Research Article pubs.acs.org/acscombsci
A Novel High-Throughput Viscometer Suraj Deshmukh,* Matthew T. Bishop, Daniel Dermody, Laura Dietsche, Tzu-Chi Kuo, Melissa Mushrush, Keith Harris, Jonathan Zieman, Paul Morabito, Brian Orvosh, and Don Patrick Core R&D, The Dow Chemical Company, Midland, MI 48674, United States ABSTRACT: A novel, rapid, parallel, and high-throughput system for measuring viscosity of materials under different conditions of shear rate, temperature, time, etc., has been developed. This unique system utilizes the transient flow of a complex fluid through pipettes. This approach offers significant practical advantages over microfluidic-based devices for viscosity screening: no cleanup is required, the method is high throughput ( 0 during active aspiration, thus fluid flow rate sign convention is Q < 0); p0 and V0 are the initial headspace pressure (1 atm) and volume; α = (πRC4/8ηLC) is the constant for Hagen−Poiseuille flow through the effective capillary; that is, Q = α(p−p0); k = p0V0 = pV (ideal gas law) describes the implicit relationship between headspace pressure and volume and has been used to substitute for p in the differential equations, A′ = A − αp0 and B = αk. In eq 2b, t1 is the duration of the active aspiration step, or the time of plunger motion before the settling period starts. Equation 2 can be iteratively solved to calculate headspace volume V versus t, from which vacuum pressure in the headspace can be calculated using the implicit relationship pV = D
DOI: 10.1021/acscombsci.5b00176 ACS Comb. Sci. XXXX, XXX, XXX−XXX
Research Article
ACS Combinatorial Science fairly simple geometries and constant fluid properties, each component of the pressure balance can be defined and solved using a time-stepping approach coupled to a simple solution finder (such as the solver add-on in Excel). As illustrated in Figure 1a, the pipet tip can be considered as two connected truncated cones with lengths L1 and L2 and taper angles ϕ1 and ϕ2. Thus, fluid height h(t) and interfacial radius r(t) at time t can be calculated using simple geometric equations based on the aspirated fluid volume Va(t), which is in turn estimated as Va(t − Δt) + Q(t)·Δt, where Q(t) is the liquid volumetric flow rate and Δt is the designated time step. V(t) is the gas phase volume at time t, which is calculated as the difference between the total volume (initial volume plus new air line volume created when the plunger moves) and the liquid volume. The geometric equations for the pipet geometry used in the current model are Va(t ) = Va,1(t ) + V2,a(t ) Va, i(t ) =
ci =
(3)
⎛π ⎞ 2 2 ⎜ ⎟ · h (t ) · [r (t ) + R i 1, i + ri(t ) · R1, i] ⎝3⎠ i
ri(t ) = R1, i·
diminishes as the interfacial diameter gets larger with increasing liquid height. In the following examples, the liquid density and viscosity were set to 0.85 g/cm3 and 650 cP, respectively, which are the reference values for the N250 Cannon viscosity standard at 23 °C. The interfacial tension was assumed to be 70 mN/m with a contact angle of 30°. The individual pressure drop components are plotted as a function of simulation time in Figure 2a. It can
(4)
(ci + Li) ci
(5)
R1, i·Li (R 2,1·R1,1)
(6)
where Va,i(t), hi(t), and ri(t) are the liquid volume, height, and corresponding interface radius within each section, i; R1,i, R2,i, and Li are the inlet and outlet radii and total length of each section; and ci is the length of the truncated conical head. It should be noted that the effect of different container sizes or different submerged depths during aspiration or dispense on the pressure measurement was found to be insignificant provided the vial diameter or well width was much larger than the tip opening diameter, as mentioned above. The overall pressure balance in the tip can now be defined by the following equation: Δpflow = pgas − pinlet − Δphead − Δpγ
Figure 2. Pressure predictions: (a) overall CFD-based pressure drop and individual pressure components (based on CFD predictions of liquid flow rate and height) and (b) comparison of the overall pressure drop based on both CFD and the time-stepping algorithm for the Cannon N250 viscosity standard at 10 μL/s plunger rate with the corresponding experimental data.
(7)
The small angle approximation can be applied to provide an equation for the estimated pressure drop due to flow within each diverging section (subscript i = 1, 2) as ⎧ −128η ·Q (t ) ·h (t ) ⎫ ⎧ 1 1 ⎫ i ⎬ ⎬·⎨ Δpflow, i = ⎨ − 3 2R1, i 3 ⎭ ⎩ 6π (R1, i − 2ri(t )) ⎭ ⎩ 2ri(t ) ⎪
⎪ ⎪
⎪
⎪
⎪ ⎪
⎪
be seen that most of the pressure drop comes from the flow equation (eq 8). The interfacial tension component is similar (but opposite in sign) to the liquid gravity head component initially, but the magnitude of the gravity head increases while the interfacial tension decreases. The resulting gas pressure curves predicted by both the CFD simulation and the timestepping algorithm are shown in Figure 2b versus experimental measurements for the same viscosity standard. The two numerical methods agree very closely and follow the shape of the experimental profile very well; however the experimental curves are offset from the predicted curves to lower pressures. The most likely cause for this difference is a lab temperature cooler than 23 °C, resulting in slightly higher experimental viscosities. Viscosity of this fluid is very sensitive to temperature (Ea/R = 5260K, where Ea is the flow activation energy); when a viscosity of 750 cP (corresponding to 20.5 °C) is used, the
(8)
where R1,i is the inlet radius of the ith diverging cylinder section, hi(t) is the height of the fluid column in that section, and η is the fluid viscosity.27 The gas pressure in the syringe system is calculated from pgas = ((p0 + patm)(V0/V(t)) − patm), where p0 and V0 are the initial pressure and volume, and patm = 1 atm. The inlet pressure due to submersion of the tip to depth htip below the fluid surface is calculated from pinlet = −ρ·g·htip, where ρ is fluid density and g = 9.8 m/s2. Similarly, the gravitational head pressure inside the tip is determined by Δphead = −ρ·g·h(t). Finally, the pressure drop across the interfacial boundary due to the surface tension (γ) is Δpγ = 2·γ· cos θ/r(t), where θ is the contact angle of the liquid with the pipet surface. The pressure effect of the interfacial tension E
DOI: 10.1021/acscombsci.5b00176 ACS Comb. Sci. XXXX, XXX, XXX−XXX
Research Article
ACS Combinatorial Science simulation results have a close quantitative agreement with experiment. The liquid flow rate and aspirated volume are plotted in Figure 3a and b along with the displacement rate and volume
Figure 4. Total mass dispensed as a function of viscosity for a series of viscosity standards and formulated products. Averages with standard deviations are shown.
The total weight of the sample dispensed as a function of viscosity can be empirically fit with a quadratic polynomial as shown in Figure 4. This can then be used as a calibration equation to determine the viscosity of Newtonian fluids with unknown viscosity. For example, if we measure the total weight dispensed for an unknown non-Newtonian fluid at the same temperature and liquid handler settings (dispense rate and time), this quadratic fitting polynomial can now be used to calculate the viscosity of that unknown fluid. In the case of nonNewtonian fluids, the total weight dispensed will be a complex function of the viscosity that will change with the shear rate. Furthermore, the shear rate itself is a complex function of the liquid flow rate and tip geometrical parameters. Hence, the measurement of complex fluid viscosity as a function of shear rate requires comparison of the experimental results with the numerical modeling results described in the earlier section. In addition to using the total mass dispensed to determine viscosity, a second approach was to use the measured pressure as a function of time instead. During the liquid handler aspiration and dispense operation, the liquid was aspirated into the pipet after an increase in the volume of the head space, which in turn induced a negative pressure drop driving the flow. The advanced liquid handlers such as the Hamilton create the negative pressure drop in microseconds and the liquid responds to it by flowing into the pipet, leading to a progressive decay in the pressure drop over time. This initial liquid response was dependent on the relaxation time of the material in the case of complex fluids. We recorded this pressure curve as a function of time and have analyzed it further to calculate rheological properties, such as viscosity, relaxation time, pot life, gel time, etc. Figure 5 shows the measured pressure as a function of time for the series of Cannon viscosity standards during a liquid aspirate operation. Figures 5a and 5b show the measured pressure for the same viscosity standards at different aspiration rates and total time settings of the liquid handler. As can be seen from Figure 5b, pressure curves for standards with viscosities above 1000 cP can only be measured at very low aspiration rate settings (2.5 μL/s) to stay below the pressure transducer limit. As can be expected from the Hagen−Poiseuille equation, decreasing the aspiration rate for a viscosity standard
Figure 3. Piston displacement versus liquid aspirationfor the Cannon N250 viscosity standard aspirated at 10 μL/s plunger rate for 400 μL total volume: (a) flow rate vs set aspiration rate and (b) aspirated volume vs piston displacement volume (CFD analysis).
due to the piston movement. There is a delay in the liquid flow response at both the start and finish of the piston motion due to inertia. This requires a waiting period at the end of the piston motion (i.e., the settling time) to enable the liquid draw to reach the specified volume. The numerical and CFD modeling approaches outlined above form an excellent basis to determine various rheological properties (especially viscosity) as a function of shear rate and time. Experimental Results: Newtonian Fluids. A number of viscosity standards with known viscosities ranging from 1 to 10 000 cP at ambient lab temperature conditions (23 °C ± 2 °C) obtained from the Cannon Instrument Company were used to understand the viscosity range and capability of this high throughput system. For the first approach, mass of the sample dispensed for these viscosity standards was measured at a set dispense rate and time setting for the liquid handler. Figure 4 shows the total mass dispensed through the liquid handler for viscosity standards at 40 °C. As can be expected, an increase in viscosity leads to a drop in the total mass dispensed, and the results show excellent reproducibility over replicate measurements. F
DOI: 10.1021/acscombsci.5b00176 ACS Comb. Sci. XXXX, XXX, XXX−XXX
Research Article
ACS Combinatorial Science
by modifying the diameter of the pipet to be larger in accordance with the Hagen−Poiseuille law described earlier. For non-Newtonian fluids, such as paints, adhesives, shampoos, etc., these pressure versus time curves were a complex function of the viscosity that itself was a function of the shear stress (or shear rate). In fact, these curves provided us with a rich source of data to calculate different rheological properties such as viscosity, relaxation time, yield stress, thixotropy, gel point, pot life, etc. This, however, requires comparing the pressure as a function of time curve with the numerical results for approximations to a generalized nonNewtonian fluid model and is out of scope for this article. While the applied pressure drop is negative in the case of the aspiration operation, a positive pressure drop was required to dispense the liquid out of the pipet. The pressure drop versus time curves showed an increasing pressure drop driving the fluid flow out of the pipet during the dispense operation (Figure 6). Figure 6a and 6b shows the dispense operation
Figure 5. Average aspirate curves (pressure vs time) for Newtonian viscosity standards in the viscosity range 1−5000 cP for two different flow rates and volumes: (a) 10 μL/s for 400 μL total and (b) 2.5 μL/s for 100 μL total.
with a constant viscosity led to a lower pressure drop at any given point in time. Conversely, the pressure drop as a function of time increased as the viscosity of the fluid increased at a given aspiration rate and pipet geometrical parameters. Since the viscosity is constant for Newtonian fluids even at different shear rates, measurement of a single pressure drop value at any set time setting can be used to create a calibration curve similar to Figure 4 above that was used for the mass dispensed approach. For Newtonian fluids with an unknown viscosity, such as oils, etc., the viscosity was easily calculated by using the calibration curve that fit measured pressure to viscosity. The pressure transducer used to record the pressure curve defined the viscosity range of the instrument, and in this particular case, the Hamilton liquid handler was better suited for lower viscosity standards (5000 cP). This can of course be modified by using different pressure transducers or
Figure 6. Average dispense pressure vs time curves obtained for viscosity standards in the range of 100−3000 cP at two different flow rates and volumes: (a) 10 μL/s for 400 μL total and (b) 2.5 μL/s for 100 μL total.
pressure curves for the same viscosity standards at two different dispense rate and total time settings for the liquid handler. As expected from the Hagen−Poiseuille equation, a higher viscosity fluid required a higher pressure drop to drive fluid flow out of the pipet at the specified dispense flow rate. Also, the same viscosity standards (example, Cannon N250 or 650 cP as shown in Figure 6) showed a significantly lower measured G
DOI: 10.1021/acscombsci.5b00176 ACS Comb. Sci. XXXX, XXX, XXX−XXX
Research Article
ACS Combinatorial Science
can be seen from the figure, a wide range of viscosities and aspiration rates (that can be correlated to different shear rates) are accessible by this tool with simple changes of geometrical parameters such as the pipet diameter and length. Experimental Results: Complex Fluids. The transient flow through a pipet (or capillary) for a complex fluid provides a complication that necessitates the use of modeling to determine rheological properties (such as viscosity as a function of shear rate) from the raw pressure curve. Utilizing CFD modeling and pressure measurements at various aspiration rates (that can be related to shear rates), times, and temperature conditions, rheological behavior of complex fluids can be determined in a high-throughput manner. Although the detailed description of the CFD modeling and viscosity calculation for complex fluids is out of scope for this article, we illustrate the potential of this methodology by using a series of paints that represent typical power-law fluids. Tips customized to closely mimic the idealized geometry shown in Figure 1b were used to record the maximum pressure value at different aspiration rates. Since this idealized tip geometry more closely mimicked the capillaries used in a capillary rheometer with a constant diameter, we used the same methodology used in a capillary rheometer to calculate the viscosity. This involved using the Hagen−Poiseuille equation to calculate the apparent viscosity from the measured pressure drop and the set aspiration rate (assumed equivalent to the flow rate).9 The apparent shear rate or the Newtonian shear rate at the wall is then also calculated using the set aspiration rate and custom capillary diameter. We have used custom tips of different diameters at two different lengths each so it is possible to determine the Bagley correction and correct the pressure measurement. However, for the purposes of illustration of the methodology in this article, we simply plot the calculated apparent viscosity vs apparent shear rate and compare it to the actual viscosity versus shear rate measured using a controlled stress rheometer. Figure 8 shows the calculated viscosity vs calculated shear rate curves for a series of proprietary benchmark paints as measured using our high-throughput viscosity measurement tool, compared with viscosity measurements for the same paints using a conventional stress-controlled rotational rheometer (Anton Paar MCR 301). The 12 different benchmark paint formulations show a typical power law fluid behavior, with viscosity decreasing as a function of shear rate. Since there are a number of assumptions in this calculation, such as the assumption of a steady state pressure value and the set aspiration rate being directly correlated with the shear rate independent of the viscosity, we expect a large error in the viscosity calculation. Surprisingly, the calculated apparent viscosity versus calculated apparent shear rate compares quite well with a conventional rheometer measurement. It is especially suited for measurements at low shear rates (1−100 1/s) for viscosities ranging from 1−100 000 cP. The shear rate and viscosity range is a direct function of the pressure transducer limit, and as mentioned above, the pressure transducer can be easily swapped to change the operating window. The rheological properties of the paints, especially the viscosity as a function of shear rate, are a critical screen for determining the final product performance, such as the flow and application behavior as the paint is picked up by a brush/ roller and applied on surfaces. The relatively good comparison of the calculated viscosity values from this technique compared
gauge pressure as the dispense flow rate was decreased from 10−2.5 μL/s. The viscosity standards used in this work are mineral or silicone oils and, hence, do not tend to accumulate on the outside of the tip, resulting in relatively clean and reproducible aspirate and dispense pressure curves. However, this is seldom true for consumer products. Especially in the case of the dispense operation, as the viscosity increased, filament formation, and excess fluid on the outside of the tip become significant issues. The pressure profiles for the two different dispense flow rate conditions, as shown in Figure 6a and 6b, both show discontinuities due to complications of drop formation and breakup during dispense. The pressure rose until the excess fluid along with the first drop was dispensed causing a release in the pressure, at which point the pressure then rose more steadily as the dispense profile became more continuous. Depending on the amount of excess fluid on the outside of the tip, the pressure curve changed significantly. The initial increase in pressure drop for non-Newtonian fluids was also a function of a number of other variables, such as relaxation time, “stringiness”, wetting behavior of the system, etc., and it can indeed be further studied to determine rheological properties for complex fluids. This can of course be minimized by having the dispense operation occur very close to the liquid surface or by submersing the tip inside the liquid container. We used primarily the aspirate operation for pressure measurements of our Newtonian fluids, such as oils, solvents, polyols, glycerin, dilute biological systems, etc., and our nonNewtonian fluids such as surfactant systems, thermosets, adhesives etc. Figure 7 below shows the true high-throughput
Figure 7. Average pressures recorded at a set time for different aspiration rates and different pipettes for the series of viscosity standards. Each data point is a composite of 8 measurements, and the elongation denotes the variability in the measurement technique.
nature of this tool: approximately 7000 pressure curves measured for different viscosity standards with different commercial and custom tips at different aspiration rates each represented only once in the figure by their maximum pressure value. Each set of hundred pressure measurements was made within an hour without any cleanup required and is plotted against the corresponding viscosity value. The elongation in any cluster of points represents the spread of the measurement for each set of 8 replicate measurement and depicts the high resolution of the tool, with a standard deviation
