The Influence of Dimensionality on the Rate of Diffusive Escape From an Energy Well.

A number of models of material behavior rely on an understanding of the rate of separation of molecular bonds under random thermal excitation or due to stress induced bias. A perusal of textbooks that include discussion of this idea leaves the impression that such bonds can be characterized by a single generalized coordinate or degree of freedom, although several studies based on wells of higher dimension have been reported in the literature; for example, see [1] and [2]. While escape of a state from a one-dimensional well provides a useful framework for qualitative understanding of bond separation, the role of dimensionality on transition rates of a bond remains relatively unexplored. If the configurations of the bound molecules are represented by points, then three coordinates are needed to describe relative positions; for points with orientations, six coordinates are required to represent the state of the bond. The purpose here is to report on a study of the influence of dimensionality of bond state representation on a particular transition, namely, the so-called off-rate for a bond under random thermal excitation. As a basis for proceeding, we adopt a particular profile for a bond well in any number of dimensions, say U(r)=kTU∧(r)=kT{-cb(1-r2a2)2, r≤a0,r>a 1 where the nondimensional number cb represents the well depth in units of kT and a is a measure of the well width. The coefficient kT is a product of the Boltzmann constant k and the ambient temperature in degrees Kelvin. Thus cb is the well depth in units of kT, which has a value of about 4 pN nm at room temperature. The profile (1) is sketched in Fig. ​Fig.11. Fig. 1 The profile of energy versus distance which provides the common basis for identification of energy bond wells on one, two, and three dimensions A one-dimensional bond for states along a line within − a < r < a by reflection of the profile (1) with respect to r = 0, a bond for states within a circle of radius a in a plane is defined by rotating the profile about an axis through r = 0, and a three-dimensional bond is defined within a spherical region of radius a by assuming that the confining energy varies with distance according to the profile along every radial line emanating from the common point r = 0. According to reaction rate theory, the off-rate η for a large number of nominally identical, noninteracting bonds is defined in the following way. If the number of intact bonds among them at time t = 0 is n 0, and if no bond is reformed once separated, then the number n of intact bonds at time t > 0 is given by n = n 0 e −tη. Equivalently, η > 0 is the ratio of the instantaneous rate of decrease in the number of intact bonds to the total number of instantaneously intact bonds. In the context of the present discussion, this is equivalent to the ratio of the net flux of bonds out of the energy well to the total number of bonds remaining within the well. This ratio is calculated here in terms of the properties of bond wells in one, two, or three dimensions, each with the profile (1). The calculation procedure for two dimensions is developed in some detail and results for the other two cases are stated by analogy. A Confining Potential in Two Dimensions. As a representative case the state of a molecular bond is assumed to have two degrees of freedom and, consequently, all accessible states can be represented by the points in a plane. Any particular state can be identified by its plane polar coordinates r and θ, where r is the length of a radial line from the point of lowest bond energy to the point representing the current state and θ is the angular orientation of that line with respect to some arbitrary but fixed reference orientation. The free energy of the bond state, here denoted by U, is assumed to depend only on the current values of the coordinates r, θ that characterize the state. Specifically, the bond energy is zero when r assumes large values in some sense, indicating that the molecules are separated beyond each other’s influence. Essentially, the polar coordinates are viewed as representing the position of one molecule of the bond pair with respect to the other molecule, but other interpretations are useful for some purposes. When the molecules are fully bound, that is, at their minimum energy configuration, the state of the bond is identified with the origin of coordinates in the plane. All other accessible states of the bond are represented by means of a surface of elevation U(r,θ) over the coordinate plane. This surface, which is assumed to be continuous and relatively smooth, has the properties that U→0 as r→∞ and that it assumes its minimum value that r = 0. An axisymmetric example of such a surface based on the profile (1) is U(r,θ)=kTU∧(r,θ)=kT{-cb(1-r2a2)2,r≤a, 0 a, 0<θ<2π 2 where cb represents the maximum depth of the bond well as a multiple of the reference thermal energy kT; as such, the parameter cb is dimensionless. The state of the bond at any instant of time corresponds to some particular point in the plane spanned by the coordinates (r, θ). Because of its immersion in a thermal environment, the state of the bond varies randomly over points in the plane as a result of the fluctuations in position that it experiences. Consequently, the configuration of the bond is described in terms of the probability of being at any particular place at a certain instant of time. According to the central limit theorem, a very large number of noninteracting identical bonds will have an actual density of states identical to the probability distribution for the state of any particular bond. On this basis, attention is redirected to the study of the time evolution of the density of states of a large number of identical noninteracting bonds; this density of states is here denoted by ρ(r, θ, t). A feature of a diffusive process is that the energy imparted to a state as a result of its immersion in a thermal environment, which appears in the form of fluctuations in state, is rapidly returned to that environment through small steps. In other words, the energy dissipation during microscopic motion is relatively rapid and the change in configuration or step size executed as a result of that excitation is very small on the scale of a. It follows that the escape of a state from the well is a rare event among all changes in configuration. Under such conditions, the evolution in time of the density of states ρ is essentially a continuous diffusion process. The density of states in the plane evolves according to a local flux of states j(r, θ, t) representing the fractional number of states crossing a line in the plane that is perpendicular to the flux vector, measured per unit time and per unit length along that line. Local conservation of states requires that the rate of change of probability density at a point is balanced by the local divergence of the flux vector according to ∂tρ+∇·j=0 3 The flux is driven by a gradient in chemical potential arising from both a gradient in density of states and a gradient in the confining potential. If it is assumed that the flux of states varies linearly with the chemical potential gradient and that the proportionality factor is the spatially uniform and isotropic diffusivity D, then j=-D(∇ρ+ρ∇U∧)=-De-U∧∇(ρe-U∧) 4 The evolution of the density of states is then determined by solving the partial differential equations (3) and (4) throughout the relevant region of the plane, given an initial density of states and suitable conditions on the boundaries of the region considered. The second or alternate form of the transport equation (4) shows that the flux of states in everywhere normal to the level curves of the surface defined by the function of position ρeU∧ in the plane at any fixed time.