Rod-climbing characterization of kaolinite suspended polyisobutylene solutions.
It is well-known that non-Newtonian fluids climb up a rotating rod, whereas Newtonian fluids do not. The Newtonian fluid near the rotating rod is pushed outward by a centrifugal force, which forms a dip near the center of the liquid surface. The opposite phenomenon is observed in a polymeric liquid; the liquid moves toward the center and climbs up the rod (1). This behavior is related to nonlinear viscoelastic effects and normal stresses, which do not exist in Newtonian fluids.
The most intensive studies on rod-climbing have been carried out by Joseph et al. (2-6). Joseph and Fosdick (2) developed a systematic construction for the shape of the free surface above a rheologically simple fluid resulting from a perturbation while in a state of rest. This construction gives a quantitative theory of rod-climbing when rod speeds are low. Furthermore, Joseph et al. (3), as well as Beavers and Joseph (4), include the effects of surface tension in order for the shapes of the free surface to agree with the measured shapes.
A qualitative and precise analysis of rod-climbing phenomena for viscoelastic polymer solutions can be used to characterize important rheological parameters of polymer solutions (3, 4). This includes the elasticity of polymer solutions, which is obtained from the climbing constant ([Beta]) and the first normal stress difference ([N.sub.1]), using a second-order fluid model. An ideal second-order fluid, which satisfies the second-order fluid model, is known to possess both a constant high viscosity and high elasticity at room temperature. Such a fluid is known as a "Boger fluid." Various Boger fluids, such as a maltose syrup + polyacrylamide solution (7), polyacrylamide in mixtures of glycerine and water (8), and polyisobutylene (PIB) in polybutene (PB) and in kerosene (9, 10), have been investigated. Recently, Solomon and Muller (11) studied transient extensional behavior of polystyrene-based Boger fluids. A mixture of PIB and PB is also well described by a second-order fluid model (12-14). The second-order fluid model is a constitutive equation which can be obtained keeping only the first two terms of the Taylor series of the retarded-motion expansion.
In this paper, in order to examine the effect of suspended particles on the characteristics of a typical Boger fluid, particle suspended PIB/PB system was prepared, and its rod-climbing constant and rheological property were investigated. Kaolinite was used as dispersed particles in a mixture of high molecular weight PIB in a low molecular weight PB.
The rod-climbing constants for various polymer concentrations, solvent viscosities, molecular weights, rod-sizes, particle concentrations and temperatures were then investigated. Shear viscosity and the first normal stress difference of the particle suspended PIB solutions were also obtained by using an RMS-800.
The most important parameter in rod-climbing is the climbing constant [Beta] = 3[[Alpha].sub.1] + 2[[Alpha].sub.2], where [[Alpha].sub.1] and [[Alpha].sub.2] are the parameters of the second-order fluid model of the deviatoric stress tensor ([Tau]) (4). The constants [[Alpha].sub.1] and [[Alpha].sub.2] are related to the zero shear values of the first and second normal stress differences [N.sub.1] and [N.sub.2] (or the first and second normal stress difference coefficients [[Psi].sub.1] and [[Psi].sub.2]) by the following equations (6):
[Mathematical Expression Omitted] (1)
where [Mathematical Expression Omitted] is the shear rate, [[Alpha].sub.1] = -[[Psi].sub.1]/2 and [[Alpha].sub.2] = [[Psi].sub.1] + [[Psi].sub.2].
In addition, the parameter [Beta] is proportional to the height of the climb in a slow steady flow (15). The shape of the free surface is expressed by a perturbation method in the following form (2):
h(r, [Omega]) = [h.sub.0](r) + [h.sub.2](r) [[Omega].sup.2] + O([[Omega].sup.4]) (2)
where [Omega] is the angular frequency of the rod, [h.sub.0](r) is the static rod climb (which is independent of [Omega]), and the higher order terms O([[Omega].sup.4]) are neglected. Considering the effect of surface tension (4), the height rise function, [h.sub.2](r), can be obtained from the following equations:
[Sigma]/r (r[h[prime].sub.2])[prime] - [Rho]g[h.sub.2] = -2 [a.sup.4]/[r.sup.4] [Beta] + [Rho][a.sup.4]/2[r.sup.2], (3)
where [h.sub.2][prime] = 0, and [h.sub.2](r)[approaches]0 as r[approaches][infinity],
Here, [Sigma] is the surface tension, a is the radius of the rod, [Rho] is the density of the liquid, g is the gravitational acceleration, and the prime denotes the derivative with respect to r. For a rheologically simple fluid, [[Tau].sub.(n)] = [[Tau].sub.1] + [[Tau].sub.2] + ... [[Tau].sub.n]. Here, [[Tau].sub.1] = [Mu][A.sub.1] for the linear, or Newtonian approximation, and [[Tau].sub.2] is given by:
[[Tau].sub.2] = [[Alpha].sub.1][A.sub.1] + [[Alpha].sub.2][[A.sub.1].sup.2] (4)
where [A.sub.1] is the Rivlin-Ericksen kinematic tensor of degree 1, and the expression [[Tau].sub.(2)] = [[Tau].sub.1] + [[Tau].sub.2] is termed the extra stress of a fluid of grade two.
The first-order solution of a simple fluid is exactly the same as that for a Newtonian fluid. The two-parameter expansion procedure is adopted, and the second-order solution is obtained as follows:
h(a, [Omega]) [approximately equal to] [h.sub.0] (a) + 2[[Pi].sup.2]a/[Sigma][-square root of S] [4[Beta]/4 + [Kappa] - [Rho][a.sup.2]/2 + [Kappa]][[Omega].sup.2] (5)
where [Kappa] = a[(S).sup.1/2] with S = [Rho]g/[Sigma].
From Eq 5 and the observation that h depends linearly with [[Omega].sup.2] in the rod-climbing experiment, [Beta] can be calculated for the known values of [Sigma] and [(dh/d[[Omega].sup.2]).sub.[Omega][approaches]0] as follows (3):
[Beta] = 3[[Alpha].sub.1] + 2[[Alpha].sub.2] = 4 + [Kappa]/4 [[Sigma][-square root of S]/2a[[Pi].sup.2] [(dh/d[[Omega].sup.2]).sub.[Omega][approaches]0] + [Rho][a.sup.2]/2 + [Kappa]] (6)
Therefore, using Eq 1, [Beta] is expressed as follows (6),
[Beta] = [[Psi].sub.1]/2 + 2[[Psi].sub.2] (7)
Thus, from the measured values of the rod climbing heights, the value of [Beta] can be determined for the particle suspended Boger fluid.
High molecular weight PIBs of various concentrations and molecular weights were prepared in low molecular weight PB with the suspended particles. The PIBs (Vistanex, MM grades L-80, L-100, L-120 and L-140) obtained from Exxon Chemicals) were highly paraffinic hydrocarbon polymers, composed of long straight-chain molecules terminally unsaturated and are characteristically lightly colored, odorless, tasteless, and nontoxic. Due to their molecular structure, PIBs are relatively inert and resistant to chemical and oxidative attack but are soluble in hydrocarbon solvents. The PBs (obtained from Daelim Chemicals, Seoul, Korea) are chemically stable liquids with moderate to high viscosities (120 poise [approximately] 400 poise). Additionally, the PBs do not form residues when they are either volatilized or thermally decomposed at sufficiently high temperatures. In this study, PB 900 was used as the solvent except in the solvent viscosity variation experiment. To investigate the effect of the particles on rod-climbing, solutions of different particle concentrations (0 to 10 %v/v) were prepared using kaolinite ([[[Al.sub.2][Si.sub.2][O.sub.5][(OH).sub.4]].sub.2]). It is the most common clay mineral having a two dimensional silicate structure (the mean particle size is 4 [[micro]meter] and the specific gravity is 2.6).
The Newtonian solvent, PB, was made viscoelastic by dissolving a small amount of PIB with reagent grade toluene as a co-solvent, which is a volatile and also good solvent for both PIB and PB. PIB is compatible with PB, but it is difficult to dissolve it in PB without a co-solvent. Small pieces of PIB were dissloved in toluene by stirring with a magnetic stirring bar in a sealed glass flask at room temperature for at least 24 hours. The solution of PIB in toluene was then mixed with PB in a 1000 ml beaker and stirred occasionally with a glass-stick for at least 2 days. A low rotational speed was applied to minimize the degradation of the long chain molecule (PIB). The kaolinite particles were then suspended in PIB solutions. A ball mill was used to disperse the particles uniformly. Finally, toluene was removed as completely as possible with a rotary-evaporator and vacuum-oven, in that order.
The rod-climbing apparatus consisted of a rod inserted vertically into a jar of the solution. The rod was driven by an electrostatic servomotor with a control system to maintain a constant speed under varying amounts of torque. The apparatus could accommodate rods with diameters of up to 1.2 cm. This limitation on rod diameter was imposed by the requirement that the diameter of the fluid container (12 cm) should be 10 times that of the rod diameter in order for the unbounded fluid approximation to be valid. The rod-climbing apparatus was enclosed in a thermostatic container and placed in a temperature-controlled chamber accurate to [+ or -]0.1 [degrees] C. The rod and climbing fluid were viewed through the front of the chamber. The angular speed of the rod was fixed for each reading and measured using a digital tachometer with an accuracy of 0.5 rpm. The climbing height of the fluid was then measured using a cathetometer (Gaertner Scientific Co.) with a reproducibility of 1 [[micro]meter]. In addition, surface tensions of the particle suspended polymer solution were measured using DCA 315 (Cahn's Dynamic Contact Angle).
RESULTS AND DISCUSSION
Figures 1 and 2 show both the shear viscosity and the first normal stress difference as a function of shear rate for different particle concentrations of L-80 PIB/PB solutions measured by using a mechanical spectrometer (RMS-800) in a cone and plate geometry at 30 [degrees] C, respectively. Without kaolinite particles, both figures give typical Boger fluid characteristics for shear viscosity and first normal stress difference with constant viscosity and a quadratic dependence of [N.sub.1] on [Mathematical Expression Omitted] for PIB in PB solutions for a wide range of [Mathematical Expression Omitted]. Even though a slight shear thickening behavior appears at a high shear rate region for a 7 %v/v concentration of the kaolinite particles in Fig. 1, it can be regarded that kaolinite suspended PIB/PB solutions show time-independent high and constant values of shear viscosity. It also indicates that the shear viscosity increases with increasing particle concentration as similar to a typical particle suspended polymer solution (16). From Figs. 1 and 2, it is found that the kaolinite suspended PIB/PB-solutions show second-order fluid behavior (Boger fluid) with high and constant values of viscosity and elasticity. In contrast to typical particle suspended polymer solutions or melts, Fig. 2 shows that the first normal stress difference increases with particle concentration. in general, it is known that the addition of inert solid particles to a polymer decreases the melt elasticity, and the first normal stress differences decrease as the filler concentration increases (17, 18). The observation that polymers with high filler concentrations have smaller first normal stress differences than those with low concentrations can be explained by the fact that the filler itself can increase the rigidity of the polymer (19).
The upper convected Maxwell model was applied to characterize the rheological properties of second-order fluids. Thus, [N.sub.1] was obtained with a relaxation time ([Lambda]) as follows (20, 21):
[Mathematical Expression Omitted] (8)
Over the broad ranges of shear rates [Mathematical Expression Omitted], [N.sub.1] exhibits a quadratic dependence on [Mathematical Expression Omitted] (note: [Eta] is independent of shear rate), as shown in Fig. 2.
Figure 3 shows the rod-climbing height (h) versus [[Omega].sup.2] for three different concentrations of PIB in kaolinite suspended PB fluids.
Higher polymer concentrations (producing higher elasticity) exhibit higher rod-climbing heights. The relationship between [Beta] and PIB concentration (C) can be investigated from the theory developed by Brunn (22), who adopted Brinkman's analysis for the dumbell (polymer) model to describe a second-order fluid. The material functions ([Eta], [[Psi].sub.1] and [[Psi].sub.2]) obtained by Brunn are:
[Eta] = [[Eta].sub.s](1 + C[[Eta]] + 0.5[(C[[Eta]]).sup.2]),
[Mathematical Expression Omitted], (9)
[Mathematical Expression Omitted]
Inserting Eq 9 into Eq 7, we obtain
[Mathematical Expression Omitted] (10)
Equation 10 implies that the rod-climbing height increases with Increasing polymer concentration and solvent viscosity ([[Eta].sub.s]). Experimentally, the rod-climbing height was observed to increase with the square of the solvent viscosity, which is in agreement with Eq 10. Rearranging Eq 10, the following equation can be obtained:
[Mathematical Expression Omitted] (11)
where B = M[[[Eta]].sup.2]/RT. Equation 11 predicts a linear correlation between [Mathematical Expression Omitted] and C, as confirmed by Fig. 4.
The rod-climbing experiment was also performed for different molecular weights of PIB. For a solution of 0.2% w/w with three different molecular weights of PIB in particle suspended PB, the climbing constant increases with increasing molecular weight, as shown in Fig. 5.
Figure 6 shows the rod-climbing heights for 0.2% w/w L-100 PIB in PB-kaolinite versus the square of the rotational speed of the rod for three different rod sizes. As expected, the rod-climbing height increases with increasing rotational speed and rod size (less than the critical rod-size ([R.sub.c])). Since [Beta] is almost identical for the several rod sizes (less than the [R.sub.c]), a rod size of 1.0 cm was selected in our experiment. When surface tension is neglected, [h.sub.2](r) in Eq 1 can be represented as follows:
[h.sub.2](r) = 4[[Pi].sup.2]/[Rho]g [2[a.sup.4]/[r.sup.4] [Beta] - [Rho][a.sup.4]/2[r.sup.2]] (12)
In order for the free surface to rise ([h.sub.2] [greater than] 0), [r.sup.2] must be less than (4[Beta]/[Rho]) (from Eq 12) for small [Omega]. Therefore, it is better to use rods with small diameters in the rod-climbing experiments.
Figure 7 shows the effect of temperature on the rod-climbing height with 0.2% w/w L-140 PIB in PB-kaolinite.
In this experiment, the temperatures (25, 30, and 35 [degrees] C) of the polymer solutions were each maintained within [+ or -]0.1 [degrees] C. As predicted, the rod-climbing height decreases with increasing temperature. Beavers and Joseph (4) explain that the climbing heights of the polymer exhibit a bell-shaped profile at the rod which is largely influenced by the temperature of the fluid.
The climbing constants of PIB solutions containing suspended particles were measured in rod-climbing experiments. The rheological properties of the polymer solution were also investigated using a RMS-800 rheometer (Rheometrics, Piscataway, USA). The measured values of beth [N.sub.1] and [Eta] from our study imply that the particle-suspended PIB solutions showed the Boger fluid behavior. Increasing the elasticity with particle concentration produces opposite results to those for typical particle-suspended polymer melts. The suspended particles in our study played the role of increasing the solvent viscosity. This increase is related to the rigidity of the polymer, and it induces the increase of the overall elasticity of the system. The rod-climbing constants were found to increase with polymer concentration, solvent viscosity, and molecular weight of the polymer but it decreased with temperature. For particle suspended PIB solutions, the climbing height drastically increased with particle concentration.
This research was supported by the Korea Science and Engineering Foundation (1998).
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|Title Annotation:||5th International Conference on Polymer Characterization|
|Author:||Choi, Hyoung J.; Lee, Hong J.; Lim, Sung T.; Jhon, Myung S.|
|Publication:||Polymer Engineering and Science|
|Date:||Mar 1, 1999|
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