Entropy Time Evolution in a Twin Flight Single Screw Extruder and its Relationship to Chaos

Entropy Time Evolution in a Twin Flight Single Screw Extruder and its Relationship to Chaos Winston Wang (wxw6@po.cwru.edu) and Ica Manas-Zloczower (ixm@po.cwru.edu) Department of Macromolecular Science Case Western Reserve University, Cleveland OH 44106 (216) 368-3596 Miron Kaufman (m.kaufman@csuohio.edu) Department of Physics Cleveland State University, Cleveland OH 44115 (216) 687-2436 Abstract The flow fields in polymer processing exhibit complex behavior with chaotic characteristics, due in part to the non-linearity of the field equations describing them. In chaotic flows fluid elements are highly sensitive to their initial positions and velocities. A fundamental understanding of such characteristics is essential for optimization and design of equipment used for distributive mixing. In this work we analyze the flow in a twin-flight single screw extruder, obtained through 3-D FEM numerical simulations. We study particle motion and, implicitly, mixing in the extruder. Here, particles are massless points whose presence does not affect the flow field or other particle motion. We visualize chaos through Poincaré sections and calculate Lyapunov exponents as a measure of divergence of initial conditions, signaling chaotic features of flow. We use entropic measures to probe disorder or system homogeneity. The time evolution of the Renyi entropy of β = 1 for the 3-D spatial distribution of particles using different initial conditions are followed. The Kolmogorov-Sinai entropy rate, calculated by the sum of positive Lyapunov exponents, is correlated with the rate of evolution of entropy. In the same context we also examine the eccentric Couette flow. We find that the Renyi entropy dependence on time is logarithmic. To gain further understanding of this numerical observation, we analyze analytically the diffusion with drift entropy and find that it also depends logarithmically on time. Using the logarithmic coefficient of the Shannon entropy (ß = 1), as a measure of the overall rate of 1

mixing, we find that the eccentric Couette device has the highest rate of mixing, followed by the twin-flight single screw extruder, and by the 1-D diffusion with drift. Introduction Mixing is an important component of many polymer-processing operations. Material processability and product properties are highly influenced by mixing quality. In turn, mixing quality is affected by the type of flow in the equipment. In systems having flows with high Reynolds number, turbulence may occur which can greatly enhance distributive mixing. The study of turbulent mixing has been undertaken mostly through statistical theory [1, 2]. As a system deviates from the laminar regime, inherent kinetic and hydrodynamic fluctuations increase in scale and magnitude. This results in dynamic instability of motion, described by kinetic/hydrodynamic equations [3]. The complexity arising from the instability leads to chaotic flows, which are characterized by a high sensitivity of fluid elements to their initial positions and velocities. Chaotic behavior is expected in systems with turbulent flow. However, low Reynolds-number flows, as often found in most polymer-processing equipment, can also exhibit chaotic features[4,5]. This chaotic behavior arises from the geometric and operational complexity present in these systems. Aref [6, and references therein] was first to articulate the importance of chaotic flows, at low Reynolds numbers. Ottino [7-9] has studied extensively the role of chaotic flows on mixing coining the concept of chaotic mixing. Zumbrunnen [10,11] was first to perform chaotic advection studies of polymer blends and composites. A fundamental understanding of characteristic chaotic features is essential for the optimization and design of equipment used for distributive mixing. Central to any attempt to achieve such an understanding of mixing processes are the statistical physics concept 2

of entropy and the nonlinear dynamics concepts of Kolmogorov-Sinai entropy rate and Lyapunov exponents. The entropy is the rigorous measure of system homogeneity. It satisfies commonsense requirements for a measure of lack of information or disorder: (i) the entropy takes its lowest value, zero, when one of the probabilities is unity and the rest are zero (i.e., total information, perfect order); (ii) the largest value for the entropy is achieved when all probabilities are equal to each other (i.e. the absence of any information, complete disorder); and (iii) the entropy is additive over partitions of the outcomes. Since we are interested to measure the degree of distributive mixing in polymeric systems, we have used [12] the entropy to measure the degree of mixing. The information entropy for a particular experimental condition with a set of M possible outcomes is: M S = p j ln p j (1) j =1 where pj is the relative frequency of outcome #j. Equation (1) is the standard formulation [13] of uncertainty. If the axiom iii above is relaxed to consider only statistically independent partitions, Renyi [14] determined that the information entropy can be replaced by a one-variable function: S(β) = 1 M 1 β ln( β p ) (2) j j =1 In the limit β 1, Equation (2) reduces to Equation (1). If β > 0 the function defined in Eq. (2) is maximized when all pj are equal to each other, while if β < 0 the function in Equation (2) is minimized when all the pj are equal thus violating axiom (ii) above. For this reason we consider in our study only nonnegative values of the parameter β. It follows that: 0 S(β) ln(m). Thus S(β)/ln(M) constitutes an index of homogeneity: it is 3

1 for total disorder or homogeneity and is small for high order or segregation. Shannon called this ratio the relative entropy. The relative entropy subtracted from unity constitutes a measure of the deviation of the concentration in each local area from the ideal distribution. We perform the relative Renyi entropy calculations on spatial distributions of the minor component in the system at specified time steps, which are a series of instantaneous "snapshots" of the state of mixing. In order to further quantify the effect of the non-linear flow on the dynamics of distributive mixing, we evaluate the rate of mixing ds/dt. Latora and Baranger [15, 16] have proposed, based on numerical analysis of several low-dimensional maps, that the rate of mixing is related to chaotic features of flow via the positive Lyapunov exponents. They have concluded that typically, the dynamics of Shannon entropy, S(t), shows three stages: the first one strongly dependent on initial conditions and the last one plateaus to an equilibrium value. For the second stage they found that ds/dt is approximately equal to the sum of the positive Lyapunov exponents. There is no mathematical proof for this relationship so we will provide further numerical verification of its validity for the flow in the single extruder and for the eccentric Couette flow device. The Lyapunov exponents quantify the rate of exponential divergence of nearby trajectories and are calculated [17] using: t / t 1 σ 1 = lim ln d n t (3) t n=1 d 0 t/ t 1 A n σ1+ σ2 = lim ln t t n= 1 A (4) 0 where d n is the distance between two particles at time step n, and A n is the area of a triangle formed at time step n. Of course we estimate the infinite sums above by using a 4

finite number of terms after we have checked in all cases for their convergence. One can in fact estimate the discrepancy between the estimate after a finite (but large) number of steps and the exact value (infinite number of time steps) to be inversely proportional to the number of time steps (which in our numerical work is as large as 1000, see below). Positive values of Lyapunov exponents reveal an exponential growth in the length and area stretch of the initial conditions. This is the signal for high sensitivity to initial conditions, which is the main characteristic of chaotic flows. Pesin [18] has proved that the sum of the positive Lyapunov exponents, which measures trajectories rate of expansion, is equal to the Kolmogorov-Sinai entropy rate K K = σ (5) positiveσ Since we estimate the Kolmogorov-Sinai entropy rate by means of the Pesin theorem, by computing the Lyapunov exponents, we relegate the definition of the Kolmogorov-Sinai entropy rate to the Appendix. We propose to verify the Latora and Baranger relation for an industrial system, namely the polymer flow in an extruder. Then we can proceed with the optimization of the polymer processing conditions and of the design of the tools (extruders). Our goal is to identify processing conditions and the geometry of the extruder that are optimal with respect to the rate of mixing, i. e. maximize K. We also examine a classical example of a chaotic system, the flow in an eccentric Couette device, to further illustrate the relationship between the rate of mixing and chaos. It is to be expected that the applied research on mixing and the non-linear dynamics of mixing will generate new directions of inquiry that may feed studies on fundamental physics questions. 5

For both the extruder and the eccentric Couette device, the time dependence of entropy is found to be logarithmic. A useful benchmark case is provided by the onedimensional diffusion with drift. The analytical calculation of the time dependence of entropy for the one-dimensional forced diffusion also reveals a logarithmic time dependence of entropy. Procedure In our group we carried out three dimensional, isothermal flow simulations for various batch and continuous mixing equipment [19-26]. In this project, we looked at the flow patterns in a twin-flight, single screw extruder. A fluid dynamics analysis package-fidap, using the finite element method was employed to solve the 3D, isothermal flow of a Newtonian fluid. No slip boundary conditions on the screw surfaces and barrel walls were used. The operating conditions were selected such that 1 clockwise revolution of the screw was made per unit time. A throttle ratio of 1/2 was used via an applied pressure difference across the inlet and outlet surfaces. The continuity equation (mass conservation) and the Navier-Stokes equation of motion for the steady state, isothermal flow of an incompressible Newtonian fluid are: V = 0 (6) ρ(v )V = P + µ 2 V (7) In equations (6) and (7), V is the velocity vector, ρ is density, P is pressure and µ is viscosity. The Cray T-90 at the Ohio Supercomputer Center running FISOLV was used to solve the field equations (6) and (7) with appropriate boundary and initial conditions. In work previously done by our group [19, 21], an algorithm was developed for tracking massless points that affect neither the flow field nor other particles. Since the 6

flow field is completely deterministic, the location of the particles can be found by integrating the velocity vectors: t 1 X(t 1 ) = X(t 0 ) + V(t)dt (8) t 0 where X(t 1 ) is the location of a particle at any time t 1, X(t 0 ) is the location of the same particle at initial time t 0, and V(t) is the corresponding velocity vector of the particle. The location of each particle is calculated every time step. For the twin flight single screw extruder, we used a coordinate system that rotates with the same angular velocity as the screw, so that one finite element model could be used for all the calculations. We also use a periodic boundary condition to simulate an extruder longer than the finite element model. Due to the no-slip boundary conditions, a particle, which runs into, or overshoots a wall, is considered stuck there and no longer moves. Poincaré sections can be used [25, 27, 28] to characterize chaotic features of flow. By converting the 3-D time dependent flow field into a 2-D cross-sectional map, the complexity of the system is reduced. A cross section in a spatially or time periodic system is considered. For the twin-flight single screw extruder, we use an axial cross section at the end of each pitch. From the particle tracking algorithm results, the intersection of the particle paths and the selected cross-sections are plotted. The resultant maps can be visually inspected for regions of good mixing and poor mixing. To determine the Renyi entropies, the spatial region is divided into M bins of equal size with a close to regular aspect ratio. The concentration of the minor component 7

in the j th bin is p j and is used in equation (2) with various ß values between 0 and 10 to calculate the Renyi entropies. For the twin-flight screw extruder, three-dimensional bins are used, that have an aspect ratio close to 1:1:1. In order to compute the Lyapunov exponents we track the positions of three particles placed initially in a right triangle with area A 0 and legs with length of d o. After a time period t, we note the new area formed by the points, A n, and the new distance d n between the first two points. We then reposition the second particle such that the distance between the particles is renormalized to d 0 while preserving the orientation. The 3 rd particle is then repositioned such that a right triangle with area A 0 is reformed. The 1 st Lyapunov exponent is calculated by using the average of ln (d n /d o ) at long times t., and the 2 nd Lyapunov exponent is calculated by using the ln (A n /A o ) at long times t. and subtracting the 1 st Lyapunov exponent. Positive values of Lyapunov exponents reveal an exponential growth in the length and area stretch of the initial conditions. This is the signal for high sensitivity to initial conditions, which is the main characteristic of chaotic flows. Results and Discussion Poincare Sections, Lyapunov Exponents and Rate of Mixing The finite element mesh shown in Figure 1 for the twin flight single screw extruder contains 16200 8-node brick fluid elements, and 8340 4-node quadrilateral boundary elements, for a total of 20535 nodes. FIMESH, a mesh generator part of FIDAP, was used to create the computer model. 8

The radius of the barrel is 4.009623, the radius of the screw is 3.534884, the flight clearance is 0.0096, and the flights are 0.44 thick. The extruder is square pitched, so one pitch corresponds to approximately eight length units. At the operating conditions used in the flow simulations (1 revolution of the screw per unit time and a throttle ratio of 1/2), the Reynolds number is less than 1/2. After checking that the inertial terms in the Navier-Stokes equation can be safely neglected at this low Reynolds number, the results presented in this paper were obtained by neglecting the inertial terms, which is equivalent to a zero Reynolds number. In this case the forces induced in the non-inertial frame of reference do not show up and thus rotating the barrel is equivalent to rotating the screw. The flow field calculations required 395 user CPU seconds over a total elapsed time of 860 seconds and 11.8 MWord of memory. We study chaotic features of flow for the twin flight single screw extruder by analyzing Poincare sections obtained by using ten identical clusters of 547 particles, each well distributed at the entrance cross section of the extruder. The location of the clusters is shown in Figure 2. They were tracked for sixty revolutions. The intersection between the particle paths and 12 axial cross-sections taken from the end of each of the first twelve pitches is shown in Figure 3. Due to symmetry only the top half of the cross sections is shown. Several chaotic features of flow are noticeable. There is evidence of a strong attractor at the position indicated by the arrow and the close up view of the region. There is also a weak attractor, which appears to be larger in area because it is slower in condensing the particle concentration within its influence. A close up of the weak attractor shows a small void region in the center. There is also a large void region pointed out in the figure, indicative of a repellor. Though we call this regions 9

attractor/repellor because particles are attracted/repelled towards these regions, we do not imply that they are equivalent to strange attractors present in dissipative systems. In order to quantify the chaotic flow, 1,050 sets of particles were tracked for 1,000 time steps with a t of 0.01. The initial distance between the particles is 0.01, and the initial area of the triangle was 0.00005. The sum of the positive Lyapunov exponents ranged from 0.0021 to 2.401 with the average sum of the positive Lyapunov exponents determined to be 0.5633. The dynamics of mixing in the twin flight single screw extruder was studied by calculating the time evolution of the 3D Renyi entropies. The axial cross section at the beginning of the extruder was saturated with 332,000 randomly distributed particles and tracked for 10 revolutions. The evolution of the 3-D Renyi entropy in two pitches of the extruder was followed. A variable number of up to 311,040 bins were used maintaining the ratio of bins in each direction. (1 radial::45 angular::32 axial). Figure 4a shows the evolution of the Renyi entropy (ß=1) for various numbers of bins. All the curves appear very similar in nature, differing only by vertical separation. This separation is almost completely dependent on the number of bins used, as shown in Figure 4b, where the curves have been normalized by the natural logarithm of the number bins; the evolution of the relative Renyi entropy (ß=1) for the various number bins collapse into one curve. One can distinguish for each curve four stages. The first stage is characterized by a sharp incline and is highly dependent on the details of the system and initial configuration. The second stage is a linear increase, the slope of which is conjectured to be equal to the Kolmogorov-Sinai entropy rate [15], which in turn is equal to the sum of positive Lyapunov exponents [18]. Indeed we find a slope of 0.5583 (11,520 bins) that 10

correlates well with the positive Lyapunov exponents sum of 0.5633. The percent error is 0.9%. This illustrates the close relationship between chaos and the rate of mixing in this system. The third stage is characterized by a more gradual incline as the system approaches equilibrium. The fourth stage is indicated by a sharp decline. This decline is associated with particles leaving the confined region (2 pitches) being examined. As particles leave the entropy drops due to the lowered number of states available. To further check the Lattora-Baranger conjecture we concentrate next on a well studied chaotic system, the eccentric Couette flow device. Swanson and Ottino [27] have studied the chaotic mixing that occurs in the eccentric Couette flow due to the oscillating boundary conditions. We duplicate the operating conditions that provided the greatest amount of chaos. The inner cylinder was rotated 720 clockwise, and then the outer cylinder was rotated 720 counter-clockwise. The cycle was then repeated as often as necessary. 360 displacement was made per unit time, so one complete cycle took four time units. The 2D isothermal steady state flow of a Newtonian fluid at zero Reynolds number was solved using FIDAP for each of the boundary conditions. Figure 5 shows the finite element mesh for the eccentric Couette flow. The mesh employs 9,600 4-node quadrilateral fluid elements and 4,160 linear boundary elements for a total of 11,760 nodes, for each boundary condition. The physical parameters of the eccentric Couette device are given in consistent dimensionless length units. The outer cylinder has a radius of 3, and the inner cylinder has a radius of 1. The inner cylinder is offset by 0.9 on the x- axis. In the eccentric Couette flow, a material line was placed along the x-axis using 1,590 particles. The particles were tracked for 100 time units (25 cycles of oscillations). 11

The spatial distributions at the end of each of the 25 periods are superimposed for the Poincaré section as shown in Figure 6. Almost the entire region is chaotic except for a small island slightly offset from the center of the system. For the eccentric Couette flow, the Renyi entropy of the material line used for the Poincaré section, was followed for 20 revolutions. 1,568 bins were used: 14 in the radial direction and 112 in the angular direction. Figure 7 shows the time evolution of the Renyi entropy (ß=1). This curve shows the same first three stages as the extruder, but not the fourth stage, as none of the particles can leave the system. The slope of the linear fit for the second stage is 0.4403. Only the 1 st Lyapunov exponent is considered for the eccentric Couette flow as a 2-D time-periodic Hamiltonian system has at most only one positive Lyapunov exponent [29]. 500 sets of particles were randomly distributed throughout the system. The sets of particles were tracked for 100 time units and renormalized every 0.01. The 1 st Lyapunov exponent ranged from 0.0530 to 0.4551 with the average determined to be 0.2801. The average value of the 1 st Lyapunov exponent deviates from the slope of the linear fit by a factor of 1.5. This may be because the eccentric Couette flow is a time-periodic system, whereas the extruder, as well as all of the mappings Latora and Baranger [15, 16] examined, are steady-state, spatially periodic systems. Thus there is possibly a different relationship between the Kolmogorov-Sinai entropy rate and the time evolution of the physical entropy for a time-periodic system. This relationship is yet to be elucidated and warrants further study averaging over many initial conditions. 12

Logarithmic Time Evolution of Renyi Entropy We find that the overall time evolution of Renyi entropy for the time interval preceding the time when particles leave the volume of the extruder under study is well described by the following logarithmic dependence: S = aln( t) + b. (9) Figure 8 shows the time evolution of the Shannon entropy in the twin-flight single screw extruder with a logarithmic curve fit for the first three stages. The logarithmic coefficient is a = 0.5581 with an r 2 value of 0.9788. A logarithmic curve fit applied to the eccentric Couette device is shown in Figure 9. The logarithmic coefficient is a = 0.5986 with an r 2 value of 0.9480. In order to get some insight in the origin of the logarithmic entropic evolution, we propose as a bench of reference for mixing, the one-dimensional diffusion with drift problem. In this case we divide a length L into M bins of size z = L/M. The probability p(z,t) to find a particle at time t in a particular bin of size z located at z is approximately: p(,) zt zρ(,) zt (10) ρ(z,t) is the distribution function obtained by solving the 1D diffusion equation with the assumption that all particles start from origin: ρ(z,t) = 2 (z v 1 D t) 4πDt e 4Dt (11) where D is the diffusion constant and ν D is the drift velocity. We use Equation (2) to compute the associated Renyi entropy: 13

S(β) = 1 M 1 β ln( p β j ) 1 1 β ln( z β 1 j=1 L /2 ρ(z,t) β dz) (12) L /2 By using Equation (11) and assuming times that are short compared to L/v D (or to L/(4D) 1/2, if v D = 0), we replace the integral limits by infinity and obtain: S(β) ln( z) + 1 2 ln(4πdβ 1 β 1 ) + 1 ln(t) (13) 2 First we note that Equation (13) is precisely analogous to the logarithmic time evolution of Equation (9). The coefficient of the logarithmic term is a = 0.5. This is independent of the value of ß. The constant term decreases monotonically with ß. In the limit ß 1, 1 β β 1 e so the Shannon entropy is: S(1) ln( z) + 1 2 ln(4πde) + 1 ln(t) (14) 2 The logarithmic coefficient provides a dimensionless measure of the rate of mixing: a = ds/dln(t). Using the logarithmic coefficients we can examine the relationship between chaos and the rate of mixing. For ß = 1 (Shannon entropy), the eccentric Couette flow, being the most chaotic system (according to the Poincaré section) has the highest rate of mixing, a = 0.5986, the twin flight single screw extruder being less chaotic shows a slower rate of mixing, a = 0.5581, and the 1-D diffusion with drift has the lowest rate of mixing, a = 0.5. The dependence of the logarithmic coefficient a on the value of ß was examined for a range of ß values. As stated above (Equation (13)), for the case of 1-D forced diffusion the logarithmic coefficient is independent of ß. Figure 10 shows the time evolution of the Renyi entropy for ß=0,1,2 in the twin-flight singe screw extruder. The values for the logarithmic coefficient decrease with increasing ß. On the other hand, for 14

the eccentric Couette flow, the time evolution of the Renyi entropy for (ß=0,1,2) with an applied logarithmic curve fit (as shown in Figure 11) shows the opposite trend. That is, the logarithmic coefficient increases with ß. We believe that this apparent discrepancy actually further characterizes the nature of distributive mixing in different systems. In Reference 12 we have demonstrated that the Renyi entropies can be tailored to specific needs: from focusing on void volumes when β = 0 to analyzing only areas of highest concentration of minor component when β >> 1. Whereas in the 1D diffusion with drift problem, all aspects of distributive mixing improve equally well, the eccentric Couette flow improves the region of worst mixing (high concentration of minor component) faster than covering the void regions. On the other hand, the twin-flight single screw extruder is more efficient at covering void regions rather than distributing regions of high concentration of the minor component. In Figure 12 we show the dependence of a on ß for the three systems analyzed in our paper. Conclusions Examination of the Poincaré sections indicate that given the 720 displacement operating condition, the eccentric Couette flow is almost completely chaotic. In the twinflight single screw extruder the geometry gives rise to chaotic features of flow. The effect these features have on the overall distributive mixing in the system was quantified via the calculations of Lyapunov exponents, namely the sums of the positive Lyapunov exponents were correlated with the time evolution of the Shannon entropy. Linear fits applied to the second stage of the time evolution of Renyi entropy for ß=1 correlate well with the average sum of the positive Lyapunov exponents in the case of the twin-flight single screw extruder. 15

The Renyi entropy was found to depend on time logarithmically for the first three stages. For ß=1, the logarithmic coefficient for the eccentric Couette flow (0.59) is higher than that of the twin-flight single screw extruder (0.56). Both these coefficients are higher than the analytically derived value for 1-D diffusion with drift (0.5). We have determined, numerically for the extruder and the Couette flow and analytically for the diffusion with drift, the dependence of the logarithmic coefficient on the Renyi parameter ß. The 1-D diffusion with drift shows no dependence on ß, indicating that all aspects of distributive mixing were equally improved. The twin-flight single screw extruder shows the logarithmic coefficient decreasing with ß, indicating that this device is not as effective at decreasing the concentration of the minor component where it was the highest than preventing void regions. The eccentric Couette flow shows the logarithmic coefficient increasing with ß, indicating the effectiveness of this flow at distributing regions of high concentration of the minor component. Acknowledgement The authors acknowledge the financial support of this work by the National Science Foundation through grants DMI-9812969 and DMI-0140412. We would also like to acknowledge the use of computing services from Ohio Supercomputer Center. Appendix The Kolmogorov-Sinai [17] entropy rate K is defined as: K = lim S (A1) t t S is the information entropy associated with a four-dimensional hyperspace: space-time. The space is partitioned in equal size cells and S is calculated by using: 16

S = p i0,i 1,..., i t ln(p i0,i 1,...,i t ) (A2) (i 0,i 1,...,i t ) where p i0,..,it is the probability for the trajectory i 0, i 1,i t, i. e. the probability that at time 0 the particle is in box i 0, at time t in box i 1, at time 2 t in box i 2 Note that K is an entropy rate, entropy per unit time. The actual calculation of the Kolmogorov-Sinai entropy rate requires knowledge not only of the particles position at a given time but the trajectory used to arrive at that position. This results in calculations on the order of M t where M is the number of spatial bins and t is the number of time intervals. A direct calculation of the Kolmogorov-Sinai entropy rate is not feasible with the computational resources currently available and thus we have resorted to the Pesin theorem [18] to estimate K by means of computing positive Lyapunov exponents. References [1.] G. K. Batchelor, The Theory of Homogenous Turbulence (Cambridge University Press, Cambridge, 1953). [2.] P. K. Yeung, J. Fluid Mech., 427, 241 (2001). [3.] Y. L. Klimontovich, Physica B, 228, 51 (1996). [4.] H. Aref, S. Balachandar, Phys. Fluids, 29, 3515 (1986) [5.] Z. Tadmor, JSW Technical Review, 14, 3 (1990). [6.] H. Aref, Phys. Fluids, 14, 1315 (2002). [7.] D.V. Khakhar, H. Rising, J. M. Ottino, J. Fluid Mech., 172, 419 (1986). [8.] D.V. Khakhar, J. G. Frangione, J. M. Ottino, Chem. Eng. Sci., 42, 2909 (1987). [9.] J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport (Cambridge University Press, Cambridge, 1989). [10.] R. I. Danescu, D. A. Zumbrunnen, Powder Technology, 125, 251 (2002). [11] O. Kwon, D. A. Zumbrunnen, J. Applied Polym. Sci., 82, 1569 (2001). [12.] W. Wang, I. Manas-Zloczower, M. Kaufman, Int. Polym. Proc., XVI, 315 (2001). [13.] C. E. Shannon, Bell System Technical Journal, 27, 379 (1948). [14.] A. Renyi, Probability TheoryNorth-Holland series in applied mathematics and mechanics 10 (North-Holland Pub. Co., Amsterdam, 1973). [15.] V. Latora, M. Baranger, Physical Review Letters, 82, 520 (1999). 17

[16.] M. Baranger, V. Latora, A. Rapisarda, Chaos, Solitons & Fractals, 13, 471 (2002). [17.] A.Wolf, J.B.Swift, H.LSwinney, J.A.Vastano, PhysicsD, 16, 285 (1985). [18.] Y. B. Pesin, Russian Math Surveys, 32, 55 (1977). [19.] H.-H. Yang, I. Manas-Zloczower, Polym. Eng. Sci., 32, 1141 (1992). [20.] I. Manas-Zloczower, Rubber Chem. Technol., 67, 504 (1994). [21.] T. Li, I. Manas-Zloczower, Int. Polym. Proc., 10, 314 (1995). [22.] T. Li, I. Manas-Zloczower, Chem. Eng. Commun., 139, 223 (1995). [23.] C. Wang, I. Manas-Zloczower, Int. Polym. Proc., 11, 115 (1996).. [24.] C. H. Yao, I. Manas-Zloczower, Int. Polym. Proc., 12, 92 (1997). [25.] H. F. Cheng, I. Manas-Zloczower, Polym. Eng. Sci., 38, 926 (1998). [26.] C. H. Yao, I. Manas-Zloczower, Polym. Eng. Sci., 38, 936 (1998). [27.] P. D. Swanson, J. M. Ottino, J. Fluid Mech., 213, 227 (1990). [28.] T. H. Lee, T. H. Kwon, Adv. Polym. Technol., 18, 53 (1999). [29.] T. Howes, P. J. Shardlow, Chem. Eng. Sci., 52, 1215 (1997). List of Figures Figure 1: Finite element mesh for the twin-flight single screw extruder Figure 2: Initial location of 10 identical clusters with 547 particles each Figure 3: Poincaré section in the top half of the twin-flight single screw extruder from the superposition of the 5470 particle positions at the end of each of 12 pitches. Figure 4a: Evolution of the Renyi entropy (ß=1) for various numbers of bins. Figure 4b: Evolution of the relative Renyi entropy (ß=1) for various numbers of bins. Figure 5: Finite element mesh for the eccentric Couette flow Figure 6: Poincaré section in the eccentric Couette flow from the superposition of the 1590 particles positions at the end of each of the 25 oscillations. Figure 7: Time evolution of the Renyi entropy (ß=1) in the eccentric Couette device. Figure 8: Time evolution of the Renyi entropy (ß=1) in the twin-flight single screw extruder with a logarithmic curve fit Figure 9: Time evolution of the Renyi entropy (ß=1) in the eccentric Couette device with a logarithmic curve fit. Figure 10: Time evolution of the Renyi entropy (ß=0,1,2) in the twin-flight single screw extruder. Figure 11: Time evolution of the Renyi entropy (ß=0,1,2) in the eccentric Couette flow. Figure 12: Logarithmic coefficient a dependence on β for diffusion (green), eccentric Couette (blue), extruder (red). 18

Figures: Figure 1: Finite element mesh for the twin-flight single screw extruder. 19

Figure 2: Initial location of 10 identical clusters with 547 particles each. 20

Figure 3: Poincaré section in the top half of the twin-flight single screw extruder from the superposition of the 5470 particle positions at the end of each of 12 pitches. 21

Figure 4a: Evolution of the Renyi entropy (ß=1) for various numbers of bins. 22

Figure 4b: Evolution of the relative Renyi entropy (ß=1) for various numbers of bins. 23

Figure 5: Finite element mesh for the eccentric Couette flow. 24

Figure 6: Poincaré section in the eccentric Couette flow from the superposition of the 1590 particles positions at the end of each of the 25 oscillations. 25

Figure 7: Time evolution of the Renyi entropy (ß=1) in the eccentric Couette device. 26

Figure 8: Time evolution of the Renyi entropy (ß=1) in the twin-flight single screw extruder with a logarithmic curve fit 27

Figure 9: Time evolution of the Renyi entropy (ß=1) in the eccentric Couette device with a logarithmic curve fit. 28

Figure 10: Time evolution of the Renyi entropy (ß=0,1,2) in the twin-flight single screw extruder. 29

Figure 11: Time evolution of the Renyi entropy (ß=0,1,2) in the eccentric Couette flow. 30

Figure 12: Logarithmic coefficient a dependence on β for diffusion (green), eccentric Couette (blue), extruder (red). 31