Diffusion reference frame

Suppose we offset this motion by applying a Galilean transformation x = x +Pt ). In the new reference frame, the system will move just as it did in the old reference frame but, because a — /pqt = / i P )t/A, its diffusion is slowed down by a Lorentz-Fitzgerald-like time factor 1-/3. Intuitively, as some of the resources of the random walk computer are shifted toward producing coherent macroscopic motion (uniform motion of the center of mass), fewer resources will remain available for the task of producing incoherent motion (diffusion). [tofI89]... 

By selecting the reference properly, the diffusion coefficients DA and DB can be made equal to each other. This value is termed the mutual diffusion (or interdiffusion) coefficient Dab- The reference frame is one across which no change in volume occurs (fixed volume) ... 

The fixed mass (M) and fixed number (N) reference frames can also be defined. The mutual diffusion is equally well described in any of these three coordinate systems ... 

When applied to a volume-fixed frame of reference (i.e., laboratory coordinates) with ordinary concentration units (e.g., g/cm3), these equations are applicable only to nonswelling systems. The diffusion coefficient obtained for the swelling system is the polymer-solvent mutual diffusion coefficient in a volume-fixed reference frame, Dv. Also, the single diffusion coefficient extracted from this analysis will be some average of concentration-dependent values if the diffusion coefficient is not constant. 

Choosing an appropriate reference frame such as a = 1 for all k, the diffusivity matrix can be written as... 

Figure 3-26 Quartz crystal growth and diffusion profile in (a) a laboratory-fixed reference frame and (b) an interface-fixed reference frame. At a given time, a given kind of curve is used to outline the crystal shape and plot the concentration profile.

Crystal growth rate may be constant, which could happen if temperature is decreasing or if there is convection. Smith et al. (1956) treated the problem of diffusion for constant crystal growth rate. In the interface-fixed reference frame, the diffusion equation in the melt is... 

The new reference frame is known as the interface-fixed reference frame, and the old reference frame is called the laboratory-fixed reference frame. The melt consumption rate u depends on whether the growth is controlled by interface reaction, or by diffusion, or by externally imposed conditions such as cooling. 

A reference frame is a frame in which a diffusion profile is measured. It is discussed in more detail in Section 4.2.1. 

In the case of one-dimensional crystal dissolution with u = Uq, if the reference frame is fixed at the faraway melt (x = oo), the melt does not flow even though the melt is generated at the interface at velocity u. (The interface moves as a rate of u.) Hence, the diffusion equation is Equation 3-9 without a velocity term ... 

Still in the case of one-dimensional dissolution, if the reference frame is fixed at the non dissolving part of the crystal x = - oo), the interface moves at a velocity of However, any point in the melt is moving at a velocity of w > That is, relative to the reference frame fixed to the nondissolving part of the crystal, the melt flows at a velocity of (w — w ). Hence, the equation to describe diffusion in the melt is the flow-diffusion equation (Equation 3-19b),... 

One-dimensional diffusive dissolution With the above general discussion, we now turn to the special case of one-dimensional crystal dissolution. Use the interface-fixed reference frame. Let melt be on the right-hand side (x > 0) in the interface-fixed reference frame. Crystal is on the left-hand side (x < 0) in the interface-fixed reference frame. Properties in the crystal will be indicated by superscript "c". For simplicity, the superscript "m" for melt properties will be ignored. Diffusivity in the melt is D. Diffusivity in the crystal is D. The concentration in the melt is C (kg/m ) or w (mass fraction). The initial concentration in the crystal is or simplified as or if there would be no confusion from the context. It is assumed that the interface composition rapidly reaches equilibrium. In the following, diffusion in the melt is first considered, and then diffusion in the crystal. 

Note that Equation 4-99 means that the solution is an error function with respect to the lab-fixed reference frame (x = x—2aVSf). In the interface-fixed reference frame, the solution appears like an error function, and its shape is often error function shape, but the diffusion distance is not simply especially... 

Using the interface-fixed reference frame (i.e., x = 0 at the interface) and defining melt to be at the right-hand side (x > 0) and crystal to be at the left-hand side (x < 0), the diffusion profile for the major component in the melt is... 

If the diffusion of a minor or trace element can be treated as effective binary (not uphill diffusion profiles) with a constant effective binary diffusivity, the concentration profile may be solved as follows. The growth rate u is determined by the major component to be n D ff, and is given, not to be solved. Use i to denote the trace element. Hence, w, and Dt are the concentration and diffusivity of the trace element. Note that Di for trace element i is not necessarily the same as D for the major component. The interface-melt concentration is not fixed by an equilibrium phase diagram, but is to be determined by partitioning and diffusion. Hence, the boundary condition is the mass balance condition. If the boundary condition is written as w x=o = Wifl, the value of Wi must be found using the mass balance condition. In the interface-fixed reference frame, the diffusion problem can be written as... 

In reading the literature on the fluid mechanics of diffusion, one encounters numerous difficulties because of the diversity of reference frames and definitions used by authors in various fields. Frequently more time is spent in translating from one system of notation to another than is spent in the actual study of the physics of the problem. It is hoped that the glossaries of terminology and symbols given here will be of use to those whose fields of research require a familiarity with the literature on diffusion. This exposition should emphasize the extreme importance of giving lucid definitions in any discussion of diffusion and mass transfer. 

Because field-gradient spin-echo measurements of D depend on no driving force such as a concentration, temperature, or velocity gradient, etc., they reflect Brownian motion of the molecules in the laboratory reference frame, and are usually referred to as self-diffusion. These attributes are further discussed in Sections 2 and 3. 

After this formal discussion of chemical diffusion, let us now turn to some more practical aspects. In order to compare the formal theory with experiment, we have to carefully define the reference frame for the diffusion process, which is not trivial in the case of binary or multicomponent diffusion. To become acquainted with the philosophy of this problem, we deal briefly with defining a suitable reference frame in a binary system. Since only one (independent) transport coefficient is needed to describe chemical diffusion in a binary system, then according to Eqn. (4.57) we have in a one-dimensional system... 

This result is important in practice since chemical diffusion experiments are normally analyzed with the help of concentration profile measurements in the volume reference frame. Thus, we obtain directly the only chemical diffusion coefficient D... 

The flux of a component in a solution can be complicated because components cannot always diffuse independently. This complication necessitates the introduction of different types of diffusion coefficients defined in specified reference frames to distinguish different diffusion systems. 

Four different types of diffusivities are summarized in Table 3.1. These include the self-diffusivity in a pure material, D the self-diffusivity of solute i in a binary system, Df, the intrinsic diffusivity of component i in a chemically inhomogeneous system, Dand the interdiffusivity, D, in a chemically inhomogeneous system. These diffusivities are applicable only in certain reference frames which are also listed in Table 3.1. In the remainder of this book, the type of diffusivity under discussion will be identified by these symbols when this information is relevant. When a diffusivity is identified in this manner, it may be assumed that the diffusion under consideration is being described in the proper corresponding frame. 

Fig. 9. Plot of normalized approach to equilibrium mass against the square root of time for a temperature-sensitive 10 x 4 PNIPAAm gel sheet swelling and shrinking between 10 and 25 °C-Shown are the curve fits to the kinetic data of theory developed from Fick s law of diffusion in a polymer-fixed reference frame [149]. The equilibrium degree of swelling is 17.0 at 10 °C and 11.1 at 25 °C the diffusion coefficients obtained from the curve fits are 2.3 x 10 7 cm2/s for swelling and 3.6 x 10 7 cm2/s for shrinking [121]...

Whether the volume transition occurs with respect to temperature, salt or solvent composition, each model works well. Furthermore, neither model outperforms the other in terms of statistical quality of fit of data to theory. Also, the diffusion coefficients obtained from either model are normally comparable, even if a correction for the difference in reference frames is applied [119, 121, 153]. Theoretically, the value of D obtained from the different models differ significantly only for very large volume changes. Thus, if the desire is to correlate different experiments or to reduce kinetic data to a single parameter either model can be used satisfactorily. 

Before closing, let us add a few comments about the diffusion constant D in the equations above. For diffusion of solutes like A and B in a solvent, it is customary to introduce diffusion constants Da and Db such that the associated fluxes J A and Jg are given relative to the flux of the solvent molecules. Since they are present in an overwhelming quantity as compared to the solutes, this will be equivalent to a center-of-mass reference frame, and if the system is stationary, to a laboratory reference... 

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