Physical Chemistry of Foods

Some Theory. We will only consider air-filled systems.
Development of theory generally begins with a simple two-dimensional
array of hexagonal cells, as depicted in Figure 17.29a, and then is extended
to more complicated systems. The most important structural parameter is
therelative densityof the system, defined asr/rm, whereris the density of
the system andrmthat of the matrix material. The simplest case then is
prediction of themodulus, often Young’s modulusE.
For a sponge in which nearly all matrix material is in beams of
thicknessband lengthL,r/rm&(b/L)^2. Applying a forceFto a beam, it
will be deflected by an amountd, and it is derived from the theory of the
bending modulus thatd!FL^3 =Emb^4. Taking into account thatE¼s=e
(whereeis the Hencky strain),s&F=L^2 ande¼d=L, we arrive for open
cells atE&F=Ld, which results in

EðspongeÞ!Em

b^4

L^4



!Em

r

rm

 2

ð 17 : 21 Þ

In an idealsolid foamall of the matrix material is in walls of thickness
band edgeL, leading tor/rm¼b/L. Furthermore, the deflection of a wall is
then given byd!FL^2 =b^3 , and we arrive at

EðfoamÞ!Em

b^3

L^3



!Em

r

rm

 3

ð 17 : 22 Þ

Generally, Eq. (17.21) is well obeyed, but for most solid foams the exponent
is smaller than three, sometimes even as small as two, because even for a
closed-cell structure most of the matrix material may be in the beams rather
than in the walls. The theory can be extended for other rheological
parameters, such as the yield stress, but these relations are mostly not well
obeyed. As a general rule, however, one may state that for any rheological
parameterZ

Z¼KZm

r

rm

n

ð 17 : 23 Þ*

a scaling law that is mostly well obeyed within the ranger/rm¼0.5–0.15 (j
¼0.5–0.85); the values of the proportionality constantKand the scaling
exponentncannot be readily predicted and vary widely among systems. In
systems of j values below 0.5 the exponent is generally smaller,
approximating unity forj<0.2.
What will happen atlarge deformationwill depend on the properties of
the matrix material. If it is purelyelasticand not very stiff (i.e., rubberlike),
buckling of beams and cell walls will occur, as illustrated in Figure 17.29b.