Tadmor, Chapter 7 deals with the basic approaches to the description and understanding of mixing in continuous and batch processes. Many of these ideas are critical to design and use of polymer and chemical process equipment and the purpose is to make you familiar with the terminology and approaches which are commonly used. These include statistical descriptions of mixing, and various distribution functions which are used to calculate average features from a process. Of particular importance are strain distribution functions(SDF) and residence time distribution functions (RTD).

Mixing is often the critical step of polymer processing and is the least understood. Consider an extruder as a model. Mixing of pellets involves solid/solid distributive mixing in the early stages of extrusion or in a mechanical mixer such as a V-mixer. This might involve mixing of a color concentrate or foaming agent concentrate. If filler is mixed such as carbon black, silica or titania mixing proceeds to Solid/Liquid mixing in the extruder barrel. Mixing of two immiscible polymers in an extruder is common (LDPE and LLDPE) which occurs mostly in the high shear parts down stream in the extruder. Mixing also can occur in down stream parts such as in a calander or film blowing apparatus.

For low-viscosity liquids mixing is commonly by molecular diffusion or eddy diffusion such as milk in coffee. For high viscosity liquids mixing is by convection (bulk diffusion). There are two types of mixing which can occur in such high viscosity systems: (FIGURE 7.1 pp. 199)

1) Increase in Surface Area usually caused by shear. This is what happens in laminar flow mixing in an extruder. This is typically a ordered rearrangement. Since surface area is related to strain, the strain distribution function is needed which is related to the residence time distribution function. (FIGURE 7.2 pp. 200)

2) Distributive mixing: Randomization of Particle distribution. This is what happened in the Brabender mixer lab. This is common for solid/liquid mixing.

For 1) the strain needs to be calculated from the residence time and the strain rate. The viscosity's of the two phases being mixed are important parameters. For mixing of a solid and a liquid the yield stress for the particle to break-up is important.

We can distinguish between Intensive mixing where domains or particles break-up and Extensive Mixing due to convective flow.

Either of these can be Distributive (rearrangement of parts) or Laminar, increase of surface area as occurs in shear stretching or squeezing of a fluid.

The first question is to consider what "goodness of mixing" means in a quantitative way.

To do this you need to consider the sample concentration of minor components at "points" in the mixture.

The phrase "Concentration at a point" involves defining what a "point" is. Continuum mechanics defines a point as a very small volume from a macroscopic sense which can be described using thermodynamic properties, i.e. a point contains many particles, and can be describe with thermodynamic parameters, T,P, velocity, concentration etc., i.e. can be describe with bulk properties so is not an atom.

In the absence of molecular diffusion and eddy diffusion each point is a pure phase, e.g. Carbon black or polymer. For a larger volume of a "point you need to define a concentration, i.e. a point is larger than a single ultimate particle

Because of these issues involved in describing what a "point" is for quantitative descriptions of mixing, you should consider materials in terms of the scale of observation. Ultimately, the size and description of a "point" depends on the application you are interested in.

Gross Uniformity: Average concentration for whole sample and the randomness of the samples, e.g. the percent blue in a blue shopping bag. For Gross uniformity the scale of observation is very large compared to the domains being mixed, "r" is big. Typically Gross uniformity is judged at the size scale of the article being produced, i.e. the shopping bag.

Texture: Correlations between domains, e.g. streaks in the shopping bag. Typically "r" for Texture is smaller than "r" for Gross Uniformity.

Scale of Segregation:

The scale of segregation is the size scale of the domains. (Texture can be a larger size scale but not a smaller size scale.) The Scale of Segregation can be described by a correlation function, p(r), for the domains. (p(r) can be a vector as in the case of streaks in a shopping bag.) Typically you desire the scale of segregation to be smaller than some specific value for a given product. Usually, the scale of segregation is larger than the ultimate particle size, r*.

Tadmor discusses the "needle method" for calculation of p_c(r). (This is identical to the calculation of a correlation function for diffraction or scattering.) Throw needles (of length "r") in a material at random, p(r) is the probability that a needle will have both ends in a minor phase, i.e. in the domains. This process is repeated for needles of different sizes "r" to construct p(r) as a continuous function.

At large distances, "r" the probability is just the square of the concentration. At smaller distances the probability becomes larger due to preferred distances of separation.

Since p(r) is an arbitrary function, it is often described as a power-series of indeterminate length:

p(r) = A + Br + Cr² + D r³ + ...

The first term, A reflects the gross uniformity, and the concentration of the minor phase. (Volume)

The second term, B = dp(r)/dr reflects the Surface Area.

The third term, C = d²p(r)/dr² reflects the shape of the domains

The higher order terms can also be described in a structural approach.

In considering how well mixed a mixture is we are immediately confronted with the definition of a random mixture. This is a question which can only be quantitatively answered using statistics.

Mixing attempts to make a random mixture. We should be careful to consider the features of a random mixture. First, a random mixture contains some clusters. That is, a highly dispersed mixture such as a suspension of charged particles which repel each other is not random. Your first inclination to draw a "random" mixture is to draw dots separated from each other. Consider how this drawing would compare with a drawing made by randomly drawing dots on a sheet of paper, i.e. with your eyes closed and the paper moving randomly about the table. In this sense it is possible to make a mixture which is more dispersed than random!

A random mixture is defined by the binomial distribution. The binomial distribution describes a system composed of a lattice of N boxes which contain two distinguishable components, i.e. x's and o's. Consider the x's to be the minor components. N = N_x + N_o, and p, the volume fraction of minor components is p = N_x/N. (p reflects how much of the x component you have added to your Branbury mixer for instance.)

We consider a subset of this system composed of n boxes. "n" is the size of observation and the size of a box is the size of a base unit of the system which can not be broken down further. The binomial distribution gives the probability of finding k boxes filled with x's in a system of n boxes with a bulk volume fraction p, b(k; n, p):

p^k is the probability for k boxes to have x's and (1-p)^n-k is the probability for the remaining boxes to have o's. This situation can be constructed n! ways so this probability must be multiplied by the number of ways it can happen. Some of these ways can not be distinguished from each other since moving one x to replace another leads to an arrangement which can not be distinguished from the original. There are k!(n-k)! of these indistinguishable duplicates so this number must be used to reduce the overall probability.

The mean number of x's in a sample of n boxes for a binomial distribution is obtained by summing k times the probability b given above for all k's from 0 to n. This yields

as might be expected.

The standard deviation is the square-root of this.

Several other distribution functions are spin-offs from the binomial for certain conditions. Poisson distribution: Very low probability for an x appearing (radio-active decay) p<<1, <k><<n.

This is the distribution used for an intensity counting system where counts are recorded and the error is the square-root of the number of counts.

The Gaussian distribution is the binomial for number of observations n is infinite and the probability is finite so np>>1. This is good for random systems with a large number of samples like a large class grade distribution plot or the conformation of a polymer coil. Here:

Here we will only use the Binomial distribution but it is useful to consider the other 2 distributions which are closely related to it.

The mixing index, M, is calculated from the sample variance observed (for a sample size n) and the binomial variance,

For a "random system" as defined by the binomial distribution, M = 1. For a completely unmixed system, M=n. It is possible for M to be less than 1 as discussed above.

Other statistical methods for describing the goodness of mixing such as the t-test are also available and you should refer to pp. 208 table 7-1.

The mixing index, M depends on the sample size, n in a part of size N. This means that there is some size-scale at which we can consider a sample is segregated, s.

This Scale of Segregation, s, can be directly obtained from the samples correlation function, R(r), by integration,

R(r) can be calculated from observation of the sample,

R(r) goes from -1 to 1, -1 is desegregated, 1 is totally segregated, 0 is no correlation (random).

Coefficient of Correlation for an arbitrary 2-phase system:

Consider a striped texture such as might occur in an extruded sheet. Consider correlations perpendicular to the stripes as in figure 7.10. In problem 7.6 you will derive the coefficient of correlation from the volume fraction rule above,

which is a straight line which crosses R(r) =0 at r = L₁L₂/(L₁+L₂) = L/2, where L is the harmonic mean of the two phase sizes.

since R(r) goes to 0 at L/2, the integral for the scale of segregation, s, goes from r=0 to L/2, and s = L/4.

For this striped texture, if L₁ = L₂ = L then s = L/4

If L₁ <<L₂,

The scale of segregation may depend on the range over which the integral of the coefficient of correlation is taken reflecting different frequencies of segregation as shown in figures 7.11 and 7.12 on pp. 214.

The scale of segregation only reflects the size of segregation and does not comment on the intensity of segregation as shown in figure 7.7 on pp. 209.

The Intensity of segregation similarly, has not association with size and only reflects the concentration difference between "Phases".

For a simple 2-phase system,

and