The strong force, also known as the strong nuclear force, is indeed responsible for both holding quarks together to form protons and neutrons, and for binding protons and neutrons together to produce atomic nuclei. This phenomenon can be understood through the concept of color charge and the theory of quantum chromodynamics (QCD).
According to QCD, quarks carry a property called color charge. However, it is important to note that "color" here does not refer to the visual perception of color; it is simply a label used to describe a property analogous to electric charge in the theory of electromagnetism. Quarks can have three different color charges: red, green, and blue (plus their corresponding anticolors: antired, antigreen, and antiblue).
The strong force arises from the exchange of particles called gluons, which carry the color charge and mediate the interactions between quarks. Gluons can interact with quarks and other gluons, creating a complex network of interactions. This exchange of gluons between quarks generates the strong force.
When quarks are close to each other within a proton or a neutron, the strong force becomes very strong, effectively binding them together. The interaction between the quarks and gluons is such that the overall color charge of a proton or neutron is "colorless" or "white" when you combine the three different colors or anticolors. This property is known as color confinement.
Similarly, the strong force is responsible for holding protons and neutrons together within the atomic nucleus. The nuclear force, which is a residual effect of the strong force, acts between nucleons (protons and neutrons) in the nucleus. It is this residual strong force that keeps the protons and neutrons together despite the electrostatic repulsion between the positively charged protons. The nuclear force is effective only over very short distances, on the order of femtometers (10^−15 meters).
In summary, the strong force, mediated by gluons, binds quarks together to form protons and neutrons, and it also acts between protons and neutrons to hold them together within atomic nuclei. The remarkable properties of the strong force, such as color confinement and the exchange of gluons, allow it to play a dual role in the structure of matter at the subatomic level.