Computer Simulation of the Pre-Heating, Gelation, and Rheology of Acid Skim Milk Systems - Food Colloids: Self-Assembly and Material Science - page 272

over the time period of the process encountered in practice. For the model this
simplifies matters considerably. Because we do not have complexes involving
several casein micelles, we can restrict the simulated system to a single casein
micelle. Moreover, as the rate at which the large casein micelles diffuse is
substantially less than that of the small whey proteins, we can reasonably
ignore their motion altogether.
In the first set of pre-heating simulations, there is one single large particle of
radius R l ¼ 8 fixed at the origin, surrounded by N s ¼ 200 small particles of size
R s ¼ 1 in a cubic box of dimensions 30 30 30 units. This corresponds to
volume fractions of 7.9% and 3.1% for the casein and whey protein compo-
nents, respectively. This relative size difference is quite a bit less than that which
occurs in reality, and also the relative volume fraction is different. The amount
of whey protein in real milk is only sufficient to cover about 40% of the
surface, 5 and so by using these values the same applies in the simulated system.
A realistic size ratio of R l /R s E

30 would result in about 60 times as many of
the small particles, and the consequent increase in computational effort in the
later gel stage would make that simulation unfeasible.
It was not the purpose of this study to give quantitative predictions of the
properties of the real systems. Rather the aim was to provide some qualitative
trends in the possible behaviour of a binary system with a rather large size
ratio, not only in relation to the complexation, but also for the subsequent
gelation and large deformation rheology. A quantitative study of a model
imitating all relevant aspects of the real system would require an extremely
long simulation. But, in any case, there are not sufficient experimental
observations available to validate or calibrate the parameters within such a
refined model. Hence, we have chosen the convenient set of parameter values
mentioned above.
The effect of pH was translated into different values of the interaction
parameter for the physical interaction strength between the small particles and
the surface of the large particle (A sl ) and the bonding probability for binding of
the whey proteins to the casein micelle (p sl ). The purpose of this preliminary
numerical model was to reproduce the salient features of the real system during
heat treatment by changing as few parameter values as possible. Hence, the
bonding distance (0.3), the small small bonding probability (p ss ¼ 0.01), the
small-small physical interaction strength and distance (A ss ¼ 5andC ss ¼ 1,
respectively, with 5k B T repulsion), the small-large interaction distance (C sl ¼ 1),
and the spring constants (k ab ¼ 250) were all kept unchanged. The time-step used
was Dt ¼ 0.00025. In reality the pH values during the heat treatment process
vary only over a relatively small range, from 6.35 to 6.9. Samples are typically
heated to 801C over a period of 10 min. The temperature plays no explicit role
in our model, other than it being responsible for controlling the rate of
denaturation. This particular set of parameter values was chosen so that small
whey aggregates would form, because of the irreversible binding, but the rate
would be slowed down due to the 5k B T barrier that has to be overcome before
binding can take place. All simulations were run over the same time period
(50,000 steps).