Extrapolation

Following the discussion around the potential of mean force in chapter 3.1.3, we had the idea to use extrapolation for reconstructing the interaction potentials from the dis-tribution functions. We remember that them-body potential of mean force is nothing but the negative logarithm of the m-body distribution function. And, that the pair potential of mean force is mainly the direct pair potential corrected by an averaged interaction due to surrounding particles. Reducing the density ρof the system, the correction to the direct pair potential becomes smaller and smaller. In the limes of zero density, one can thus write

g⁽²⁾(r)−−→^ρ→0 e^−βu(2)(r). (4.1) In other words, the pair potentialβu⁽²⁾(r)can be obtained by

βu⁽²⁾(r) =−ln

lim

ρ→0

h

g⁽²⁾(r)ⁱ

. (4.2)

Likewise, the following relation should hold βu⁽²⁾(r) = _lim

ρ→0

h−_lng⁽²⁾(r)ⁱ

= lim

ρ→0

h

βw⁽²⁾(r)ⁱ, (4.3)

but having tested this method on the known potentials of the referenceMonte-Carlo simulation (later in chapter 4.2.1), we obtained better results when we extrapolate the distribution function first and take the logarithm afterwards according to eq. (4.2).

Using thesuperposition approximation of Kirkwood (chapter 3.1.2) we can also ob-tain triplet potentials from correlation functions. As previously discussed for pair interactions, thesuperposition approximationworks increasingly well with decreasing density. But we need to take an additional factor e^−βu(3) into account if there is a triplet potentialβu⁽³⁾(r,s,t)present. So, we get in the limit of zero density

g⁽³⁾(r,s,t)−−→^ρ→0 g⁽²⁾(r)g⁽²⁾(s)g⁽²⁾(t)e^{−βu(3)(r,s,t)}. (4.4) Therefore, the triplet potential can be calculated as follows:

βu⁽³⁾(r,s,t) =−ln lim

Similar to eq. (4.3), we could also obtain the triplet potential by extrapolating the corresponding potentials of mean force However, we again achieved better results by proceeding according to eq. (4.5).

Unfortunately, we cannot use these relations to determine the interaction potentials directly because measurements become very difficult at densities close to zero. Ap-proaching zero density, the statistical accuracy determining the distribution function dramatically deteriorates. This can be exemplified with a short quantitative consider-ation. As explained in chapter 3.1.5, the calculation of distribution functions consists essentially of counting particles. Decreasing the density by a factor of 1/2 leads to an equal decrease by 1/2 of central particles and, for the pair distribution function, to a further decrease by 1/2 of neighbouring particles. All in all, the number of entries in the histogram is down by a factor of 1/4 for the pair distribution function. Moreover the count for the triplet distribution function is reduced by a factor of 1/8! The cor-responding numerical errors of the distribution functions thus increase by a factor of 2 and 2√

2 respectively. In a simulation, one can compensate for this loss of counts with longer observation times and larger systems. In an experiment however, the observation time is limited by the stability of the colloidal suspension and the system size is mainly limited by the optical apparatus.

Our idea to overcome these problems is to perform a series of measurements at fi-nite densities and extrapolate the distribution functions to zero density. We will show

the details of the procedure only for the pair distribution function. For the triplet dis-tribution function, it is essentially the same. At sufficiently low densities ρ, we can write the density dependency of the pair distribution functiong⁽_ρ²⁾(r)in powers ofρ

g⁽ρ²⁾(r) =α0(r) +α₁(r)ρ+α2(r)ρ²+α3(r)ρ³+. . . (4.7) The lowest order coefficient of the expansionα₀(r)is obviously the pair distribution function at zero density g⁽_ρ²₌⁾₀(r) which is exactly the function we are interested in because of its direct relation to the pair potential.

The Scheme

To calculate the coefficient α₀(r), we need a set of n (pair) distribution functions g⁽ρ²_j⁾(r)at densitiesρ_j (j= 1, . . .n). We then have to take into account that any distri-bution function is only known at discrete distancesr_i. At each of these distances we thus end up withn data points consisting of the densityρ_j and the value g⁽_ρ²_j⁾(r_i)of the distribution function.

ρ₁,g⁽_ρ²₁⁾(r_i),

ρ₂,g⁽_ρ²₂⁾(r_i), . . . ,

ρ_n,g⁽_ρ²_n⁾(r_i). (4.8) We now have to find suitable coefficientsα_k(r_i)for the expansion up tomth-order inρ g⁽ρ²_j⁾(r_i) =α₀(r_i) +α₁(r_i)ρ+· · ·+α_m(r_i)ρ^m. (4.9) In this notation, we can easily see that our problem is equivalent to fitting an m-th order polynom

y= a₀+a₁x+a₂x²+· · ·+a_mx^m (4.10) to a set of data points

(x₁,y₁),(x₂,y₂), . . . ,(x_n,y_n) with n≥m+1. (4.11) Usually, the quality of a fitting curve is measured by adding up the square errors at each data point. The best fitting curve is thus the function f(x)that minimises the sum of the square errorsΠ

Π =

∑

n i=1

[y_i− f(x_i)]² (4.12)

=

∑

n i=₁

y_i−(a₀+a₁x_i+a₂x²_i +· · ·+amx^m_i )²=minimum.

It is therefore necessary that the derivatives with respect to the unknown coefficients Expanding the above equations, we get

∑

n above equations. As we remember, the coefficienta₀corresponds to the lowest order coefficientα₀(r_i)at distancer_i in eq. (4.9). And,α₀(r_i)_equalsg⁽_ρ²₌⁾₀(r)from which we can simply obtain the pair potential by taking the negative logarithm. So far, we have a scheme that theoretically works. But how do we expect it to perform practically?

Certainly, our approach works only for sufficiently small densities at which we can neglect higher order terms in the density expansion of eq. (4.7). But how do we know what ‘sufficiently small densities’ actually are? We have to admit that we do not have any theoretical predictions for the size and the importance of each correction term.

The only way to answer this question is to apply this method to a set ofMonte-Carlo simulations with known potentials. We have done that and we will discuss the valid-ity and the limitations of this method in great detail later in chapter 4.2.1. In general, we can expect that we should surely stay away from the crystallisation transition and from any dense liquids with longer ranging order effects. Such strongly correlated

systems can certainly not be described by a low order polynomial density expansion.

Also, higher order expansions become very soon very impractical. We have therefore restricted our studies to linear, quadratic and cubic extrapolation. As we will see in section 4.2.1, this range of extrapolation is sufficient up to moderate densities which are well accessible in the experiment.

Im Dokument From Triplet Correlations to Triplet Interactions in Colloidal Suspensions (Seite 72-76)