Physics 136 Class Notes on Application of Statistical Mechanics to Polymers The tools we have allow us to study many interesting classical systems One of these is the statistical mechanics of polymers long molecules Although this has been studied for decades it has become particularly exciting recently because biopolymers such as DNA allow the investigation of individual polymers In turn the statistical mechanics of such polymers is important in the biological function In these notes I will introduce some very simple issues and discuss some related experiments on single strands of DNA Polymers and random walks The simplest model of a polymer is a chain of like monomers of length a where each link is completely free to rotate in any direction A polymer of N such links is equivalent to a random walk of N steps of length a One dimensional random walk In one dimension the probability distribution of arriving at X after N steps starting from the origin is given by the binomial distribution P X N Pbin m N 1 with X 2m N a m forward steps minus N m backwards steps Using the properties of the binomial distribution we have hX i 0 X 2 N a 2 We also showed that for large N the probability distribution approaches a Gaussian In this limit we are usually interested in X a and so can also replace the discrete possible X by a continuum The probability density p X such that p X d X gives the probability of ending between X and X d X is 1 X2 p X 2 exp 2 2 1 2 1 PN with 12 N a 2 The large N results also follow from the central limit theorem Since X i 1 xi is the sum of N independent random variables for large N the distribution is Gaussian with variance N xi2 N a 2 Three dimensional random walk For a three dimensional polymer the large N result is similarly Gaussian Now RE X Y Z that the probability of ending up at x coordinate X is 1 X2 p X x2 N xi2 N a 2 3 exp 2 2 2 x x P xEi so 3 with similar results for p Y p Z Then the probability density for ending up at RE is 1 R2 E p R p X p Y p Z 4 exp 2 2 2 3 with 2 N a 2 3 Thus a long polymer will form a ball of radius of order N a much q smaller than the stretched length L N a The ball is often characterized by the radius of gyration RG E xi xE j 2 with the average over all links i j and over fluctuations RG is also of order N a E albeit one that must be evaluated numerically We start with We can obtain an exact expression for p R Z Z P E d 2 x1 d 2 x2 d 2 x N i xEi R E Z Z p R 5 2 2 2 d x1 d x2 d x N 1 Radial Probability N 5 Exact Gaussian 0 2 0 4 0 6 0 8 R L 1 0 E Figure 1 Comparison of exact and Gaussian expressions for the radial probability distribution 4 R 2 p R for a 3d random walk or ideal polymer R where here d 2 x denotes the integral P over the surface of a sphere of radius a This expression is hard to E However if we take the Fourier transform evaluate because of the constraint xEi R Z E E d3 R p E q ei qE R p R 6 the constraint gives exp i P i qE E xi and the integral factorizes R p E q d 2 xei qE Ex 4 a 2 N 7 This trick of using a Fourier transform to simplify the manipulation of a delta function constraint is often useful The integral in the numerator of Eq 7 is Z 1 4 a 2 sin qa 2 a 2 d cos eiqa cos 8 qa 1 Inverting the Fourier transform gives Z E 1 p R 2 3 e i qE RE sin qa qa N d 3 q 9 The integration over the angles of qE is easily done to give E p R 1 2 2 Z 0 sin q R qR 2 sin qa qa N dq 10 It is useful to introduce Q qa and L N a when the expression reduces to E p R 1 2 2 a 3 Z 0 sin N Q R L N Q R L sin Q Q N d Q 11 This is now easy to evaluate and plot in Mathematica for example Note that result depends on the fractional E 0 for R L 1 the extension R L and if you evaluate Eq 11 for various N you will find that p R 2 fully stretched case It is probably best to plot 4 R p R giving the probability of finding the endpoint at distance R from the origin Remarkably the Gaussian approximation is quite accurate even for N as small as 5 see Fig 1 Elasticity of the ideal polymer Entropy method microcanonical ensemble Consider a polymer tethered to two points on the x axis one at x 0 and the other at x X Left to itself the most probable configuration would be X 0 to stretch the polymer to endpoint separation X therefore decreases the entropy The force can be calculated as F S T X 12 with S X k B ln X and X the number of microstates consistent with the fixed endpoints We could show this for example by tethering the polymer to a piston that changes the volume of a gas and then by maximizing the total entropy show that this expression gives the right force to balance the force on the piston from the gas The number of states X is simply proportional to the probability of the polymer with a free end arriving at X microcanonical ensemble probability number of states X2 X exp 13 2N a 2 3 So S X 3 kB 2 X 2 N a2 14 and the force is 3k B T X 15 a L introducing the stretched out length L N a again We have used the Gaussian expression for p X which is only good for X not too large i e not approaching L and so the force expression is good for small X only Notice that we get Hooke0 s Law with a spring constant proportional to k B T The force arises completely from entropic effects Similar arguments can be used to calculate the elasticity of ideal rubber crosslinked polymers where again the elasticity is entropy dominated F X Energy method Gibbs like ensemble We can also do the calculation by fixing the force F rather than the endpoint X and the temperature T For an applied force F the energy of a configuration with endpoint separation X is E F X The partition function is then …
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