Langmuir 1992,8, 2932-2937
2932
Casein Micelles: Diffusivity as a Function of Renneting Time C. G.de Kruif Netherlands Institute for Dairy Research, NIZO, P.O. Box 20, 6710 BA Ede, The Netherlands Received April 30,1992. In Final Form: July 8, 1992 The diffueivity of K-casein micelles in renneted skim milk was measured by dynamic light scattering. By use of the Stokes-Einstein relation, the apparent particle radius was calculated. Aa a result of the enzymatic proteolysis of the K-casein peptides that provide the steric stabilization of the casein micelles, the hydrodynamic radius diminishes. However, as a result of the increasing attractive interactions the apparent micelle radius increases. Depending on the length scale, one can distinguish collectivediffusion and self-diffusion. For both phenomena, theoretical descriptions exist. Combining this theory with an adhesive-hard-sphere model for the casein micelles allowe a quantitative prediction of the experimental resulta. In this way a clear insight into the initial stages of the renneting proces is gained and this can be used for further modeling and control of the cheesemaking process. gelation, however, is a find stage. As a prelude the gradual change in colloidal stability of the micelles is observable In his book Die Atome, Jean Perrin’ described the by experimental techniques such as light scattering, classical experiment on measuring the diffusivity of gum transmission measurements,and viscosity measurements. arabic particles. He thus verified the famous theoretical In order to account for the observed changes Payens,8 result of Einstein2 that the mean square displacement of Walstra? and Dalgleish’O made use of the classical Von a spherical Brownian particle is linear in time. Together Smoluchoweki theory for the kinetics of flocculation. In with the result of Stokes on the fluid dynamic drag on a this theory the colloidal stability is presumed to change sphere, this theory forms a basis for determination of the from stable to unstable instantaneously or a t least on a hydrodynamic size of noninteracting colloidal particles. time scale which is short compared to colloidal diffusivity. If particles start to interact either through excluded During renneting of (skim) milk micelles the changes are volume interactions at high volume fraction or through gradual and this requires a more refined description of direct particle interactions because of the non-hard-sphere the colloidal dispersion properties in the time span character of the particles, the theory becomes more If this “induction” or “lag” period is preceding gelation. complex. Both for hard-sphere particles and for colloidal understood and described adequately, it will be possible particles that exhibit ashort-rangedattraction in addition to describe the gelation process more reliably. This in to a hard-core repulsion, the theory is well d e ~ e l o p e d . ~ ~turn will help to control the cheesemaking process. In fact measurement of the diffusivity of the particles is Before the addition of chymosin the viscosity of skim a very sensitivemeans of measuring the particle potential? milk can adequately be described by that of a dispersion Here we will illustrate this on such a familiar and of hard spheres.11J2 In other words the casein micelles extremely stable and beautiful colloidal system as milk, can be viewed as hard spherical particles in a continuous which contains casein proteins present in micellar aggresolvent medium consisting of water + salta + whey gates of about 100 nm radius. In cheesemaking these proteins. The hard-spheres character must be interpreted micelles are destabilized, which then leads to a gelation. in such terms that the repulsive interaction pair potential Here we will followthis procees by measuringthe diffusivity of two casein micelles rises to several kT units of thermal of the micelles. energy over a distance which is much smaller than the An important step in cheesemaking is the addition of particle radius of 100 nm. The thickness of the K-caseinan enzyme to the milk. Very often one uses rennet, which (GMP) brush is 7 nm, and the mutual interdigitation of is an extract from the stomachof young calvesand contains two brushes will be a fraction of 7 nm. the enzyme chymosin as a main component. On the On addition of chymosin the effective hydrodynamic addition of rennet to (skim)milk, enzyme chymosin splits thickness decreases but also, and more importantly, off the glyco macro peptide (GMP) part of the K-casein attractive interactions become operative. These attracprotein that provides the colloidal stability of casein tione must be short-ranged as well and may originate in micelles. At low chymosin concentration this hydrolysis van der Waah interactions or more specifically in so-called appears to be firat order with respect to the enzyme hydrophobic interactions, or hydrogen bonding and ion concentrationand the destabilization of the casein micelles bonding. In view of the stability of renneted micelles at is gradual. 4 “C,hydrophobic interactions appear to be present. In normal cheesemaking practice, it takes 20-30 min before the colloidal destabilization leads to a gelation. The For a description of the macroscopic behavior of such a dispersion we can use the adhesive-hard-sphere (AHS) (1)Perrin, J. Die Atome; Steinkopff: Dresden, 1914. model (seeFigure 1). Although this particle pair potential (2)Einstein, A. Investigations on the Theory of the Brownian Movement; Furth, R.,Ed.; Dover: New York, 1966. 1. Introduction
(3)Batchelor, G. K.J. Fluid Mech. 1989, 131, 166. (4) Batchelor, G. K. J. Fluid Mech. 1972,62, 246. (6) Jansen, J. W.;De Kruif, C. G.; Vrij, A. J. Colloid Interface Sci. 1986,114, 601. (6) Cichocki, B.; Felderhof, B. U. J. Chem. Phye. 1990,93,442. (7) Rouw, P. W.; De Kruif,C. G. J. Chem. Phye. 1988,88,7799.
(8) Payem, T. A. J. Adu. Colloid Interface Sci. 1989, 30,31. (9) Wabtra, P. J. Dairy Sci. 1990, 79,1966. (10)Dalgleieh, D. G.; Robson, E.W. J. Dairy Res. 1984,61, 417.
(11)Griffin, M.C.A.; Price, J. C.; Griffin, W. C. J. Colloid Interface Sci. 1989, 128, 223. (12)Jeurnink, Th. J. M. Neth. Milk Dairy J., in press.
0743-7463/92/2408-2932$03.00/0 @ 1992 American Chemical Society
Langmuir, Vol. 8, No.12, 1992 2933
Diffueiuity of Casein Micelles
Figure 1. Particle pair interaction between two casein micelles modeled as an adhesive-hard-spherepotential. The width of the well A is smaller than the length of the K-casein peptide that stabilizes the micelles. The well depth c is proportional to the amount of K-casein split off and increases linearly in time. seems to be an oversimplification of reality, it is not, as long as the range of the interactions is short compared to the particle size. The exact shape of the pair potential is not important for the theory presented below. This adhesive-hard-sphere theory allows a consistent description of macroscopic dispersion properties as a function of "stickiness" and volume fraction. Using AHS theory, we can describe such seemingly unrelated properties as dispersion viscosity13and light transmis~ionl~ with the same consistent set of parameters. Here we will present experimental results obtained with dynamic light scattering (DLS) experiments. In DLS experiments one measures the diffusivity or "mobility" of the colloidal particles. In a generalized Stokes-Einstein model the diffusivity depends on the ratio of a driving force over a fluid dynamic friction factor. For a single particle this reduces to the well-known Stokes-Einstein result for diffusion
D = kT/6uvao where k is the Boltzmann constant, Tis the temperature, 7 is the solvent viscosity, and a0 the particle radius. In the next section we will present a general discussion on the diffuivity of AHS as a function of adhesiveness and volume fraction. For a not infinitely dilute dispersion, one can recognizedifferentdiffusioncoefficientsdepending on the length and time scale of the diffusion. In the experimental section we will present diffusivity data obtained in this laboratory and elsewhere. In the last section we will show that the AHS model gives a consistent interpretation of different macroscopic properties. 2. Theoretical Background 2.1. The Adhesive-Hard-SphereModel. We will treat casein micelles as (adhesive) hard spheres. This simplification is justified by the following. As noted in the Introduction the effective viscosity of casein micelles as a function of volume fraction follows that of hard spheres. It must be realized that the stabilization of casein micelles at room temperature is dominatedby the steric interactions of the GMP part of the K-casein peptide. Coulombic repulsions must play a minor role in view of the high ionic strength of milk. The length of the GMP part of the K-caseinis about 7 nm, while the effective radius of a casein micelle is of the order of 100 nm. So both repulsion and/ or attraction take(s) place over a distance of 7% or less (13) De Kruif, C. G.; Jeurnink, Th. J. M.; Zoon, P. Neth. Milk Dairy J., in prea~. (14) De Kruif, C. G. Proceedings of the Food Colloids Conference, Luntaren, 1992.
of the particle radius. This allows us to apply the AHS model, which requires that the range of the interaction potential is short compared to the particle radiua. It was further shown by Regnaut and F i a v e ~that ~ ~ in that situation the details of the shape of the potential become irrelevant for the equilibrium properties. Thus the potential-well preceding the hard core may have the shape of a square well or of a triangle. It is realized that the width of the well strongly influences the transport coefficients like a diffusion coefficient, due to the rapid variation of hydrodynamicinteractions with particle separation. So the actual width of the well may be different from the value adopted in this paper. It ia important however, that it is constant within the experiment. Baxter16used a mathematicallyconvenient potential in order to solve the statistical mechanics equations. Here we will use the square-wellmodel, which both has a certain mathematical convenience and also reflects the physics of the system. In Figure 1 we sketched the square-well potential on the casein micelle scale. The square-well potential of width A and depth t is given by +a,
V(r)=
-e,
0,
0 < r < 2a0 2a0 < r < 2a, 2a0+ A < r
+A
The well depth t appears to depend on the degree of proteolysis of the K-casein hairs. Since the interaction ia entropic in nature, we assumed13 that the depth of the well is proportional to the logarithm of the relative amount of the GMP still present. Therefore t
P - 1 - [PI
kT where h is a proportionality constant depending on the choice of the well width (h = 1.61, [PI is the amount of GMP split off, while [P,] is the total amount and thus the final amount split off. The proteolysis of K-casein by the enzyme chymosin can be described in normal practice situations by a first-order kinetics. We found that
where [El is the concentration of chymosin added relative to the standard amount [&I. So [El/[&] is of order 1. The reaction constant k is of order s. The second osmotic virial coefficient of a colloidal dispersion can be calculated from
B; = 2rJrz(l- e-"(')/'')
(3)
Substitution of the square-well potential and performing the integration yields
B, = B ; / v ~=~4 + 12 -(I A
- e++'>
U
(4)
Bz=4-1/~ where V H S is the volume of the hard sphere. The
parameter T was introduced by BaxteP and i~ a measure, just as Bz, for the attraction. For hard spheres B2 4 and T -w
(7
> 0).
The volume fraction of micelles 4 is calculated from the amount of skim milk powder dissolved and from the specificvolume as determined from the Einstein Viscosity. (15) Fbgnand, C.; Ravey, J. C. J. Chem. Phys. 1989,91, 1211. (16) Baxter, R. J. J. Chem. Phys. 1968,49,2770.
2934 Langmuir, Vol. 8, No. 12,1992
de Kruif
Due to the proteolysis reaction the volume fraction of micellesdecreases since the effectivehydrodynamic radius of the micelles decreases. To first-order approximation this is given by
4 = 4o(l-
ff)
(5)
with ff=3
f a0
[PI -
P-1
where f is the hydrodynamiclength of the GMP, which is much smaller than ao, the initial hard sphere radius. We will use f = 7 nm as is found in refs 17 and 18. 2.2. Diffusivity. We consider a dispersion of spherical colloidal particles (radius ao) in a continuous background, the solvent. As a result of the continuous stochastic bombardment of the colloidal particle by the solvent molecules, the particle executes a Brownian motion. At any moment in time the kinetic energy content in any direction of such a particle will be of order kT. This energy will be dissipated as hydrodynamicfriction energy. The time scale of this event is estimated from
A
0 0 0 -----
U
collective diffusion
du m5 = -6rqa0V where m is the particle mass and u is the 'initial" Brownian velocity. This velocity decays exponentially on a time scale tB
z m/6?rqao
This time scale is called the Brownian time scale tg. After many such stochastic movements the particle has a mean square displacement ( h2) = 6Dt ( t B
