The physics of squeaky shoes

In these unsettling and trying times, I wanted to write about the physics of a challenge I'm facing in my professional life: super squeaky shoes.  When I wear a particularly comfortable pair of shoes at work, when I walk in some hallways in my building (but not all), my shoes squeak very loudly with every step. How and why does this happen, physically?  

The shoes in question.

To understand this, we need to talk a bit about a friction, the sideways interfacial force between two surfaces when one surface is sheared (or attempted to be sheared) with respect to the other.  (Tribology is the study of friction, btw.)  In introductory physics we teach some (empirical) "laws" of friction, described in detail on the wikipedia page linked above as well as here:

1.  For static friction (no actual sliding of the surfaces relative to each other), the frictional force \(F_{f} \le \mu_{s}N\), where \(\mu_{s}\) is the "coefficient of static friction" and \(N\) is the normal force (pushing the two surfaces together).  The force is directed in the plane and takes on the magnitude needed so that no sliding happens, up to its maximum value, at which point the surfaces start slipping relative to each other.
2. For sliding or kinetic friction, \(F_{f} = \mu_{k}N\), where \(\mu_{k}\) is the coefficient of kinetic or sliding friction, and the force is directed in the plane to oppose the relative sliding motion.  The friction coefficients depend on the particular materials and their surface conditions.
3. The friction forces are independent of the apparent contact area between the surfaces.  
4. The kinetic friction force is independent of the relative sliding speed between the surfaces.

These "laws", especially (3) and (4), are truly weird once we know a bit more about physics, and I discuss this a little in my textbook.  The macroscopic friction force is emergent, meaning that it is a consequence of the materials being made up of many constituent particles interacting.  It's not a conservative force, in that energy dissipated through the sliding friction force doing work is "lost" from the macroscopic movement of the sliding objects and ends up in the microscopic vibrational motion (and electronic distributions, if the objects are metals).  See here for more discussion of friction laws.

Shoe squeaking happens because of what is called "stick-slip" motion.  When I put my weight on my right shoe, the rubber sole of the shoe deforms and elastic forces (like a compressed spring) push the rubber to spread out, favoring sliding rubber at the rubber-floor interface.  At some point, the local static friction maximum force is exceeded and the rubber begins to slide relative to the floor.  That lets the rubber "uncompress" some, so that the spring-like elastic forces are reduced, and if they fall back below \(\mu_{s}N\), that bit of sole will stick on the surface again.  A similar situation is shown in this model from Wolfram, looking at a mass (attached to an anchored spring) interacting with a conveyer belt.   If this start/stop cyclic motion happens at acoustic sorts of frequencies in the kHz, it sounds like a squeak, because the start-stop motion excites sound waves in the air (and the solid surfaces).  This stick-slip phenomenon is also why brakes on cars and bikes squeal, why hinges on doors in spooky houses creak, and why that one board in your floor makes that weird noise.  It's also used in various piezoelectric actuators. 

Macroscopic friction emerges from a zillion microscopic interactions and is affected by the chemical makeup of the surfaces, their morphology and roughness, any adsorbed layers of moisture or contaminants (remember: every surface around you right now is coated in a few molecular layers of water and hydrocarbon contamination), and van der Waals forces, among other things.  The reason my shoes squeak in some hallways but not others has to do with how the floors have been cleaned.  I could stop the squeaking by altering the bottom surface of my soles, though I wouldn't want to use a lubricant that is so effective that it seriously lowers \(\mu_{s}N\) and makes me slip.  

Friction is another example of an emergent phenomenon that is everywhere around us, of enormous technological and practical importance, and has some remarkable universality of response.  This kind of emergence is at the heart of the physics of materials, and trying to predict friction and squeaky shoes starting from elementary particle physics is just not do-able.