Computer Simulation of the Pre-Heating, Gelation, and Rheology of Acid Skim Milk Systems - Food Colloids: Self-Assembly and Material Science - page 271

with b i being the point at the particle surface to which the Hookean spring of
strength k ab is attached. Due to their non-central nature the bonds also induce
particle rotation. If a bond is stretched during deformation of the sample
beyond a maximal length, it is removed; the energy stored in the stretched bond
is immediately dissipated.
The size of the cubic or rectangular simulation box determines the volume
fraction of the particles. Periodic boundary conditions are used to avoid edge
effects. All parameters corresponding to sizes or distances are normalized to the
radius of the small particles (R s ¼ 1), and all quantities corresponding to
energies are normalized to units of k B T (k B T ¼ 1).
By choosing the time step much larger than the relaxation time of the
particles, we can neglect the second-order term in Equation (1). So we have

" #

S i þ X
neighbours

P ij þ X
bonds

dr i

dt ¼ 1

ð 4 Þ

B ij :

6 pZ R a

The differential equations are solved numerically using an Euler forward
method:

" #

S i ð t Þþ X
neighbours

P ij ð t Þþ X
bonds

r i ð t þ Dt Þ¼ r i ð t Þþ D t
6 pZ R a

B ij ð t Þ ð 5 Þ

:

The stochastic force S i gives the random translational displacements that, on
average, obey Einstein's law for an isolated particle. The dimensionless root-
mean-square displacement in the absence of interactions is (2D T Dt) 1/2 in each
direction. The translational diffusion coefficient, D T ¼ 1/6 pZ R s (taking k B T ¼ 1),
for the small particles is normalized to unity. The large particles (R l ¼ 8) move
substantially slower than the small ones. As well as the translational motion,
each individual particle also undergoes rotational diffusion. The rotational
motion is governed by the diffusion coefficient D R :
D R ¼ 3D T
4R a

ð 6 Þ

:

The implementation of rotational diffusion in the model is similar to that of
translational diffusion. Rotational diffusion of clusters results from the com-
bined translations and rotations of individual particles. Rheological properties
are studied by affine shear deformation of the image box at a constant low
shear-rate and applying the Lees-Edwards boundary conditions.

18.3 The Pre-Heating Phase
18.3.1 Methodology
During the heating of milk, the denaturated whey proteins can bind to each
other and to casein micelles. As the casein micelles themselves do not bind
directly to each other, the solution still remains as a stable suspension, at least