ(r) and reference two-particle distribution function g_R⁽²⁾(r,r⁰), eqs. (1.129), (1.130) and (1.131) provide a mathematical framework for determining the unknown profiles of σA(r), σB(r) and σ0(r) for a given molecular model.

For homogeneous systems, the number of self-consistent density relations decreases, be-cause σ_A=σ_B ≡σ_G. They read

τ −1 = σ_GI₁^∗ (1.132)

ν−τ = 0 (1.133)

withI₁^∗ =R

f_AB(r,r⁰)g_R⁽²⁾(r,r⁰)dr⁰. The density ratioν=ρ/σ_Gis the number of hard spheres divided by the number of chain ends of a particular type, say the chain-end particle type with bonded site A (then σ_G = σ_B). Therefore, ν can also be described as the average number of hard spheres per chain-associate. Please note that monomers, i.e. non-bonded spheres, are in this context counted as chain-associates of unity length. For brevity, it is customary to speak of ν simply as average chain “length”.

Differentiating eq. (1.132) with respect to the number density ρ, and substitutingτ with ν, as specified in eq. (1.133), one obtains for the intrinsic Helmholtz energy, eq. (1.124), and pressure, eq. (1.125),

β(A[{ρ}]− A_R[ρ])

N =−2 lnν+ 1− 1

ν (1.134)

β(p−p_R)

ρ =−

1− 1

ν 1 + ∂lnI₁^∗

∂lnη

(1.135) with ν = 1 +σ_GI₁^∗, as given with eqs. (1.132) and (1.133).

TPT2: second order theory

The second-order theory (TPT2) additionally considers the second row ofc_A(r), eq. (1.115), and of c_B(r), eq. (1.116), and the first row of c_AB(r), eq. (1.117). These rows represent 3-mer subtotals. Similar to the dimer subtotal in eq. (1.101), the 3-mer subtotal can be expressed in terms of reference two- and three-particle distribution functions g_R⁽²⁾(r,r⁰) and g_R⁽³⁾(r,r⁰,r⁰⁰). The factorisation of the attractivef_AB-bonds and the remainingσ₀-field points and σ_A-field points of the 3-mer-associate, as well as the σ_B-field points, respectively, lead to

cA(r) = Z

fAB(r,r⁰)σA(r⁰)g_R⁽²⁾(r,r⁰)dr⁰ +

Z

f_AB(r,r⁰)σ₀(r⁰)f_AB(r⁰,r⁰⁰)σ_A(r⁰⁰)

×





A B

A B

B A

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+

B A

A B

B A A B

+ ...





| {z }

g⁽³⁾_R (r,r⁰,r⁰⁰)−g⁽²⁾_R (r,r⁰)g_R⁽²⁾(r⁰,r⁰⁰)

dr⁰dr⁰⁰

(1.136)

c_B(r) = Z

f_BA(r,r⁰)σ_B(r⁰)g_R⁽²⁾(r,r⁰)dr⁰ +

Z

f_BA(r,r⁰)σ₀(r⁰)f_BA(r⁰,r⁰⁰)σ_B(r⁰⁰)

×

g_R⁽³⁾(r,r⁰,r⁰⁰)−g_R⁽²⁾(r,r⁰)g_R⁽²⁾(r⁰,r⁰⁰) dr⁰dr⁰⁰

(1.137)

c_AB(r) = Z

σ_B(r⁰)f_AB(r⁰,r)f_AB(r,r⁰⁰)σ_A(r⁰⁰)

×

g_R⁽³⁾(r⁰,r,r⁰⁰)−g_R⁽²⁾(r⁰,r)g_R⁽²⁾(r,r⁰⁰) dr⁰dr⁰⁰

(1.138)

For brevity, we introduce the notation G⁽³⁾_R (r,r⁰,r⁰⁰) = g_R⁽³⁾(r,r⁰,r⁰⁰)−g_R⁽²⁾(r,r⁰)g_R⁽²⁾(r⁰,r⁰⁰) (as used in literature [27]). A more comprehensive description and graph theoretical represen-tation of these correlation functions is given in 1.A.

Using eq. (1.82), (1.122) and (1.123), the self-consistent density relations in eq. (1.136), (1.137) and (1.138), turn to

τ_A(r)−1 = Z

f_AB(r,r⁰)σ_A(r⁰)g_R⁽²⁾(r,r⁰)dr⁰ +

Z

f_AB(r,r⁰)σ₀(r⁰)f_AB(r⁰,r⁰⁰)σ_A(r⁰⁰)G⁽³⁾_R (r,r⁰,r⁰⁰)dr⁰dr⁰⁰

(1.139)

τ_B(r)−1 = Z

f_BA(r,r⁰)σ_B(r⁰)g⁽²⁾_R (r,r⁰)dr⁰ +

Z

f_BA(r,r⁰)σ₀(r⁰)f_BA(r⁰,r⁰⁰)σ_B(r⁰⁰)G⁽³⁾_R (r,r⁰,r⁰⁰)dr⁰dr⁰⁰

(1.140)

ρ(r)

ρ₀(r)−τ_A(r)τ_B(r) = Z

σ_B(r⁰)f_AB(r⁰,r)f_AB(r,r⁰⁰)σ_A(r⁰⁰)G⁽³⁾_R (r⁰,r,r⁰⁰)dr⁰dr⁰⁰ (1.141) To solve this system of equations for the densitiesσ₀(r),σ_A(r) andσ_B(r), one needs the num-ber density profile ρ(r) and reference two- and three-particle distribution functionsg_R⁽²⁾(r,r⁰) and g⁽³⁾_R (r,r⁰,r⁰⁰).

For homogeneous systems, the constraint σ_A = σ_B = σ_G applies. The self-consistent density relations then considerably simplify to

τ −1 =σ_GI₁^∗+σ_Gσ₀I₂^∗ (1.142)

ν−τ =σ_Gσ₀I₂^∗ (1.143)

with I₁^∗ = R

f_AB(r,r⁰)g_R⁽²⁾(r,r⁰)dr⁰ and I₂^∗ = R

f_AB(r,r⁰)f_AB(r⁰,r⁰⁰)G⁽³⁾_R (r,r⁰,r⁰⁰)dr⁰dr⁰⁰. By differentiating eq. (1.142) with respect to number density ρ and inserting eq. (1.143) into

the so-obtained derivative, one can rewrite the pressure, eq. (1.125), as β(p−p_R)

ρ =−

1− 1

ν 1 + ∂lnλ₁

∂lnη +σ_Gσ₀I₂^∗ ν−1

∂lnλ₂

∂lnη

(1.144) with λ₁ = I₁^∗ and λ₂ = I₂^∗/(I₁^∗)². The intrinsic Helmholtz energy is given with eq. (1.124).

The density ratios ν and τ are determined using eqs. (1.142) and (1.143).

TPT3: third order theory[45]

A further extension to third order (TPT3) takes the third row of c_A(r), eq. (1.115), and c_B(r), eq. (1.116), and the second row ofc_AB(r), eq. (1.117), into account, where the 4-mer subtotal are regarded. To formulate the self-consistent density relations, we additionally need the reference four-particle distribution functiong_R⁽⁴⁾(r,r⁰,r⁰⁰,r⁰⁰⁰). We adopt the concise notation G⁽⁴⁾_R (r,r⁰,r⁰⁰,r⁰⁰⁰), which represents an elaborate term consisting of two-, three- and four-particle distribution functions. We give more detail on this quantity in 1.A. The self-consistent density relations are

τ_A(r)−1 = Z

f_AB(r,r⁰)σ_A(r⁰)g_R⁽²⁾(r,r⁰)dr⁰ +

Z

f_AB(r,r⁰)σ₀(r⁰)f_AB(r⁰,r⁰⁰)σ_A(r⁰⁰)G⁽³⁾_R (r,r⁰,r⁰⁰)dr⁰dr⁰⁰ +

Z

f_AB(r,r⁰)σ₀(r⁰)f_AB(r⁰,r⁰⁰)σ₀(r⁰⁰)f_AB(r⁰⁰,r⁰⁰⁰)σ_A(r⁰⁰⁰)G⁽⁴⁾_R (r,r⁰,r⁰⁰,r⁰⁰⁰)dr⁰dr⁰⁰dr⁰⁰⁰ (1.145) τ_B(r)−1 =

Z

f_BA(r,r⁰)σ_B(r⁰)g_R⁽²⁾(r,r⁰)dr⁰ +

Z

f_BA(r,r⁰)σ₀(r⁰)f_BA(r⁰,r⁰⁰)σ_B(r⁰⁰)G⁽³⁾_R (r,r⁰,r⁰⁰)dr⁰dr⁰⁰ +

Z

f_BA(r,r⁰)σ₀(r⁰)f_BA(r⁰,r⁰⁰)σ₀(r⁰⁰)f_BA(r⁰⁰,r⁰⁰⁰)σ_B(r⁰⁰⁰)G⁽⁴⁾_R (r,r⁰,r⁰⁰,r⁰⁰⁰)dr⁰dr⁰⁰dr⁰⁰⁰ (1.146) ρ(r)

ρ₀(r) −τ_A(r)τ_B(r) = Z

σ_B(r⁰)f_AB(r⁰,r)f_AB(r,r⁰⁰)σ_A(r⁰⁰)G⁽³⁾_R (r⁰,r,r⁰⁰)dr⁰dr⁰⁰ + 2

Z

σ_B(r⁰)f_AB(r⁰,r)f_AB(r,r⁰⁰)σ₀(r⁰⁰)f_AB(r⁰⁰,r⁰⁰⁰)σ_A(r⁰⁰⁰)G⁽⁴⁾_R (r⁰,r,r⁰⁰,r⁰⁰⁰)dr⁰dr⁰⁰dr⁰⁰⁰ (1.147) The factor 2 in the second row in eq. (1.147) arises from applying the product rule due to the presence of two σ0-field points in each 4-mer.

For homogeneous systems, again, the constraint σA = σB = σG applies, leading to the self-consistent density relations

τ−1 =σGI₁^∗+σGσ0I₂^∗+σGσ₀²I₃^∗ (1.148) ν−τ =σ_Gσ₀I₂^∗+ 2σ_Gσ²₀I₃^∗ (1.149) with

I₁^∗ = Z

f_AB(r,r⁰)g⁽²⁾(r,r⁰)dr⁰ (1.150)

I₂^∗ = Z

f_AB(r,r⁰)f_AB(r⁰,r⁰⁰)G⁽³⁾_R (r,r⁰,r⁰⁰)dr⁰dr⁰⁰ (1.151) I₃^∗ =

Z

f_AB(r,r⁰)f_AB(r⁰,r⁰⁰)f_AB(r⁰⁰,r⁰⁰⁰)G⁽⁴⁾_R (r,r⁰,r⁰⁰,r⁰⁰⁰)dr⁰dr⁰⁰dr⁰⁰⁰ (1.152) By differentiating eq. (1.148) with respect to number density ρ and using eq. (1.149), we obtain the pressure

β(p−p_R) =−ρ

1− 1

ν 1 + ∂lnλ1

∂lnη + σGσ0I₂^∗ ν−1

∂lnλ2

∂lnη + σGσ₀²I₃^∗ ν−1

∂lnλ3

∂lnη

(1.153) with λ₁ =I₁^∗, λ₂ = I₂^∗/(I₁^∗)² and λ₃ =I₃^∗/(I₁^∗)³. The intrinsic Helmholtz energy is given as eq. (1.124). The density ratios ν and τ are evaluated from eqs. (1.148) and (1.149).

TPTn: n−th order theory

It is possible to formulate a general theory of n−thorder in the single-chain approximation.

One uses the first n rows of cA(r) and cB(r), eq. (1.115), and the first (n − 1) rows of c_AB(r), eq. (1.117), considering eachs-mer-associate with s≤(n+ 1). Evaluating then−th order theory thus requires reference particle distribution functions up to (n+ 1)−th order.

We formulate the self-consistent density relations, for a more concise notation limited to homogeneous systems, as

τ −1 =σ_G

n

X

i=1

σⁱ⁻¹₀ I_i^∗ (1.154)

ν−τ =σ_G

n

X

i=2

(i−1)σ₀ⁱ⁻¹I_i^∗ (1.155) The factor (i−1) in eq. (1.155) arises from applying the product rule due to the presence of (i−1)σ0-field points in each (i+ 1)-mer.

By differentiating eq. (1.154) with respect to number density ρand using eq. (1.155), we restate the pressure, eq. (1.125), as

β(p−p_R) = −ρ

1− 1 ν

"

1 + ∂lnλ₁

∂lnη +

n

X

i=2

σ_Gσⁱ⁻¹₀ I_i^∗ ν−1

∂lnλ_i

∂lnη

#

(1.156) with λ₁ =I₁^∗,λ_i =I_i^∗/(I₁^∗)ⁱ fori >1. The intrinsic Helmholtz energy is given in eq. (1.124), and the density ratios ν and τ are determined from eqs. (1.154) and (1.155).

Wertheim provides a low-density approximation for TPT2 [27]. This approximation, however, is not limited to a second order theory. In the low-density limit, each integral I_i^∗ with i > 2 reduces to zero. The self-consistent density relations, eq. (1.154) and (1.155), simplify to

τ −1 =σ_GI₁^∗ (1.157)

ν−τ =0 (1.158)

and eq. (1.156) turns in this case into β(p−p_R) = −ρ

1− 1

ν "

1 + ∂lnλ₁

∂lnη +

n

X

i=2

1− 1

ν ⁱ⁻¹

∂λ_i

∂lnη

#

(1.159)

Im Dokument Analysis and extension of Wertheim's thermodynamic perturbation theory (Seite 49-58)