Acknowledgment

The authors thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the Uni-versity of Stuttgart.

-1%

0%

1%

2%

3%

4%

5%

0 0.1 0.2 0.3 0.4 0.5

percent error

(a)ν = 4

0%

2%

4%

6%

8%

10%

12%

14%

0 0.1 0.2 0.3 0.4 0.5

percent error

(b)ν = 8

0%

5%

10%

15%

20%

25%

0 0.1 0.2 0.3 0.4 0.5

percent error

(c) ν = 16

0%

10%

20%

30%

40%

50%

60%

0 0.1 0.2 0.3 0.4 0.5

percent error

(d)ν= 64

Figure 3.4.4: Flexible hard-sphere chains of varying number of tangentially bonded spheres ν: deviations in pressure from three theoretical approaches, TPT1 (filled spheres), TPT2 (triangles), and TPDT2 (filled squares), from results of own molecular simulations.

3.A Correlation functions G

⁽³⁾_hs

(123) and G

⁽⁴⁾_hs

(1234)

In this section a method is discussed which allows to determine orientationally averaged par-ticle distribution functions. We have so far introducedG⁽³⁾_hs(123), eq. (3.37), andG⁽⁴⁾_hs(1234), eq. (3.16). Both of them consist of i-particle distribution functions

g_hs⁽ⁱ⁾(12...i) = N! (N −i)!

1 ρⁱZ_hs

Z exp

−βUhs(r^N)

dr^{N >i} (3.55) with i ∈ {2,3,4}. Further, Z_hs is the configurational part of the partition function of the hard-sphere fluid and we use the short-hand notation r^N ={r1,r2, ...,rN}.

The 3-particle distribution function for a hard-sphere (reference) fluid, according to g⁽³⁾_hs(123) = N!

(N −3)!

1 ρ³Z_hs

Z exp

−βUhs(r^N)

dr^{N >3} (3.56) is required, both, in the definition ofG⁽⁴⁾_hs(1234) (eq. (3.16)) and in the definition ofG⁽³⁾_hs(123) (eq. (3.37)). We note, however, that both correlation functions appear in integrals eqs. (3.35), (3.36), and (3.15) with Mayer functions as prefactors. The prefactors constrain the relevant configuration forG⁽³⁾_hs(123) as three particles in a chain-like structure corresponding to three attractively interacting hard spheres (1-2-3), whereas for G⁽⁴⁾_hs(1234) the relevant configura-tions are for two pairs of dimers (1-2 and 3-4).

The 3-particle distribution function can be simply evaluated by Monte Carlo simulation at defined variables {N,V,T} as

g_hs⁽³⁾(n, m) = 1 V⁽³⁾(n, m)N ρ²

×

* _N X

i

N_i(n) (N_i(m)−δ_nm) +

N V T

(3.57)

for two shells (n, m) ∈ N0 with thickness ∆r. N_i(n) denotes the number of particles j that are located in shell n around particle i with n ≤ (r_ij/∆r) < (n+ 1). The product

N_i(n) (N_i(m)−δ_nm) thus equals the number of all possible combinations of particles located in shellnandmwith respect to particlei. (3.57) gives an orientational averaged distribution for 3-particle configurations. The Kronecker Delta function

δnm =

(1, n=m

0, n6=m (3.58)

considers the case when n=m. Further, V⁽³⁾(n, m) represents an effective volume function normalizing the right hand side of eq. (3.57). For brevity we introduce the shorthand notation

K_nm^ij = (n−1)ⁱ−nⁱ

(m−1)^j −m^j

(3.59) The volume function reads

V⁽³⁾(n, m) = 4

3π∆r³ 2

K_nm³³ (3.60)

for both integrals eqs. (3.35) and (3.36) (respectively (3.15)). However, for eq. (3.35) we have to consider the case |n−m| < (σ/∆r) + 1 for which particles partially overlap. The molecular model, as we have defined it, allows only anglesφ ≥φGG between two sites within a particle, with φ_GG being the limiting angle for which particles don’t overlap. Therefore, the volume function changes for this case into

V⁽³⁾(n, m) = π∆r²2

"

1 2∆r²

K_nm⁴² +K_nm²⁴ + 16 9 K_nm³³

−σ²K_nm²²

# (3.61)

In the case of eq. (3.36) (respectively eq. (3.15)) we have two separate dimer-like configura-tions that do not interact attractively with each other (1-2 and 3-4). The volume function V⁽³⁾(n, m) therefore does not account for intermolecular particle overlap and is completely given as (3.60). Shell n then denotes the distance between the hard spheres within a dimer and shell m equals the distance between the different dimers.

We have simplified (3.57) such thatg_hs⁽³⁾(n, m) is now a 2-dimensional problem. The third dimension (distance between particles in shell n and m) is eliminated by orientationally averaging. The same can be done to the 4-particle distribution function

g_hs⁽⁴⁾(1234) = N! (N−4)!

1 ρ⁴Z_hs

Z exp

−βU_hs(r^N)

dr^{N >4} (3.62)

The equation for sampling the correlation function from molecular simulations then reads g_hs⁽⁴⁾(n, m) = 1

V⁽⁴⁾(n, m)N ρ³

×

* _N X

i

X

j∈Ni(m)

[(N_i(n)−δ_nm) (N_j(n)−δ_nm)−f_ij] +

N V T

(3.63)

We here consider a pair of particles i and j. The first sum goes over all particles in the system. The second sum considers only particles j that are located in shell m of particle i, respectively which are included in N_i(m). Shell n denotes the distance between the hard spheres within a dimer and shell m equals the distance between the different dimers. We further introduced variable f_ij counting the number of particles s being present in shell n of both particles, i and j at the same time. We thus subtractf_ij configurations that would falsely contribute as 4-particle configurations. The effective volume function V⁽⁴⁾(n, m) is determined by Monte Carlo algorithm.

The last step to determine particle distribution functions with hard-spheres at contact distance, is to adjust a correlation function to the results of g⁽³⁾_hs(123) and g⁽⁴⁾_hs(1234) and to extrapolate the dimer shell (here n) to distance σ. Fig. 3.A.1 shows this procedure for g⁽⁴⁾(r₁₂ = r₃₄ = σ, m) as an example. We could not find values for g⁽⁴⁾(1234) in literature, such that we could only validate the following limiting cases; the dimers are at contact distance (which is equal to a flexible chain with length ν = 4) and the dimers are far enough away from each other, such that g⁽⁴⁾_hs(1234) = (g⁽²⁾_hs(σ))². Fig. 3.A.2 illustrates the result shown in Fig. 3.A.1 but now extrapolated to contact distances of (1-2) and (3-4) for ρσ³ = 0.7 as an example. The values for the two limiting cases, for r_inter = 1 and for r_inter → ∞ are also shown and the obtained results are in good agreement to these limiting cases.

Figure 3.A.1: We illustrate the extrapolation ofg⁽⁴⁾(r_intra, r_inter) (—) tor_intra =σ at particle density ρσ³ = 0.7. r_intra (shell n in our notation) denotes the distance between the hard-spheres that build up a dimer, and r_inter (shell m in our notation) is the distance between two hard-spheres that are not part of the same dimer.

6 8 10 12 14 16 18 20 22 24

1 1.5 2 2.5 3 3.5 4

ghs( , inter)

Figure 3.A.2: We show the extrapolation result (r_intra = σ) for g⁽⁴⁾_hs(r_intra, r_inter) (•) plotted over the intermolecular dimer-dimer distance rinter. We further introduced the two limiting cases; dimers at contact distancer_inter= 1 (), and two uncorrelated dimers in a hard-sphere fluid with r_inter→ ∞ and g⁽⁴⁾_hs(1234) = (g_hs⁽²⁾(σ))² (−−).

3.B Graphical derivation of G

⁽⁴⁾_hs

(1234)

We start with the dimer-dimer mother-graphs of graph sum eq. (3.15) and (and eq. (3.11)).

Here we illustrate the zigzag line between two attractively interacting hard spheres as a combination of a solid and a dashed line. The graphs thus read

Undoing the summation over analytically identical graphs leads to

Resummation with the help of the definition of the Mayer-function e_R =f_R+ 1 gives

(3.64) Graphs that have additional structures solely attached to the dimers can be similarly disas-sembled and recomposed. From

we get

(3.65)

The next type of graphs have structures that connect the two dimers such that an additional e_R-bond arises as

Undoing the summation over analytically identical graphs leads to

Resummation gives

(3.66)

(3.67)

One more type of graphs has structures that cause two additional e_R-bonds as

Undoing the summation over analytically identical graphs thereby leads to

and recomposing gives

(3.68) Every graph that we have shown so far is not further reducible, in accordance to Wertheim’s formalism. The recomposed graphs in (3.65), (3.66), (3.67), and (3.68) can be further divided

until the prefactor is 1/8 (due to symmetry), which is equal to 1 divided by the symmetry number of the original graph of four hard spheres being connected to each other by an generalized f-bond.

Now taking the graph sum, eq. (3.15), and introducing the recomposed graphs of eq. (3.64), (3.65), (3.66), (3.67), and (3.68), the first step is manipulating the sum such that we obtain graphs with solely e_R-bonds.

(3.69)

The first row of the calculation in 3.69 represents a subsum of the recomposed graph sum (3.15). We add (and subtract) reducible graphs, so that f_R-bonds can be turned into e_R -bonds (withe_R=f_R+ 1). This first step is only applied to graphs of eq. (3.15) of which each hard sphere within a dimer-like configuration is connected to every other dimer hard sphere by f_R- and/or e_R-bonds. The second row’s graphs in 3.69 contain a 1-bond on the right side, while the third row’s graphs have a 1-bond on the left side. Adding up the positive terms of the first three rows, we get the closed, graphical representation of the 4-particle distribution function g⁽⁴⁾_hs(1234) in the fourth row after factorizing the monomer densities ρ₀ and attractive Mayer functions f_AA. The negative parts will be used in the next step to manipulate the remaining recomposed graphs of (3.15). Taking the negative second row in 3.69 and adding the remaining graphs in eq. (3.15) with a 1-bond on the right side we get

(3.70)

The third and fourth row arise from the relations

−e_R(13)e_R(24) +f_R(13)f_R(24) =−e_R(13)−f_R(24)

for columns 1-3 and 6,

−e_R(13)e_R(24) +f_R(13)e_R(24) =−e_R(24)

for columns 4 and 9, and

−eR(13)eR(24) +eR(13)fR(24) =−eR(13)

for columns 5 and 7. The same procedure can be applied on the negative third row in (3.69), as

(3.71)

At this point we have reformulated eq. (3.15) such that it consists of the last row in (3.69) and the last two rows of both, (3.70) and (3.71). The next step is manipulating the rows in (3.70) and (3.71). We start with the third row in (3.70) and introduce again additional

graph sums such that

(3.72)

leading to two graph sums (third and fourth row in 3.72) that can be formulated as closed analytical expressions, in terms of 2- and 3- particle distribution functions g_hs⁽²⁾(12), g_hs⁽²⁾(14) and g⁽²⁾_hs(34), and g_hs⁽³⁾(134), respectively. The fourth row in 3.70 demands a more sophisti-cated approach. Here, we have to turn twof_R-bonds into e_R-bonds, introducing more graph sums to obtain a closed analytical expression. To provide insight in this manipulation we

summarize the required steps as

(3.73)

The third row in (3.71) can be directly identified as

(3.74)

containing the 2- and 3-particle distribution functions g⁽²⁾_hs(34) and g_hs⁽³⁾(124). Continuing with the fourth row in (3.71) we can apply the same procedure as shown in (3.72), resulting in

(3.75)

Summarizing the derived expressions in (3.72), (3.73), (3.74) and (3.75), we obtain correla-tion funccorrela-tion G⁽⁴⁾_hs(1234) as

G⁽⁴⁾_hs(1234) = g⁽⁴⁾_hs(1234)−g_hs⁽²⁾(12)

g_hs⁽³⁾(134) +g⁽³⁾_hs(234)

−g_hs⁽²⁾(34)

g_hs⁽³⁾(123) +g_hs⁽³⁾(124) +g⁽²⁾_hs(12)g⁽²⁾_hs(34)

g⁽²⁾_hs(13) +g⁽²⁾_hs(14) +g_hs⁽²⁾(23) +g_hs⁽²⁾(24)−1

(3.76) Assuming that G⁽⁴⁾_hs(1234) is integrated over position vectors 1, 2, 3 and 4, we can further simplify eq. (3.76) as

G⁽⁴⁾_hs(1234) =g⁽⁴⁾_hs(1234)−4g_hs⁽³⁾(123)g_hs⁽²⁾(34) +g_hs⁽²⁾(12)g_hs⁽²⁾(34)

4g_hs⁽²⁾(13)−1

(3.77)

Modelling association and chemicle bonding are still up to today a very difficult task. An ac-curate prediction requires a model that allows to describe non-spherical relations, as caused by short ranged, highly directional interactions. Theories need to have a physic-based backbone for this purpose. Wertheim’s Thermodynamic Perturbation Theory provides a mathematical and graph-theoretical framework to deal with such complex relations. Graph cancelation due to steric hindrances is a powerful method to condense important configura-tional informations in terms of graph sums. Prohibited inter- and intramolecular associated structures can thus be simply eliminated by neglecting graph sums with such configura-tions. Applying TPT for a well-chosen molecular model (including bonding constraints and well-suited reference fluid) allows to predict real fluid properties and behaviours qualita-tively accurate. With this work an accessible guide of Wertheim’s brilliant TPT is given that encourages people who are not familiar with the underlying concepts to work on this promising attempt to describe complex associative interactions and non-spherical molecular configurations based on a physical model.

This work shows for the case of hard-chain fluids that low-density predictions depend strongly on the physical differences between reference and target fluid, while for higher densities the theory treats differences more indulgent. This is reasonable if one imagines the limit of having solely two chains with large chain lengths in a system. Choosing the first-order expansion TPT1 to predict the behaviour of the two chains, one is certainly aware of the lack of information with regard to higher-body effects. TPT1 only considers two-body effects of two sequenced particles within a chain. Next-to-neighbor effects, for example, are completely neglected which disregards contributions to the theory arising from additional configurational freedom. The second-order theory TPT2 considers 3-body interactions between next-to-neighbour chain members and yields a major improvement to the theory. However, it lacks informations regarding next-to-next-neighbor effects (and so on). This work shows that the magnitude of the improvement that comes from considering higher order terms in the single-chain framework descreases with increasing order, as shown in Chapter 2. This is why this study also includes contributions which go beyond the single-chain approximation. The single-chain approximation has a significant influence in this low density limit, as shown in Chapter 3. It completely neglects chain-to-chain contributions. This issue emerges especially with increasing chain length, because the low-density prediction becomes more and more inaccurate.

In this study we analyse TPT with regard of predicting hard-chain properties with respect to a reference hard-sphere fluid. The final conclusion of the analysis leads inevitably to the statement that a better-suited reference fluid has to be chosen when aiming to accuratly predict (non-spherical) hard-chain structures. An hard-sphere reference does not lead to an accurate description of highly directional forces for low densitities, because the theory is based on a purely reference fluid. All the more it is stunning, however, how well hard-chains are quantitatively described. A more empirical approach that is suggested by the result of this study is to insert the exact second virial coefficient into the developement of the theory.

This leads to an exact prediction of the low-density limit, which seems to be impossible when using a n-th order theory with reasonable n.

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