The Fixman Potential

Before we derive the appropriate expressions to calculate cL4/d from constrained simulations, we note an important difference between sampling in constrained and unconstrained simulations. There are two ways to gather statistics at (x) = . In unconstrained simulations, the positions are sampled according to exp —iiU while the momenta are sampled according to exp —j3K. If a constraint force is applied to keep fixed the positions are sampled according to A( (x) — x) exp —iiU. The momenta, however, are sampled according to a more complex statistical ensemble. Recall that 

In constrained simulations, = 0 so that is not an independent variable but rather a function of q and pg. Let us discuss the implications of this fact. Consider an arbitrary function /(x) and the following average  

The Hamiltonian function of a system which is constrained with (x) = and = 0 is given by 

This means that the average of / (f) can be computed using a constrained simulation for a modified Hamiltonian 

The second term in this potential is the so-called Fixman potential. With this potential we simply have 

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However, several authors took a slightly different route [17, 23, 25-27, 29] and derived an expression which does not explicitly introduce the Fixman potential. 

If the Fixman potential is not added to the Hamiltonian, the system is sampled differently and therefore correction terms must be added to (4.27). First, the weight Ze r in (4.24) must be reintroduced. Second, the Lagrange multiplier is different since the Fixman potential I /(2/3) In is not used. Considering (4.20) for A, it is possible to show that the Lagrange multiplier AF simply needs to be replaced by... 

The Fixman potential is not needed here since we are sampling with Hamiltonian Jtf. The second term is (4.31) can be expressed in terms of... 

Consider the constrained Hamiltonian with the Fixman potential... 

Equation (59) remains valid in the presence of constraints, such as fixed bond lengths and/or bond angles. In this case, if i and j are involved in a constraint, fy is the Lagrange multiplier force acting on i from j due to the constraint. If the rigid model is sampled in the simulation and properties of the flexible model in the limit of infinite stiffness are desired, fy must additionally incorporate contributions from the Fixman potential. 

This equation is a very important result because it shows that constrained simulations can be used to calculate dA/d . A possible algorithm would consist in running a constrained simulation with the Hamiltonian [which contains the extra Fixman potential 1/(2/ ) In Zc(q)], calculate the rate of change of -Z/J with at each step. Finally by averaging this rate of change the derivative of A can be computed. 

Imai (56) constructed a theory fear the intrinsic viscosity and sedimentation constant of ring polymers using the Fixman method shown in 2.3.2 for a model similar to the Hearst-Harris model which will be described in 4.2.2. This model reduces to the Rouse model in a limiting case. The excluded volume potential is included in the form of Eq. (2.26) and the same type of calculation as described in 2.3.2 was performed for a steady shear flow. Dynamic mechanical properties were not treated, although the extension to include this case is only a matter of tedious calculations. 

The effects of constraints on static properties of chain molecules (e.g., the equilibrium distribution of torsional angles about a particular bond) have been carefully considered [14, 74, 75]. The introduction of fixed bond lengths or angles requires the introduction of an additional potential term (the so-called Fixman potential ) in the equation of motion in order to compensate rigorously for the effect of the constraints. It has usually been argued that the... 

In the previous section, the diffusion equation for motion of a large number of particles was developed [see eqn. (211)]. When the solvent is at rest and both hydrodynamic forces and inter-reactants potential energy terms can be ignored, the equation becomes much simpler. This equation provides the basis for the analysis by Wilemski and Fixman. They chose to consider just one excited A molecule in a volume, V, together with m quencher molecules. The co-ordinates of all these molecules are rA and rQj, rQ2... rQ,n at a time f. Initially, thefluorophorand quencher molecules were positioned at rA° and rqj... r m. As a shorthand notation, these co-ordinates are called (r) and r0, respectively. The fluorophor was excited at time f°. 

To extend this analysis further to include the cations as well as anions, all that is required is to use the density nNtM and the potential energy UNrM. Rather than represent reaction by a boundary condition, there is much advantage using the Wilemski and Fixman approach of sink terms [ 51 ]. Let the anion and cation p react when they overlap in the region, say, where i r, — p j = R, and specify it by iix — 5 (i — jry — p i). The rate of... 

At this point it seems of interest that also Fixman (107) used such a scaled Gaussian distribution. This author treated the influence of the excluded volume on the intrinsic viscosity, using the subchain model, as well. However, in this work the influence of the excluded volume is treated more correctly. The Gaussian distribution is only scaled in order to make the correction terms due to the excluded volume potential as small as possible. Fixman arrived at the following lower limit of the Flory-Fox parameter ... 

Sections VIIA and D. In Fixman s formulation the initial rate and capture distance are arbitrary parameters, but the theory of this chapter gives these. The initial rate is the low-friction transition state limit of Eqs. (2.25) and (2.29). The Green s function of that approach will be complex, and should contain contributions not only at Aj., but also at from both Eqs. (2.34) and (6.1). Note that in both formulations the potential P(/ ) plays an important role. 

In summary, a formulation of the equilibrium cyclization and ring opening rates of a polymer chain has been obtained using the one-dimensional representation of the potential. These, when combined with the steady-state rate of the Wilemski-Fixman-Doi theory, " give an appropriate description of the cyclization kinetics. 

Ogasa and Imai (47) solved the same problem for dynamic properties in the non-free draining case by using a method different from Fixman s. The nature of the calculation is briefly shown here although the actual calculation is very complicated and tedious. The treatment is based on a trial spring potential S corresponding to a bond length expanded by a factor a (not x, y or z in this case) ... 

The molecular weight at which local motions become independent of molecular weight is a function of the chain structure and potentials [6]. Fixman s work shows that increasing the barrier for conformational transitions increases the chain length required to achieve asymptotic relaxation times [27]. 

According to M.C. Williams, in concentrated solutions intermo-lecular interactions are developed, whose intensity derive from both the potential function and chain segments distribution. Without specifying the nature of intermolecular forces, the author started from the theory of the polymer solutions equilibrium, elaborated by M. Fixman [1008], and proposed the following relation ... 

A theory developed by Zimm, like Fixman s treatment of the expansion factor discussed above (which it antedates), is based on a perturbation expansion for weak interactions. For interacting polymer chains, Zimm assumes that intermolecular intersegmental potentials of average force between segments i i in molecule 1 and I2 in molecule 2 are additive... 

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