Effective length factor of a polymer in solution If one wants to calculate the force needed to buckle a polymer in solution with Euler buckling, what would the effective length factor be? The polymer is free to move and rotate in solution as it sees fit (though it is still subject to stresses), but the ends cannot move very far with relation to each other, nor can they rotate freely with respect to each other. 
 A: It's not an easy question to answer within the limitations of space. This is a full-fledged research problem. I was working on a similar problem 2 years ago without any success. I would suggest you to look at this thesis where you fill find the necessary details and then reframe your question for specifics.
https://publishup.uni-potsdam.de/opus4-ubp/frontdoor/index/index/docId/3557
This thesis deals with the buckling instabilities of filaments in biological systems and uses a renormalization group technique to calculate the critical length of a filament to buckle under an applied force and vice versa. Filaments of the cytoskeleton are semiflexible polymers, i.e., their bending energy is comparable to the thermal energy such that they can be viewed as elastic rods on the nanometer scale, which exhibit pronounced thermal fluctuations. Like macroscopic elastic rods, filaments can undergo a mechanical buckling instability under a compressive load. In cells, compressive loads on filaments can be generated by molecular motors. This happens, for example, during cell division in the mitotic spindle. 
The author here investigates how the stochastic nature of such motor-generated forces influences the buckling behavior of filaments and reviews briefly the buckling instability problem of rods on the macroscopic scale and introduce an analytical model for buckling of filaments or elastic rods in two spatial dimensions in the presence of thermal fluctuations and presents an analytical treatment of the buckling instability in the presence of thermal fluctuations based on a renormalization-like procedure in terms of the non-linear sigma model where the author integrates out short-wavelength fluctuations in order to obtain an effective theory for the mode of the longest wavelength governing the buckling instability and calculates the resulting shift of the critical force by fluctuation effects and find that, in two spatial dimensions, thermal fluctuations increase this force. 
You will find many more details pertaining to the problem you are studying in the document. 
