Student Research Projects

Experimental A Cell-Model Description of Hyper-Concentrated Suspensions

James S. Burns, Ph.D.

Susan B. Kirschner

James Powell

 

ABSTRACT

The bulk tensile viscosity of large arrays of aligned, high-modulus, high aspect-ratio, rod-like, cylindrical fibers is a constitutive figure useful to composite manufacturers interested in exploiting the axial extensibility of ordered staple-reinforced laminar preforms. These preforms, when embedded in a highly-viscous coupling fluid, form a hyper-concentrated suspension for which little applicable modeling has been accomplished. Shear coupling between long fibers during bulk axial extension was studied using a series of tests that employ mechanically-driven rods embedded in a viscous Newtonian fluid (40,000 cps). The local fluid velocity field created by an axially translating fiber and this field's screening by adjacent fibers was studied by measuring viscous shear tractions. A momentum flux-capture theory previously applied to elastic solids was reformulated for creeping-regime composite deformation. The effectiveness of momentum screening for 4, 5 and 6-neighbor cells of shear-pled rods was measured at three different effective volume fractions and compared with this theory. Results of this comparison suggest that even at low volume fractions of around 0.15 a cell-type linear superposition model may suffice for describing bulk properties based on local shear-coupling interactions. At structural volume fractions of near 0.60, the cell-model approach should permit accurate modeling of bulk tensile viscosity.

 

INTRODUCTION

According to (Tucker and Advani, 1994), a fiberous suspension is termed concentrated, if it can be classified as such using the following volume-fraction criteria:

Dilute: (1)

Semi-dilute: (2)

Concentrated: (3)

where a=L/D and L and D are the length and diameter of the fibers. They further illustrate that "practical" short-fiber composites fall nearly always in the concentrated regime. What does one call a suspension of long, aligned fibers for which the volume fraction is very much less than the reciprical of the aspect ratio? (Pipes, 1994) uses the neologism "hyper-concentrated" to describe these Aligned Long Fiber Array (ALFA) suspensions.

Aside from the theory of Pipes, several in the semi-dilute volume-fraction range by (Batchelor, 1971) and (Goddard, 1976) and the theory of (Shaqfeh and Fredrickson, 1990), little study has been devoted to ALFA materials. Batchelor used a slender-body approximation and a cell model approach, as well. Shaqfeh used a slender-body approximation and an operator expansion approach to sum the momentum-wave scattering of moving cylinders in an array. Practical application requires one to truncate the scattering summations, which results in an under-estimation of a suspension's mechanical diffusivity. The Pipes models used first-order scaling arguments to calculate shear interaction between fibers in a steady elongational flow and assumes that some cut-off distance exists at which the disturbance caused in the fluid phase by the motion of a single, typical fiber has decayed to insignificance. This distance defines the boundary of a representative cell. In this work, the interaction cut-off assumptions will be explored using both modeling and experimentation. If valid for FVF near 0.6, this assumption would lead to a linear superposition model for the bulk tensile compliance of ALFA arrays.

 

EXPERIMENTAL APPROACH

The experimental apparatus used in this work is illustrated in Figure 1. It consists of a long acrylic tube, capped at the bottom, and filled with a varying number of long, identical aluminum rods and a viscous, interstitial, silicone oil (Anon., 1993). Typically, one rod in the array is attached to the cross-head of a mechanical test frame, and the remainder (along with the tube) are secured to the base of the frame. The gripped rod is pulled from the tube at a controlled rate, and measurements are taken, using a load cell, of the force required to do so. Evenly -spaced unit cells of 4, 5, and 6 neighbors were constructed by potting the rod ends in a wafer of epoxy. These fixed arrays, which form a sort of cage, were placed in the silicone-filled tube and a separate centrally placed rod was extracted. The object of these tests was to determine the "efficiency " that each "cage" of rods exhibits in absorbing the momentum imparted to the interstitial fluid by the moving central rod. This efficiency can be formulated as the ratio of the measured withdrawal force to that expected from in the presence of a solid annular shell at the same average distance from the center rod as the outer cage. Information from these tests will help determine the validity of cell model approaches mentioned above.

 

FLUX CAPTURE

The notion of "flux capture" by the outer rods has been used by (Wagner and Eiton, 1993) to examine shear coupling between individual fibers in a solid composite specimen. Their approach assumes that a momentum flux radiates outward from a central fiber along trajectories normal to iso-strain contours. Where these lines are intercepted by another fiber, a traction results. The more lines that are intercepted, the greater the traction. This approach fails to account for the disturbance in the stress field, and resulting curvature in the flux lines, caused by the presence of the outer fibers. Such an effect would result in greater flux capture than predicted by the models. Work by (Grubb, Li and Phoenix, 1993) attempts to accommodate this shortcoming by proposing that the outer fiber cage possess an "effective" annular surface with which to capture momentum. This radius of this surface was determined by them from the distance (found from experiment) at which a shear stress in the matrix caused by the central fiber decays to some arbitrarily low value.

Figure 1. Experimental apparatus for the examination of array fluid mechanics.

Figure 2. Idealized flux-capture and geometry in an aligned array.

The macroscopic system of rods must obey the principle of similitude, if it is to accurately portray the physics of its microscopic counterpart. The Reynolds number (Re) for the flow induced by the relative motion of parallel fibers can be specified as an upper bound:

(4)

where V is a system-characteristic maximum fiber velocity, L is a system-characteristic maximum length scale, and n is a minimum anticipated kinematic viscosity. For an ALFA material composed of AS-4 fiber and PEKK thermoplastic, the upper bound value during a tensile elongation test is around 10-6. Exact similitude is usually not crucial when building models of systems with such low Reynolds numbers. Provided the model value is still much less than 1, say around 10-2, creeping flow assumptions are still valid. This suggests that the crosshead speed of the test should satisfy the following variation of Equation 4:

(5)

An examination of the model rod radius and of the properties of the fluid limits the maximum crosshead speed to around 0.32 m/sec: an easily achieved target.

 

A CHECK WITH CLASSICAL SOLUTION

Withdrawal of a single, concentric rod from tubes with a range of diameters was performed to provide a baseline for nearest-neighbor tests and to check the standard theoretical solution for the velocity profile Vz(r) and withdrawal force. For a Newtonian fluid, the classic fully-developed velocity profile and volume flux are given as:

(6)

where m is the fluid viscosity and

(7)

If one balances the volume-flux due to the pressure term with that due to the dragging motion of the rod, one may determine the axial pressure gradient.

(8)

and Vo is the inner surface drag velocity, Rrod and Rtube are the annular channel dimensions.

The force F due to rod surface shear stress for fully developed flow is:

(9)

where Vz is the sum of both the drag and pressure induced and wetted rod length is L. Table I summarizes predicted and measured values of the withdrawal force for several values of b .

Table I. Finite outer-tube diameter effects test matrix.

Rrod/Rtube	Fiber volume fraction	Predicted Force/ Length (N/m)	Number of Tests	Measured Force/ Length (N/m)
1/17	0.0017	2.87	3	2.97
1/8	0.0078	4.77	3	4.83
1/4	0.0318	10.63	3	10.53
1/2	0.1340	57.33	-	-

 

HYDRODYNAMIC SCREENING

Hydrodynamic screening is a term used to describe the concentration-dependent transition of suspension behavior from one in which local fluid velocities are important to one in which global or average velocities are important. Figure 3 illustrates the concentration-dependence of the flow within a cluster of rods to that induced outside the cluster. For increasing rod-concentrations the inner and outer velocity fields are expected to become less well correlated. Screening was experimentally characterized for a variety of rod spacing using the previously described test apparatus. Rod extraction tests were performed to help quantify the distance-dependence of hydrodynamic screening for various rod concentrations and configurations.

 

MODELING

An idealized 6-neighbor cell-model geometry was illustrated in Figure 2. To quantify the inadequacy of the currently popular approach, assume that the motion of the center rod (up and out of the page) establishes a radially-symmetric velocity field in the interstitial fluid. The outer rods are treated as if they form an incomplete annular shell around the center fiber; one that does not disturb the symmetry of the axial velocity. This shell is represented as a series of arc segments and will be designated Model 1. Each segment has an average radius corresponding to the mean of the fiber surface separation distance as given by Equation 10.

Model 1 (10)

Figure 3. Hydrodynamic screening of local axial velocity field.

where (11)

S is the interfiber spacing, r is the fiber radius, and q1 and q2 are described in the Figure 2. Only a fraction of the momentum flux leaving the surface of the center rod can be captured by the incomplete shell. To solve for the shear tractions on the inner rod, one simply calculates the shear tractions on the rod for a complete annular shell, and, then, multiplies by the fraction of the perimeter of the shell actually present. This fraction, f, is obtained for cells of n neighbor rods as:

(12)

A slightly more refined model may be formulated by calculating the incremental shear stress in the fluid between segments of the rod surfaces and integrating over the length of each surface to find the average shear stress. In this model, Model 2, the actual separation, R, is employed from Equation 13.

Model 2 (13)

Table II. Summary of Experimental and Model Behaviors

		Withdrawal Force/Length in N/m
Neighbors		Model 1		Model 2
(FVF)	Measured	Incomp.Shell	Comp. Shell	Incomp.Shell	Comp. Shell	RSM
6 (0.10)	8.282	1.96	6.16	2.01	6.31	7.74
5 (0.10)	8.329	1.81	6.51	1.83	6.58	7.82
4 (0.10)	8.340	1.52	6.66	1.55	6.78	6.67
6 (0.25)	12.167	5.51	10.8	5.74	11.3	10.23
5 (0.25)	12.111	5.23	11.9	5.4	12.2	11.11
4 (0.25)	13.037	4.44	12.2	4.65	12.8	10.76
6 (0.60)	39.049	22.63	28.4	25.63	32.1	31.24
5 (0.60)	35.565	23.17	33.8	27.5	39.2	33.99
4 (0.60)	53.232	21.46	37.5	25.63	44.5	35.51

Finite element modeling offered flexible incorporation of more realistic fluid-mechanical assumptions in the region between fibers. The MARC finite element package was used to model a representative two-dimensional momentum transport problem. The withdrawal force per unit length was calculated for 15 different combinations of three volume fractions, three cell geometries, and three assumed values for a far-field no-slip boundary location. These 15 combinations conform to the Box-Behnken experimental design. In Figure xx, a plotted slice of the corresponding three-variable response surface shows the predicted withdrawal forces per unit rod length. The response surface showed minimal dependence on the far-field boundary condition over the range of volume fraction and geometry of interest. Little dependence on number of neighbors is evident in the figure. The coefficents for the quadratic model are listed in Table V.

Table III. Coefficients of a 3-factor response surface model

b0	4.069906
b1	20.7007046
b2	7.03944698
b3	-3.3600644
b4	-5.347125
b5	-0.2436392
b6	0.01374513
b7	86.8692208
b8	-0.6104563
b9	0.16703297

Figure 4. Withdrawal force per unit rod length.

 

DISCUSSION

The cell model, as outlined above, is an illustrative but inaccurate picture of the true nature of the fluid velocity field. Table IV reveals that these both Model 1 and 2 routinely underpredict the withdrawal force per unit length of the center rod. For the highest volume fractions, experimental evidence in Table IV suggests that nearly all the momentum flux is captured by the outer rods. The conclusion one may draw is that the disturbance in the velocity field caused by the presence of the outer rods is sufficient, even at a volume-fraction of 0.1, to curve the flux lines. At a volume fraction of 0.6, a virtually complete capture of the momentum flux occurs and complete isolation from any velocity field external to the cell results.

Typical micrographic evidence suggests that the probability that a rod will have less than four neighbors is small over the volume-fraction range of interest here. Construction of shear compliance of typical 4, 5, and 6-neighbor clusters of rods, as well as the number of each type of cluster in an array, one might begin to form a realistic model of bulk compliance.

These values are found by determining the y-intercept of each of the traces. Increasing scatter is observed as the volume-fraction increases. This scatter is caused by increasing sensitivity of the shear coupling between the center rod and its neighbors to small errors in the center rod position at higher volume fractions. This can be illustrated with a simple planar array containing just three fibers. The fibers are assumed square and finite-width effects are ignored to simplify calculations. For the ideal, equally spaced configuration, the effective center rod withdrawal force per unit length in a Newtonian fluid is proportional to the sum of the two independent Couette-type terms:

(14)

where V1 = V3 = 0 and the spacing perturbations d = 0.

The withdrawal force per unit length possess a global minimum at d = 0 and increases rapidly for large spacing perturbations. Such scatter would be natural for the non-ideal geometries present in real ALFA systems.

 

REFERENCES

Tucker III, C. L. and S. G. Advani, 1994, Processing of Short-Fiber Systems, Flow and Rheology in Polymer Composites Manufacturing: Chapter 6, Elsevier Science, B. V.

Pipes R. B. Coffin D. W. Shuler. S. F. Simacek P., 1994, "Non-Newtonian Constitutive Relationships for Hyperconcentrated Fiber Suspensions", Journal of Composite Materials V 28 N 4. P 343-351.

Batchelor, G. K., 1971, "The Stress Generated in a Non-Dilute Suspension of Elongated Particles by Pure Straining Motion", J. Fluid Mech., 46, 4, 813-829.

Goddard, J. D. , 1976, "Tensile Stress Contribution of Flow-Oriented Slender Particles in Non-Newtonian Fluids", Journal of Non-Newtonian Fluid Mechanics, 1, 1-17.

Shaqfeh, E. S. G., Fredrickson, 1990, "Hyrodynamic Stress in a Suspension of Rods", G. H., Phys. Fluids, A, 2(1):7-24.

Anon., 1993, "Information About Dow Corning Silicone Fluids", Dow Corning Product Document Numbers 22929 and 22230, Dow Corning Corp., Midland, Michigan.

Wagner H. D., Eiton A., 1993, Comp. Sci. and Tech. 46, 353.

Grubb, D. T., Z. F. Li, and S. L. Phoenix, "Experimental Studies of Fiber Interactions with Multi-Fiber Model Composites", 8th Annual Technical Conference, American Society for Composites, 592-601.