### Explorations in Chocolate Texture

#### Overview of the Science

###### A spring is a useful analogy for describing the texture of food.
• Based on Hooke’s Law, the elongation of a spring is proportional to the force applied, resulting in a linear plot of force vs. elongation.

$$k = \frac{F}{\Delta L}$$

where F is the force applied and ΔL is the elongation.

• Three-dimensional foods can be described in terms of stress (F/A) and strain (ΔL/L0):

$$E = \frac{F}{A}\frac{L_0}{\Delta L}$$

where E is the elasticity, A is the area over which the force is applied, and L0 is the initial size of the object.

For example, image that a piece of tofu is cm thick. When a force of N is applied over an area of cm2, then it depresses by mm. This implies that the the stress is Pa, the strain is , and the elasticity is Pa.

• ###### Calculate the elasticity of a material, based on physical measurements of its deformation.
• The stress of a material is equal to the force applied to a material, divided by the area.
• The strain of a material is a dimensionless way to describe the amount of deformation.
• The elasticity of a material is equal to the stress divided by the strain. Stiffer materials, which are more elastic, require more force for the same deformation.
• ###### Calculate how the elasticity of a material scales with changes in the bonds at the microscopic level.
• The elasticity of a material is due to energy stored in the bonds between its molecules. More energy can be stored by increasing the density of bonds, or the energy stored in each bond. For a gel, the energy is roughly equal to thermal energy (kBT). The density of the bonds is equal to the reciprocal of the cross-link spacing, l, cubed:

$$E = \frac{k_B T}{l^3}$$

When l = nm and T = °C, then the elasticity is kPa.

• ###### Draw an approximation of how the microscopic structure of food changes when its elasticity changes.
• Extra firm tofu has a smaller cross-link spacing than silken tofu.
• Well-done steak has a smaller cross-link spacing that rare steak.
• The elasticity of a gel is inversely proportional to the density of cross-links.