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Portable imaging viscometry for quantitative complex fluid measurements
Experimental Thermal and Fluid Science 107 (2019) 29–37
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Portable imaging viscometry for quantitative complex fluid measurements a
a
Soheila Shabaniverki , Antonio Alvarez-Valdivia , Jaime J. Juárez a b
a,b,⁎
T
Department of Mechanical Engineering, Iowa State University, 2529 Union Drive, Ames, IA 50011, United States Center for Multiphase Flow Research and Education, Iowa State University, 2519 Union Drive, Ames, IA 50011, United States
ABSTRACT
Portable optical microscopy systems offer a pathway for serving as analytical platforms that characterize the properties of complex fluids. Measuring the diffusion characteristics of particles dispersed in complex fluids, known as microrheology, is one possible microscopy-based approach. In this study, we are the first to demonstrate that microrheology can be used to analyze diffusion profiles measured in a portable microscopy viscometer constructed from off-the-shelf parts. We perform measurements in two polyvinylpyrrolidone (PVP) solutions with 5 wt% and 10 wt% concentrations to demonstrate how this platform can be utilized as a viscometer for low-viscosity solutions. Our experiments consist of tracking the trajectory of diffusing colloidal particles dispersed in the PVP solutions and measuring their mean square displacement (MSD). The microrheological analysis used to characterize the MSDs are used to calculate the complex modulus over a frequency range of 0.07–33 Hz. The portable microscope viscometer measured a complex modulus of 0.23 mPa for the 5 wt% PVP solution and 0.36 mPa for the 10 wt% PVP solution. We compared our measurements to a scientific grade microscopy system, where the measured complex moduli were 0.24 mPa and 0.49 mPa at 5 wt% and 10 wt%, respectively. Our analysis of the results showed the need to account for sources of dynamic error caused by discrete sampling at limited exposure time. The viscosity of each sample appears to be lower in comparison to the case where dynamic error is accounted for. The static error for the portable microscope viscometer is measured to be 10–100 nm, which is an order of magnitude larger than the static error measured on our scientific grade microscope. Despite the larger static error, the portable microscope viscometer measured the viscosity of our samples to be within 4–6% of comparable measurements performed on the scientific grade microscope. Overall, we demonstrate that the portable microscope viscometer is capable of many of the same measurements as the scientific grade microscopes, which has implications for a variety of portable (e.g., smartphone) image-based microscopy experiments such as flow cytometry, viscometry and forensic analysis.
1. Introduction Portable microscopy systems serve as a platforms for low-cost health care monitoring [1], imaging flow cytometry [2], and forensic analysis of trace evidence [3]. Establishing portable microscopy as a viable technique for obtaining quantitative analytical data can open the door for in-situ measurements that enable food quality monitoring [4], fluid viscosity measurements [5], or perform experiments on surface coatings [6]. A common challenge that ties many of these applications together is the need to characterize the properties of complex fluids exhibiting a non-Newtonian response. Complex fluids are estimated to comprise up to 60% of the materials handled by the chemical process industry [7] and there is a need to perform online viscosity measurements to characterize the complex fluids produced by these processes [8]. Instruments for measuring the properties of complex fluids can cost $20,000 for a high-end viscometer and $200,000 for a research-grade rheometer [9]. In an effort to make viscometers and rheometers more affordable, instrumentation for measuring complex fluid viscosity is being developed using off-the-shelf components. These portable complex fluid measurement instruments rely on optical video cameras to capture pinching events when the complex fluid is pulled apart [10] or allowed to drip [11]. ⁎
Another method for measuring complex fluid properties relies on monitoring the transport of a complex fluid in a microfluidic channel [12]. In this format, a microfabricated channel can be combined with integrated pressure sensors [13] or microparticle image velocimetry [14] to perform complex fluid measurements. Microfluidic viscometers offer several advantages over other portable formats in that only a small sample volume is necessary for measurements and they overcome the torque limitations of viscometers and rheometers, enabling measurement of low-viscosity fluids [15]. A limitation of microfluidic viscometers is the need for access to cleanroom facilities for fabricating these devices. Furthermore, many of the microfluidic viscometer demonstrations utilize scientific grade microscopes imaging, which ultimately increases the cost of the viscometry platform. As an alternative to microfluidic viscometry, this study examines the feasibility of using a portable microscope as a viscometer platform. This platform tracks the mean square displacement (MSD) of particles dispersed in a complex medium using optical video microscopy and extracts the viscosity of the fluid based on a microrheological analysis of the data. This study represents the first time that low-viscosity microrheology has been performed in a portable microscope viscometer platform. As a model fluid for this study, we utilize 5 wt% and 10 wt% polyvinylpyrrolidone (PVP) as a test complex fluid solution. Our
Corresponding author at: Department of Mechanical Engineering, Iowa State University, 2529 Union Drive, Ames, IA 50011, United States. E-mail address: [email protected] (J.J. Juárez).
https://doi.org/10.1016/j.expthermflusci.2019.05.009 Received 29 December 2018; Received in revised form 7 May 2019; Accepted 16 May 2019 Available online 17 May 2019 0894-1777/ © 2019 Elsevier Inc. All rights reserved.
Experimental Thermal and Fluid Science 107 (2019) 29–37
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Nomenclature a B C Do D D E f G G’ G” G g h
havg kB MSD m NA n
PVP r2 2 rmeas r2 T V Γ εm εo
particle radius electrostatic interaction magnitude electrolyte concentration diffusion coefficient average hindered diffusion coefficient particle diameter charge of an electron hydrodynamic hinderance buoyant mass storage modulus loss modulus complex modulus gravitational acceleration distance between the particle center and the boundary surface average height Boltzmann’s constant mean square displacement pre-factor fitting parameter Avogadro’s number power fitting parameter
p
s
η max
κ−1 ρp ρm σ Ψ1 Ψ2 ω
polyvinylpyrrolidone actual mean square displacement measured mean square displacement mean square displacement corrected for dynamic error ambient temperature migration velocity gamma function dielectric constant of the medium permittivity of free space static error of portable microscope static error of scientific grade microscope viscosity of the medium maximum measurable viscosity Debye length particle density medium density exposure time of the camera dimensionless lag time surface potential of the particle surface potential of the boundary Local shear rate
fluorescent polystyrene beads (ThermoFisher Scientific, F8852) with peak excitation and emission spectra of 505 nm and 515 nm, respectively. One microliter of fluorescent bead stock solution is taken directly from the manufacturer’s container and stirred into the PVP solution for 10 min. The dispersion consisting of particles and PVP was poured into a petri dish, which was placed in 4 °C refrigerator for 3 h. This gave the sample time to cool and sediment. When the sample was placed on the microscope stage, we waited for a 15 min period to give the dispersion an opportunity to settle before the experiment. For each sample, we performed same-day observations of 10 different particles in each sample.
microrheology experiments cover a frequency range of 0.07–33 Hz and measure complex moduli down to approximately 0.23 mPa, which is nearly one order of magnitude more sensitive than a commercial rheometer [16]. The portable microscope viscometer presented in this paper represents an alternative to commercial rheometers for obtaining quantitative data of low-viscosity fluid properties. Furthermore, where previous studies have focused on quantifying single, static images using portable optical microscopy systems [17–19], we demonstrate that capturing video with these types of microscopy platforms allows for the measurement of dynamic data comparable to scientific grade microscopy systems. The portable microscope viscometer described here could serve as a platform that introduces microrheology as a tool for quantifying the viscosity and rheology of industrial fluids. For example, the portable microscope viscometer could be a tool for rapid inline monitoring of complex fluids, which is used to assess quality and control processing of petroleum products [20], pharmaceutical production [21], and food production [22]. This platform could also have an impact on the industrial use of nanofluids. At present, the measurement of nanofluid properties at temperatures and concentrations encountered in industry is challenging due to the high cost of quantitative instruments [23]. With improved optics [24] and particle tracking tools [25], the portable microscope viscometer could be used to measure nanofluid properties. This platform would provide the experimental basis for developing models [26,27] that guide the development of complex fluids for industrial applications.
2.2. Microscopy Microrheology experiments were initially performed using a scientific grade research microscope (Olympus, IX70) and a scientific CMOS camera (QImaging, Optimos). The petri dish is placed on a sample stage where it is illuminated by an LED-based fluorescence module (wLS, QImaging) with a FITC filter set. All experiments on the scientific grade microscope are performed with a 40x Olympus long working-distance objective lens. Video tracking data for single particle trajectories, with an image size of 100 × 100 pixels (22 μm × 22 μm), are captured using an open source microscopy software (μManager). A thousand frames of data are recorded for each experiment at a frame rate of 30 frames/s and exposure time of 30 ms. To obtain sufficient statistics to measure viscosity, we performed at least ten different observations of each sample and the resulting video files are analyzed using the particle tracking algorithm developed by Crocker and Grier [28] written in MATLAB. The experiments performed on the scientific grade microscope were designed to be compared to similar experiments performed on the portable microscope viscometer. The supplementary information contains additional details on the off-the-shelf materials used to fabricate this platform. Imaging of particle trajectories is done using a machine vision-grade CMOS camera (FLIR Machine Vision, BFLY-U3-23S6C-C). The camera is attached to a 1-inch diameter lens tube (Thorlabs, SM1L30) using a C-mount adapter (Thorlabs, SM1A9). The camera and lens tube assembly are attached to a 40x Lecia objective lens (Leica, Part Number 506149) using an M25 thread adapter (Thorlabs, SM1A12). The protocol for capturing images using this camera is
2. Materials and methods 2.1. Sample preparation Twenty milliliters of deionized water, sourced from an ARIES High Purity Water System (Aries Filterworks) with a 0.2 μm filter, are added to a beaker with a stir bar. The beaker is placed on a stirring hot plate and heated to a temperature of 50 °C. An aqueous solution of polyvinylpyrrolidone (PVP), is prepared as a model, low-viscosity complex fluid. The PVP, with a molecular weight of 40 kDa, is sourced from Alfa Aesar (product number J62417). Two different amounts of PVP powder (1 gr and 2 gr) are used to create either a 5 wt% or 10 wt% aqueous solutions. To these solutions, we add a dispersion of 1 µm diameter 30
Experimental Thermal and Fluid Science 107 (2019) 29–37
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similar to the protocol followed for the microrheology experiments done on the Olympus microscope. The only difference is that we used 150 × 150 pixel (20 μm × 20 μm) to better match the physical size of the images due to differences in the size of the CMOS sensors on the cameras. The lens tube assembly is mounted to a base fabricated from acrylic (OPTIX, 1AG2123A) and attached to a solid wooden base using long bolts and washers (Fig. 1). The sample stage is also fabricated from acrylic and attached to two stepper motors with ‘screw and nut’ sliders to make it easier to focus. The stepper motors are regulated by a stepper motor driver controlled by a A3967 EasyDriver integrated circuit. The driver is controlled using an Arduino programmed using the opensource Arduino IDE software. A leveling tool was used to help position the stage at an even height to prevent particle migration due to sample tilting.
where the double bar symbol indicates that this quantity is related to diffusion parallel to a boundary, h is distance between the particle center and the boundary surface, and the function, f , represents the hydrodynamic hinderance to diffusion near a boundary. This function is given as [30],
f (h a) = 1
D
The diffusion coefficient for a single particle far from any boundary is given by the Stokes-Einstein equation [29], (1) −23
J/K is Boltzmann’s constant, T is the ambient where kB = 1.38 × 10 temperature of the experiment, η is the viscosity of the medium and a is the particle radius. When the particle approaches a boundary, the diffusion coefficient of the particle as it travels parallel to the boundary is hindered,
D = Do f (h a)
45 a 256 h
4
1 a 16 h
5
(3)
(4)
= Do f
where the average hinderance function is evaluated at an average height located at the position of mechanical equilibrium. The average diffusion coefficient for a particle is measured using the mean square displacement (MSD) of a trajectory obtained through image analysis of optical video microscopy data. The average MSD for a particle diffusing in two dimensions is expressed as, r 2 = 4Do f . In a complex fluid, we expect that the diffusion coefficient of the particle will depend on the local shear rate, ω, induced by the particle as it fluctuates due to Brownian motion. If the viscosity is a power law fluid [32], = m n 1, where m and n are obtained through fitting the model to viscometry data, then the MSD measured in experiment is expressed as,
3.1. Diffusion in complex fluids near a wall
kB T 6 a
3
As the particle diffuses over the surface, it samples the potential energy well that represents the net energy interaction between the particle and the surface. The diffusion coefficient measured by optical video microscopy represents the net effect of the potential energy on the diffusing particle. We can define the average diffusion coefficient as [31],
3. Theory
Do =
9 a 1 a + 16 h 8 h
r2 = 4
(2)
kB T 6 a (m n 1)
f t
(5)
Fig. 1. A schematic of the portable microscope system used to measure the viscosity of PVP solutions with diffusing colloidal probes. The videos are captured through an objective lens using a machine vision CMOS camera. Details on construction may be found in the supplementary information. 31
Experimental Thermal and Fluid Science 107 (2019) 29–37
S. Shabaniverki, et al.
where the expression for the power law fluid is substituted for viscosity. 1 t for microrheological The shear rate is approximated as measurements in non-Newtonian media [33]. Using this approximation for shear rate, the MSD for a particle diffusing in a power law fluid near a wall is,
expected power law viscosity of a complex fluid. The MSD measured using trajectory data can also be used to characterize rheological parameters such as storage (G′) and loss (G″) moduli, which are critical for understanding the dynamic mechanical response of the fluid [37]. This is done using an approach known as microrheology [38,39] where the diffusion induced by Brownian motion causes the particle to locally shear the fluid. The storage and loss moduli of the bulk fluid are expressed as [33],
(6)
r 2 = 4D t n For simplicity, the parameter, D , is,
D = kB T f
(7)
6 am
Eq. (6) is similar to established models of MSD for particles diffusing in complex fluid media [34]. Eq. (6) can be used to fit MSD data to obtain some rheological information about the medium. The exponent, n, found in Eq. (6) is directly related to the power law exponent for the fluid. Eq. (7) is coupled directly to hydrodynamics of the diffusing particle and the power law pre-factor, m. Eq. (6) represents one possible model which can be used to fit the MSD data for a particle diffusing in a power law fluid. However, this expression does not account for possible sources of error. In Section 4, we elucidate the effect that errors have on MSD data and compare extracting viscosity data using Eq. (6) to a model that incorporates error.
B G
B = 64 G=
m oa
4 3 a ( 3
p
kB T e m)
2
tanh
e 1 e 2 tanh 4kB T 4kB T
6 aD
a
)
[1 +
(15)
( )]
)=
dln r 2 (1 d lnt
)
(16)
Image analysis is subject to measurement noise, which arise from static effects like mechanical vibrations in the microscope stage or fluctuations in light intensity [41]. The experimental noise is additive and it influences the measured MSD by increasing its apparent magnitude. The measured and actual MSDs are related to each other by, 2 rmeas = r2 +
(17)
2
where ε is the static error that arises from experiments and is the actual MSD of the particle. Measuring the static error in our portable system can yield insight into the limitations of our portable system. To do this, we utilize a procedure outlined in our previous work [42] where we image 1 µm polystyrene beads immobilized on a glass substrate. In this procedure, the static error represents the value of the MSD measured at a lag time equal to the exposure time of the camera, t = σ. This value, ε2, is then subtracted from the experimentally measured MSD to find the actual MSD. The finite exposure time of the experimental imaging system can also cause a distortion effect that leads to errors in localizing the center of the diffusing particle [43]. This distortion requires a second correction, which is found by comparing the data to a model of the expected
r2
(10) (11)
where κ is the Debye length, e is the charge of an electron, NA is Avogadro’s number, C is the electrolyte concentration, εm is the dielectric constant of the medium and εo is the permittivity of free space. The parameter, B, represents the electrostatic interaction of the system, which depends on the surface potential of the particle (Ψ1) and the boundary (Ψ2). The buoyant mass, G, depends on particle volume and the difference between particle density (ρp) and medium density (ρm). The parameters in Table 1 are substituted into Eqs. (9)–(11) to estimate the average height sampled by particle based on Eq. (8). The resulting height obtained by evaluating Eq. (8) is substituted into Eq. (3) to find the average hinderance factor as, f = f (havg a) . The value of f is used to decouple D in Eq. (7) to estimate the power law pre-factor as,
kB T f
kB T f r 2 (1
3.4. Sources of experimental error
−1
m=
(14)
In the low-shear limit, the Cox-Merz rule gives the viscosity of the fluid as = G" ( ) [40]. The average hinderance factor is included in Eq. (15) to correct for the presence of a wall.
(9)
m o kB T
( ) 2)
(1
where havg represents the average height sampled by the diffusing particle when the electrostatic energy interaction between the particle and the boundary is in mechanical equilibrium with gravity. The parameters, κ, B and G are expressed as [36],
=
G" ( ) = |G ( )| sin (
where < r > is the MSD evaluated at a value of t = 1/ω, ω is the local shear rate of the diffusing particle, Γ is a gamma function approximated by Γ(1 + α) ≈ 0.457(1 + α)2 − 1.36(1 + α) + 1.90, and the parameter α is given by,
(8)
2e 2NA C
(13)
2
In order to decouple D from the hydrodynamics of the particle diffusing near a wall, we estimate the average value of the hydrodynamic hinderance factor. This is done by evaluating Eq. (3) using the average expected height of the diffusing particle in dynamic equilibrium with its environment. The average height is estimated as [35], 1ln
( ) 2)
|G ( )| =
3.2. Estimating hydrodynamic hinderance
havg = a +
G' ( ) = |G ( )| cos (
Table 1 Parameters used to calculate the average height sampled by the particle using Eq. (8). The resulting height is substituted into Eq. (3) to calculate the average hinderance function, f .
(12)
where D is obtained by fitting MSD data to a model for expected particle dynamics in a power law fluid. 3.3. Microrheology The analysis discussed above can quickly yield insight into the 32
Parameter
Equation
Value
T (K) e/10−19 (C) NA/1023 (1/mol) C (mM) κ −1 (nm) εm a (nm) Ψ1 (mV) Ψ2 (mV) ρp (kg/m3) ρm (kg/m3) f
1, 6, 7, 9, 10, 12, 15, 21 9, 10 9 9 8, 9 9, 10 1–3, 6–8, 10–12, 15, 21 10 10 11 11 5–7, 12, 15
298 1.6 6.02 0.1 30 80 500 −50 −50 1055 999 0.698
Experimental Thermal and Fluid Science 107 (2019) 29–37
S. Shabaniverki, et al.
particle dynamics in a complex fluid [41],
r2 ( + 1)2 + n + ( 1)2 + n 2 = 4D (1 + n)(2 + n)
2+n
2
s
max
4.1. Effect of exposure time Fig. 2A shows a comparison of measured static error as a function of camera exposure time for both the scientific grade microscope and the portable microscope viscometer. The average value of static error is larger for the portable microscope viscometer in comparison to the scientific grade microscope, which is likely the result of minimal antivibration equipment on the portable microscope combined with a noisier camera sensor. We also observe that the static error decreases with increasing exposure time. Increasing the exposure time allows for more light to fall on the CMOS sensor, making it easier to localize particles. Our measurements indicate that the static error for the portable system is highly repeatable, but we also observed large standard deviations associated with the data collected from the scientific grade microscope. The magnitude of the static error on the scientific grade microscope is consistently smaller than the portable microscope. This is because the scientific grade microscope is mounted on an anti-vibration table and the camera used for imaging is Peltier cooled, which reduces the overall noise from this microscope. Using the data from Fig. 2A, we can estimate the relationship between static error and exposure time by curve fitting the data from the portable microscope viscometer and the scientific grade microscope,
2kB T 3 a 2
(21)
4.2. Mean square displacement measurements Fig. 3 shows the average MSD as a function of lag time obtained from optical microscopy measurements done on the scientific grade microscope (top row) and portable microscope viscometer (bottom row). The experiments for each data set were obtained in the same way by using similar exposure times (∼30 ms) as described in the methods section. The trends shown in both data sets at 5 wt% PVP (Fig. 3A and 3B) appear to be comparable with each other, with the standard deviation increasing with time for both systems. However, the error appears to increase to a greater degree for the portable system at higher lag times when compared to the scientific grade microscope. The data for 10 wt% PVP in Fig. 3D shows that the portable microscope viscometer exhibits a larger standard deviations at lower lag times than the
0.619
ms
(20)
ms
where ε2 is the square of the relationship between static error and exposure time from Eqs. (19) and (20). This parameter can also be experimentally measured using the protocol described above. Fluid viscosities that exceed this limit cannot be passively measured by diffusing particles because their motion would be indistinguishable from static error. In Fig. 2B, we compare the maximum accessible viscosity for both the portable microscope and the scientific grade system by substituting Eqs. (19) and (20) into Eq. (21). This analysis shows that the scientific grade system, with its lower static error, can theoretically measure higher viscosity fluids. In order for the results to be comparable between the portable system and the scientific grade system, the viscosity of the fluid cannot exceed the value that is measurable on the portable system. This analysis serves as a guide for determining the limitations of viscosity measurements with the portable microscope viscometer.
4. Results and discussion
= (0.08µm)
0.973
where p and s stand for portable microscope viscometer and scientific grade microscope, respectively. It is evident that the static error decreases with increasing exposure time. This is expected since increasing exposure time allows for more light to hit the CMOS sensor, which aids in localizing particle centers. The static error also places practical limits on the measureable viscosity of a fluid using microrheology. As the viscosity of a fluid increases, the diffusion coefficient of the particle will decrease. This has the effect of slowing the particle dynamics to the point where the displacement observed by the camera could not be distinguishable from camera noise. The upper limit for fluid viscosity that can be measured using diffusing particles is [46],
(18)
where r 2 is the MSD corrected for dynamic error, n is the exponent from Eq. (7), and = t is a dimensionless quantity normalized by the exposure time. During image acquisition of a particle diffusing in a viscous fluid, the characteristic relaxation time for the diffusing particle is often less than the interval between images. If particle dynamics slow down in comparison to the camera exposure time, the particle will appear to blur on the image [44,43]. This effect leads to an overestimate for measured MSD of diffusing particles [45]. In other words, diffusing particles will appear to diffuse faster (i.e., super-diffusion) than they act are. Decreasing the exposure time helps with this issue, but this also reduces the amount of available light, which makes it difficult to track the particle. The model for MSD presented in Eq. (18) helps correct for the effect of a finite exposure time [41].
p
= (1.4 × 10 3µm)
(19)
Fig. 2. (A) The static error measured on the portable (circles) and scientific grade (triangles) microscopes as a function of exposure time. The bars represent the standard deviation calculated from at least 10 measurements. (B) The maximum viscosity measurable on the portable (solid line) and scientific grade (dashed lines) microscopes calculated using Eqs. (19)–(21).
33
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S. Shabaniverki, et al.
Fig. 3. The two-dimensional mean square displacement corrected for static error obtained on portable (circles) and scientific grade (triangles) microscopes. The measurements were conducted in 5 wt% (A and B) and 10 wt% (C and D) aqueous polyvinylpyrrolidone solutions. The bars represent the standard deviation calculated from at least 10 measurements.
data in Fig. 3B. The 10 wt% PVP measurement on the scientific grade microscope (Fig. 3C) exhibits increasing standard deviation with lag time, which is consistent with the 5 wt% observations from Fig. 3A. However, the error appears to be smaller at lower lag times in comparison to the portable microscope viscometer due to the low static error observed in the scientific grade microscope.
We also compare the percent difference for the values of m and n obtained using both the scientific grade microscope and portable microscope viscometer. The value for m appears to be insensitive to the platform used to perform the measurement. The largest percent difference between the two m values occurs at 10 wt%, before dynamic error is accounted for. The power factor, n, appears to be the variable that is most sensitive to the type of platform used. We find that n is consistently higher when measured on the portable system. The difference between measured n values becomes less apparent when dynamic error is accounted for. The power factor, n, was also critical in determining the possible effects of migration due to sample tilting or convection. Migration is a form of directed motion which, with diffusion, has a characteristic MSD of [49],
4.3. Measurement of viscosity The parameters, m and n, for the power law fluid viscosity model can be obtained by fitting the MSD data to either Eq. (7) or Eq. (18). The parameters for Eq. (7) are obtained using the curve fitting algorithms available in Sigma Plot. Eq. (18) is computationally more expensive to solve for, but it represents a critical correction due to camera distortion. In order to do this, we adapted the non-linear solver algorithm available on Microsoft Excel [47] to find the parameters for Eq. (18). The fits to both equations were adjusted to achieve a coefficient of determination of r2 = 0.99 or better. The parameters obtained by fitting our data to these equations are shown in Table 2 for the 5 wt% PVP solution and Table 3 for the 10 wt% PVP solution. When dynamic error is accounted for, we observe that the parameter, m, does not appear to change significantly at 5 wt%. The value of m is larger at 10 wt%, indicating that this solution is more viscous. The correlation between increasing polymer concentration and m value has been observed in other aqueous polymer solutions [48]. When we account for dynamic error at 10 wt%, we find that the change in m appears to be more pronounced when compared to 5 wt%, indicating that dynamic error is more significant at higher concentrations. For both cases, we observe a slight increase in the value for n when dynamic error is accounted for.
r 2 = 4D t + (Vt )2 where V is the average migration velocity of the particle. If migration were present in our sample, we would expect n to scale with a value of 2. However, our measurements indicate that n has a value that is consistently close to 1. This indicates that migration caused by sedimentation driven by sample tilting or convection driven by temperature or airflow changes are negligible in our system. Our analysis also illustrates that complex fluids will appear to exhibit a lower viscosity than expected, based on the power factor n, before accounting for dynamic error. This is because the distortion caused by dynamic error will cause the particle to exhibit super diffusive behavior. In this case, the particle will appear to diffuse at a faster rate. This leads to a situation where the particle will appear to be diffusing in a medium of lower viscosity. This error is particularly evident if Eq. (7) is used to find the viscosity of the fluid. Accounting for
Table 2 Fitting parameters found by fitting data from the 5 wt% PVP data to Eq. (7) and Eq. (8). The percent difference compares m and n values found by the two microscopes. 5 wt% PVP Without Dynamic Error (Eq. (7)) 2
Scientific Grade Microscope Portable Microscope Percent difference comparison
n
With Dynamic Error (Eq. (18)) n
D* (μm /s )
m (mPa-s )
n
D* (μm2/sn)
m (mPa-sn)
n
0.089 0.084 –
3.429 3.619 5.41
1.088 1.121 2.97
0.081 0.077 –
3.774 3.959 4.79
1.136 1.169 2.82
34
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Table 3 Fitting parameters found by fitting data from the 10 wt% PVP data to Eq. (7) and Eq. (8). The percent difference compares m and n values found by the two microscopes. 10 wt% PVP Without Dynamic Error (Eq. (7))
Scientific Grade Microscope System Portable Microscope System Percent difference comparison
With Dynamic Error (Eq. (18))
D* (μm2/sn)
m (mPa-sn)
n
D* (μm2/sn)
m (mPa-sn)
n
0.0406 0.0383 –
7.4970 7.9469 5.83
1.0958 1.2266 11.26
0.0342 0.0357 –
8.9184 8.5264 4.49
1.180 1.261 6.64
dynamic error by using Eq. (18) as a model for particle MSD leads to a reduction in measured viscosity and it allows for the values measured on the portable microfluidic viscometer to closely match those measured on a scientific grade system.
In terms of sensitivity, the portable microscope measured the complex modulus to be 0.23 mPa for the 5 wt% PVP solution and 0.36 mPa for the 10 wt% PVP solution, both measured at a shear rate of 0.066 Hz. The scientific grade microscope measured a similar complex modulus (0.24 mPa) at 5 wt%, but found the complex modulus to be 0.49 mPa at 10 wt%. For comparison, both of these results indicate that the portable microscope has a high degree of sensitivity to sub-millipascal complex moduli, whereas macroscale rheology experiments using rheometers or viscometers are limited to 1 mPa at similar shear rates [16]. As the fluid viscosity increases, we expect that the portable microscope viscometer measurements will deviate from comparable measurements made on a scientific grade microscope due to the static error limitations discussed in Section 4.1.
4.4. Complex modulus In order to connect the power law viscosity parameters to the expected mechanical response of the fluid, we substitute Eqs. (6) and (7) into Eq. (15) to derive the complex modulus for a power law fluid. This expression is,
|G | =
3m n 2 (1 + n)
(22)
where α from Eq. (16) is equal to the power factor, n, from the power law model. Eq. (22) is plotted in Fig. 4 for both the 5 wt% PVP and 10 wt% PVP solutions. To plot this curve, we use the parameters obtained from the portable microscope data after correction for dynamic error. Eq. (22) provides a simple way to calculate the complex modulus using parameters obtained from a MSD analysis of particles diffusing in a power law fluid. The value obtained for the curve representing Eq. (22) is compared to complex moduli calculated with Eqs. (13)–(16) using ensemble average MSD data for 5 wt% PVP and 10 wt% PVP solutions as inputs for MSD. The MSD data for both the scientific grade and portable microscopes are initially corrected for dynamic error using Eq. (18) before calculating the complex modulus. The results shown in Fig. 4 confirms the analysis of the data in Tables 2 and 3, indicating that both methods produce comparable values after dynamic error is accounted for. A coefficient of determination is obtained by comparing the curve representing Eq. (22) to the data from both microscopes. The coefficient of determination shows a strong correlation between Eq. (22) and the data for both solutions at all shear rate ranges between 0.066 Hz and 33 Hz.
5. Conclusion In this article, we demonstrated the development of a portable microscope viscometer for complex fluids. This platform is based on optical video microscopy measurements of MSD of colloidal particles dispersed in a model complex fluid. These measurements enable the simultaneous measurement of viscosity and rheology of the fluid. This work represents the first that microrheology has been performed in a portable microscopy format. Our results are validated by performing the same measurements on a scientific grade microscope. As part of this validation, we demonstrated the critical importance of accounting for sources of static and dynamic error for interpreting the results from our portable microscope system. The maximum viscosity measurable by this technique is limited by the static error of the system. The blurring effect caused by discrete exposure time leads to dynamic error in our samples. Our MSD results show that the power law fluid pre-factor, m, is insensitive at low polymer concentrations, but begins to deviate at higher concentrations due to the slowing of particle diffusion. The power factor, n, is insensitive to polymer concentration for the range of data explored in this work. Correcting our data for dynamic error Fig. 4. The complex modulus for (A) 5 wt% PVP and (B) 10 wt% PVP solutions is calculated using the average mean square displacement shown in Fig. 3 for both the portable (circles) and scientific grade system (triangles). For comparison, the dynamically corrected values of m and n found by the portable microscope are used as inputs into Eq. (22) for comparison. The coefficient of determination was evaluated by comparing both data sets to Eq. (22).
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increases the apparent viscosity measured by our system. This correction also has the effect of improving the agreement between the scientific grade microscope and portable microscope viscometer data. The MSD data was used to calculate the complex modulus over a frequency range of 0.066–33 Hz down to a value of approximately 0.23 mPa. We found close agreement between the complex moduli calculated using the data collected on both microscopes. A simple model of the complex modulus strongly correlates with the data when parameters from MSD measurement are used. Overall, this work demonstrates that portable microscopes are capable of the same degree of accuracy that scientific grade microscopes for microrheology at low-viscosity. As the viscosity of complex fluid increases, the portable microscope measurements are expected to deviate from comparable measurements made on a scientific grade microscope due to the static error limitations. The research presented in this article serves as a guide for quantitatively interpreting viscosity data obtained by portable microscopy platforms such as smartphones or endoscopes. Our future work aims to explore using the portable microscope as a platform for the measurement of particlesurface forces or in-situ viscosity measurements of plant-based complex fluids or biological fluids.
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Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgements JJJ and AAV acknowledge financial support from the United States National Science Foundation (award number 1560138) through the Multiscale Sensing Actuation and Imaging (MoSAIc) research experience for undergraduates program in the Department of Mechanical Engineering at Iowa State University. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.expthermflusci.2019.05.009. References [1] D.N. Breslauer, R.N. Maamari, N.A. Switz, W.A. Lam, D.A. Fletcher, Mobile phone based clinical microscopy for global health applications, PLOS One 4 (2009) e6320, https://doi.org/10.1371/journal.pone.0006320. [2] H. Zhu, S. Mavandadi, A.F. Coskun, O. Yaglidere, A. Ozcan, Optofluidic fluorescent imaging cytometry on a cell phone, Anal. Chem. 83 (2011) 6641–6647, https://doi. org/10.1021/ac201587a. [3] D.A. Stoney, C. Neumann, P.L. Stoney, Discrimination and classification among common items of evidence using particle combination profiles, Forensic Sci. Int. 289 (2018) 92–107, https://doi.org/10.1016/j.forsciint.2018.05.024. [4] S.K.J. Ludwig, H. Zhu, S. Phillips, A. Shiledar, S. Feng, D. Tseng, L.A. van Ginkel, M.W.F. Nielen, A. Ozcan, Cellphone-based detection platform for rbST biomarker analysis in milk extracts using a microsphere fluorescence immunoassay, Anal. Bioanal. Chem. 406 (2014) 6857–6866, https://doi.org/10.1007/s00216-0147984-4. [5] S. Kim, K.C. Kim, E. Yeom, Microfluidic method for measuring viscosity using images from smartphone, Opt. Lasers Eng. 104 (2018) 237–243, https://doi.org/10. 1016/j.optlaseng.2017.05.016. [6] K. İçöz, Image processing and cell phone microscopy to analyze the immunomagnetic beads on micro-contact printed gratings, Appl. Sci. 6 (2016) 279, https://doi.org/10.3390/app6100279. [7] C.F. Zukoski, Particles and suspensions in chemical engineering: Accomplishments and prospects, Chem. Eng. Sci. 50 (1995) 4073–4079. [8] J.D. Martin, J.N. Marhefka, K.B. Migler, S.D. Hudson, Interfacial rheology through microfluidics, Adv. Mater. 23 (2011) 426–432, https://doi.org/10.1002/adma. 201001758. [9] J.M. Perkel, Rheometry Buyers’ Guide, Rheometry Buy. Guide. (n.d.). http://www. labcompare.com/10-Featured-Articles/36766-Rheometry-Buyers-Guide/ (accessed September 29, 2018). [10] B. Hallmark, M. Bryan, E. Bosson, S. Butler, T. Hoier, O. Magens, N. Pistre, L. Pratt, B.A. Ward, S. Wibberley, D.I. Wilson, A portable and affordable extensional rheometer for field testing, Meas. Sci. Technol. 27 (2016) 125302, https://doi.org/10. 1088/0957-0233/27/12/125302. [11] J. Dinic, L.N. Jimenez, V. Sharma, Pinch-off dynamics and dripping-onto-substrate (DoS) rheometry of complex fluids, Lab. Chip. 17 (2017) 460–473, https://doi.org/ 10.1039/C6LC01155A.
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37
Contents lists available at ScienceDirect
Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Portable imaging viscometry for quantitative complex fluid measurements a
a
Soheila Shabaniverki , Antonio Alvarez-Valdivia , Jaime J. Juárez a b
a,b,⁎
T
Department of Mechanical Engineering, Iowa State University, 2529 Union Drive, Ames, IA 50011, United States Center for Multiphase Flow Research and Education, Iowa State University, 2519 Union Drive, Ames, IA 50011, United States
ABSTRACT
Portable optical microscopy systems offer a pathway for serving as analytical platforms that characterize the properties of complex fluids. Measuring the diffusion characteristics of particles dispersed in complex fluids, known as microrheology, is one possible microscopy-based approach. In this study, we are the first to demonstrate that microrheology can be used to analyze diffusion profiles measured in a portable microscopy viscometer constructed from off-the-shelf parts. We perform measurements in two polyvinylpyrrolidone (PVP) solutions with 5 wt% and 10 wt% concentrations to demonstrate how this platform can be utilized as a viscometer for low-viscosity solutions. Our experiments consist of tracking the trajectory of diffusing colloidal particles dispersed in the PVP solutions and measuring their mean square displacement (MSD). The microrheological analysis used to characterize the MSDs are used to calculate the complex modulus over a frequency range of 0.07–33 Hz. The portable microscope viscometer measured a complex modulus of 0.23 mPa for the 5 wt% PVP solution and 0.36 mPa for the 10 wt% PVP solution. We compared our measurements to a scientific grade microscopy system, where the measured complex moduli were 0.24 mPa and 0.49 mPa at 5 wt% and 10 wt%, respectively. Our analysis of the results showed the need to account for sources of dynamic error caused by discrete sampling at limited exposure time. The viscosity of each sample appears to be lower in comparison to the case where dynamic error is accounted for. The static error for the portable microscope viscometer is measured to be 10–100 nm, which is an order of magnitude larger than the static error measured on our scientific grade microscope. Despite the larger static error, the portable microscope viscometer measured the viscosity of our samples to be within 4–6% of comparable measurements performed on the scientific grade microscope. Overall, we demonstrate that the portable microscope viscometer is capable of many of the same measurements as the scientific grade microscopes, which has implications for a variety of portable (e.g., smartphone) image-based microscopy experiments such as flow cytometry, viscometry and forensic analysis.
1. Introduction Portable microscopy systems serve as a platforms for low-cost health care monitoring [1], imaging flow cytometry [2], and forensic analysis of trace evidence [3]. Establishing portable microscopy as a viable technique for obtaining quantitative analytical data can open the door for in-situ measurements that enable food quality monitoring [4], fluid viscosity measurements [5], or perform experiments on surface coatings [6]. A common challenge that ties many of these applications together is the need to characterize the properties of complex fluids exhibiting a non-Newtonian response. Complex fluids are estimated to comprise up to 60% of the materials handled by the chemical process industry [7] and there is a need to perform online viscosity measurements to characterize the complex fluids produced by these processes [8]. Instruments for measuring the properties of complex fluids can cost $20,000 for a high-end viscometer and $200,000 for a research-grade rheometer [9]. In an effort to make viscometers and rheometers more affordable, instrumentation for measuring complex fluid viscosity is being developed using off-the-shelf components. These portable complex fluid measurement instruments rely on optical video cameras to capture pinching events when the complex fluid is pulled apart [10] or allowed to drip [11]. ⁎
Another method for measuring complex fluid properties relies on monitoring the transport of a complex fluid in a microfluidic channel [12]. In this format, a microfabricated channel can be combined with integrated pressure sensors [13] or microparticle image velocimetry [14] to perform complex fluid measurements. Microfluidic viscometers offer several advantages over other portable formats in that only a small sample volume is necessary for measurements and they overcome the torque limitations of viscometers and rheometers, enabling measurement of low-viscosity fluids [15]. A limitation of microfluidic viscometers is the need for access to cleanroom facilities for fabricating these devices. Furthermore, many of the microfluidic viscometer demonstrations utilize scientific grade microscopes imaging, which ultimately increases the cost of the viscometry platform. As an alternative to microfluidic viscometry, this study examines the feasibility of using a portable microscope as a viscometer platform. This platform tracks the mean square displacement (MSD) of particles dispersed in a complex medium using optical video microscopy and extracts the viscosity of the fluid based on a microrheological analysis of the data. This study represents the first time that low-viscosity microrheology has been performed in a portable microscope viscometer platform. As a model fluid for this study, we utilize 5 wt% and 10 wt% polyvinylpyrrolidone (PVP) as a test complex fluid solution. Our
Corresponding author at: Department of Mechanical Engineering, Iowa State University, 2529 Union Drive, Ames, IA 50011, United States. E-mail address: [email protected] (J.J. Juárez).
https://doi.org/10.1016/j.expthermflusci.2019.05.009 Received 29 December 2018; Received in revised form 7 May 2019; Accepted 16 May 2019 Available online 17 May 2019 0894-1777/ © 2019 Elsevier Inc. All rights reserved.
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Nomenclature a B C Do D D E f G G’ G” G g h
havg kB MSD m NA n
PVP r2 2 rmeas r2 T V Γ εm εo
particle radius electrostatic interaction magnitude electrolyte concentration diffusion coefficient average hindered diffusion coefficient particle diameter charge of an electron hydrodynamic hinderance buoyant mass storage modulus loss modulus complex modulus gravitational acceleration distance between the particle center and the boundary surface average height Boltzmann’s constant mean square displacement pre-factor fitting parameter Avogadro’s number power fitting parameter
p
s
η max
κ−1 ρp ρm σ Ψ1 Ψ2 ω
polyvinylpyrrolidone actual mean square displacement measured mean square displacement mean square displacement corrected for dynamic error ambient temperature migration velocity gamma function dielectric constant of the medium permittivity of free space static error of portable microscope static error of scientific grade microscope viscosity of the medium maximum measurable viscosity Debye length particle density medium density exposure time of the camera dimensionless lag time surface potential of the particle surface potential of the boundary Local shear rate
fluorescent polystyrene beads (ThermoFisher Scientific, F8852) with peak excitation and emission spectra of 505 nm and 515 nm, respectively. One microliter of fluorescent bead stock solution is taken directly from the manufacturer’s container and stirred into the PVP solution for 10 min. The dispersion consisting of particles and PVP was poured into a petri dish, which was placed in 4 °C refrigerator for 3 h. This gave the sample time to cool and sediment. When the sample was placed on the microscope stage, we waited for a 15 min period to give the dispersion an opportunity to settle before the experiment. For each sample, we performed same-day observations of 10 different particles in each sample.
microrheology experiments cover a frequency range of 0.07–33 Hz and measure complex moduli down to approximately 0.23 mPa, which is nearly one order of magnitude more sensitive than a commercial rheometer [16]. The portable microscope viscometer presented in this paper represents an alternative to commercial rheometers for obtaining quantitative data of low-viscosity fluid properties. Furthermore, where previous studies have focused on quantifying single, static images using portable optical microscopy systems [17–19], we demonstrate that capturing video with these types of microscopy platforms allows for the measurement of dynamic data comparable to scientific grade microscopy systems. The portable microscope viscometer described here could serve as a platform that introduces microrheology as a tool for quantifying the viscosity and rheology of industrial fluids. For example, the portable microscope viscometer could be a tool for rapid inline monitoring of complex fluids, which is used to assess quality and control processing of petroleum products [20], pharmaceutical production [21], and food production [22]. This platform could also have an impact on the industrial use of nanofluids. At present, the measurement of nanofluid properties at temperatures and concentrations encountered in industry is challenging due to the high cost of quantitative instruments [23]. With improved optics [24] and particle tracking tools [25], the portable microscope viscometer could be used to measure nanofluid properties. This platform would provide the experimental basis for developing models [26,27] that guide the development of complex fluids for industrial applications.
2.2. Microscopy Microrheology experiments were initially performed using a scientific grade research microscope (Olympus, IX70) and a scientific CMOS camera (QImaging, Optimos). The petri dish is placed on a sample stage where it is illuminated by an LED-based fluorescence module (wLS, QImaging) with a FITC filter set. All experiments on the scientific grade microscope are performed with a 40x Olympus long working-distance objective lens. Video tracking data for single particle trajectories, with an image size of 100 × 100 pixels (22 μm × 22 μm), are captured using an open source microscopy software (μManager). A thousand frames of data are recorded for each experiment at a frame rate of 30 frames/s and exposure time of 30 ms. To obtain sufficient statistics to measure viscosity, we performed at least ten different observations of each sample and the resulting video files are analyzed using the particle tracking algorithm developed by Crocker and Grier [28] written in MATLAB. The experiments performed on the scientific grade microscope were designed to be compared to similar experiments performed on the portable microscope viscometer. The supplementary information contains additional details on the off-the-shelf materials used to fabricate this platform. Imaging of particle trajectories is done using a machine vision-grade CMOS camera (FLIR Machine Vision, BFLY-U3-23S6C-C). The camera is attached to a 1-inch diameter lens tube (Thorlabs, SM1L30) using a C-mount adapter (Thorlabs, SM1A9). The camera and lens tube assembly are attached to a 40x Lecia objective lens (Leica, Part Number 506149) using an M25 thread adapter (Thorlabs, SM1A12). The protocol for capturing images using this camera is
2. Materials and methods 2.1. Sample preparation Twenty milliliters of deionized water, sourced from an ARIES High Purity Water System (Aries Filterworks) with a 0.2 μm filter, are added to a beaker with a stir bar. The beaker is placed on a stirring hot plate and heated to a temperature of 50 °C. An aqueous solution of polyvinylpyrrolidone (PVP), is prepared as a model, low-viscosity complex fluid. The PVP, with a molecular weight of 40 kDa, is sourced from Alfa Aesar (product number J62417). Two different amounts of PVP powder (1 gr and 2 gr) are used to create either a 5 wt% or 10 wt% aqueous solutions. To these solutions, we add a dispersion of 1 µm diameter 30
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similar to the protocol followed for the microrheology experiments done on the Olympus microscope. The only difference is that we used 150 × 150 pixel (20 μm × 20 μm) to better match the physical size of the images due to differences in the size of the CMOS sensors on the cameras. The lens tube assembly is mounted to a base fabricated from acrylic (OPTIX, 1AG2123A) and attached to a solid wooden base using long bolts and washers (Fig. 1). The sample stage is also fabricated from acrylic and attached to two stepper motors with ‘screw and nut’ sliders to make it easier to focus. The stepper motors are regulated by a stepper motor driver controlled by a A3967 EasyDriver integrated circuit. The driver is controlled using an Arduino programmed using the opensource Arduino IDE software. A leveling tool was used to help position the stage at an even height to prevent particle migration due to sample tilting.
where the double bar symbol indicates that this quantity is related to diffusion parallel to a boundary, h is distance between the particle center and the boundary surface, and the function, f , represents the hydrodynamic hinderance to diffusion near a boundary. This function is given as [30],
f (h a) = 1
D
The diffusion coefficient for a single particle far from any boundary is given by the Stokes-Einstein equation [29], (1) −23
J/K is Boltzmann’s constant, T is the ambient where kB = 1.38 × 10 temperature of the experiment, η is the viscosity of the medium and a is the particle radius. When the particle approaches a boundary, the diffusion coefficient of the particle as it travels parallel to the boundary is hindered,
D = Do f (h a)
45 a 256 h
4
1 a 16 h
5
(3)
(4)
= Do f
where the average hinderance function is evaluated at an average height located at the position of mechanical equilibrium. The average diffusion coefficient for a particle is measured using the mean square displacement (MSD) of a trajectory obtained through image analysis of optical video microscopy data. The average MSD for a particle diffusing in two dimensions is expressed as, r 2 = 4Do f . In a complex fluid, we expect that the diffusion coefficient of the particle will depend on the local shear rate, ω, induced by the particle as it fluctuates due to Brownian motion. If the viscosity is a power law fluid [32], = m n 1, where m and n are obtained through fitting the model to viscometry data, then the MSD measured in experiment is expressed as,
3.1. Diffusion in complex fluids near a wall
kB T 6 a
3
As the particle diffuses over the surface, it samples the potential energy well that represents the net energy interaction between the particle and the surface. The diffusion coefficient measured by optical video microscopy represents the net effect of the potential energy on the diffusing particle. We can define the average diffusion coefficient as [31],
3. Theory
Do =
9 a 1 a + 16 h 8 h
r2 = 4
(2)
kB T 6 a (m n 1)
f t
(5)
Fig. 1. A schematic of the portable microscope system used to measure the viscosity of PVP solutions with diffusing colloidal probes. The videos are captured through an objective lens using a machine vision CMOS camera. Details on construction may be found in the supplementary information. 31
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where the expression for the power law fluid is substituted for viscosity. 1 t for microrheological The shear rate is approximated as measurements in non-Newtonian media [33]. Using this approximation for shear rate, the MSD for a particle diffusing in a power law fluid near a wall is,
expected power law viscosity of a complex fluid. The MSD measured using trajectory data can also be used to characterize rheological parameters such as storage (G′) and loss (G″) moduli, which are critical for understanding the dynamic mechanical response of the fluid [37]. This is done using an approach known as microrheology [38,39] where the diffusion induced by Brownian motion causes the particle to locally shear the fluid. The storage and loss moduli of the bulk fluid are expressed as [33],
(6)
r 2 = 4D t n For simplicity, the parameter, D , is,
D = kB T f
(7)
6 am
Eq. (6) is similar to established models of MSD for particles diffusing in complex fluid media [34]. Eq. (6) can be used to fit MSD data to obtain some rheological information about the medium. The exponent, n, found in Eq. (6) is directly related to the power law exponent for the fluid. Eq. (7) is coupled directly to hydrodynamics of the diffusing particle and the power law pre-factor, m. Eq. (6) represents one possible model which can be used to fit the MSD data for a particle diffusing in a power law fluid. However, this expression does not account for possible sources of error. In Section 4, we elucidate the effect that errors have on MSD data and compare extracting viscosity data using Eq. (6) to a model that incorporates error.
B G
B = 64 G=
m oa
4 3 a ( 3
p
kB T e m)
2
tanh
e 1 e 2 tanh 4kB T 4kB T
6 aD
a
)
[1 +
(15)
( )]
)=
dln r 2 (1 d lnt
)
(16)
Image analysis is subject to measurement noise, which arise from static effects like mechanical vibrations in the microscope stage or fluctuations in light intensity [41]. The experimental noise is additive and it influences the measured MSD by increasing its apparent magnitude. The measured and actual MSDs are related to each other by, 2 rmeas = r2 +
(17)
2
where ε is the static error that arises from experiments and is the actual MSD of the particle. Measuring the static error in our portable system can yield insight into the limitations of our portable system. To do this, we utilize a procedure outlined in our previous work [42] where we image 1 µm polystyrene beads immobilized on a glass substrate. In this procedure, the static error represents the value of the MSD measured at a lag time equal to the exposure time of the camera, t = σ. This value, ε2, is then subtracted from the experimentally measured MSD to find the actual MSD. The finite exposure time of the experimental imaging system can also cause a distortion effect that leads to errors in localizing the center of the diffusing particle [43]. This distortion requires a second correction, which is found by comparing the data to a model of the expected
r2
(10) (11)
where κ is the Debye length, e is the charge of an electron, NA is Avogadro’s number, C is the electrolyte concentration, εm is the dielectric constant of the medium and εo is the permittivity of free space. The parameter, B, represents the electrostatic interaction of the system, which depends on the surface potential of the particle (Ψ1) and the boundary (Ψ2). The buoyant mass, G, depends on particle volume and the difference between particle density (ρp) and medium density (ρm). The parameters in Table 1 are substituted into Eqs. (9)–(11) to estimate the average height sampled by particle based on Eq. (8). The resulting height obtained by evaluating Eq. (8) is substituted into Eq. (3) to find the average hinderance factor as, f = f (havg a) . The value of f is used to decouple D in Eq. (7) to estimate the power law pre-factor as,
kB T f
kB T f r 2 (1
3.4. Sources of experimental error
−1
m=
(14)
In the low-shear limit, the Cox-Merz rule gives the viscosity of the fluid as = G" ( ) [40]. The average hinderance factor is included in Eq. (15) to correct for the presence of a wall.
(9)
m o kB T
( ) 2)
(1
where havg represents the average height sampled by the diffusing particle when the electrostatic energy interaction between the particle and the boundary is in mechanical equilibrium with gravity. The parameters, κ, B and G are expressed as [36],
=
G" ( ) = |G ( )| sin (
where < r > is the MSD evaluated at a value of t = 1/ω, ω is the local shear rate of the diffusing particle, Γ is a gamma function approximated by Γ(1 + α) ≈ 0.457(1 + α)2 − 1.36(1 + α) + 1.90, and the parameter α is given by,
(8)
2e 2NA C
(13)
2
In order to decouple D from the hydrodynamics of the particle diffusing near a wall, we estimate the average value of the hydrodynamic hinderance factor. This is done by evaluating Eq. (3) using the average expected height of the diffusing particle in dynamic equilibrium with its environment. The average height is estimated as [35], 1ln
( ) 2)
|G ( )| =
3.2. Estimating hydrodynamic hinderance
havg = a +
G' ( ) = |G ( )| cos (
Table 1 Parameters used to calculate the average height sampled by the particle using Eq. (8). The resulting height is substituted into Eq. (3) to calculate the average hinderance function, f .
(12)
where D is obtained by fitting MSD data to a model for expected particle dynamics in a power law fluid. 3.3. Microrheology The analysis discussed above can quickly yield insight into the 32
Parameter
Equation
Value
T (K) e/10−19 (C) NA/1023 (1/mol) C (mM) κ −1 (nm) εm a (nm) Ψ1 (mV) Ψ2 (mV) ρp (kg/m3) ρm (kg/m3) f
1, 6, 7, 9, 10, 12, 15, 21 9, 10 9 9 8, 9 9, 10 1–3, 6–8, 10–12, 15, 21 10 10 11 11 5–7, 12, 15
298 1.6 6.02 0.1 30 80 500 −50 −50 1055 999 0.698
Experimental Thermal and Fluid Science 107 (2019) 29–37
S. Shabaniverki, et al.
particle dynamics in a complex fluid [41],
r2 ( + 1)2 + n + ( 1)2 + n 2 = 4D (1 + n)(2 + n)
2+n
2
s
max
4.1. Effect of exposure time Fig. 2A shows a comparison of measured static error as a function of camera exposure time for both the scientific grade microscope and the portable microscope viscometer. The average value of static error is larger for the portable microscope viscometer in comparison to the scientific grade microscope, which is likely the result of minimal antivibration equipment on the portable microscope combined with a noisier camera sensor. We also observe that the static error decreases with increasing exposure time. Increasing the exposure time allows for more light to fall on the CMOS sensor, making it easier to localize particles. Our measurements indicate that the static error for the portable system is highly repeatable, but we also observed large standard deviations associated with the data collected from the scientific grade microscope. The magnitude of the static error on the scientific grade microscope is consistently smaller than the portable microscope. This is because the scientific grade microscope is mounted on an anti-vibration table and the camera used for imaging is Peltier cooled, which reduces the overall noise from this microscope. Using the data from Fig. 2A, we can estimate the relationship between static error and exposure time by curve fitting the data from the portable microscope viscometer and the scientific grade microscope,
2kB T 3 a 2
(21)
4.2. Mean square displacement measurements Fig. 3 shows the average MSD as a function of lag time obtained from optical microscopy measurements done on the scientific grade microscope (top row) and portable microscope viscometer (bottom row). The experiments for each data set were obtained in the same way by using similar exposure times (∼30 ms) as described in the methods section. The trends shown in both data sets at 5 wt% PVP (Fig. 3A and 3B) appear to be comparable with each other, with the standard deviation increasing with time for both systems. However, the error appears to increase to a greater degree for the portable system at higher lag times when compared to the scientific grade microscope. The data for 10 wt% PVP in Fig. 3D shows that the portable microscope viscometer exhibits a larger standard deviations at lower lag times than the
0.619
ms
(20)
ms
where ε2 is the square of the relationship between static error and exposure time from Eqs. (19) and (20). This parameter can also be experimentally measured using the protocol described above. Fluid viscosities that exceed this limit cannot be passively measured by diffusing particles because their motion would be indistinguishable from static error. In Fig. 2B, we compare the maximum accessible viscosity for both the portable microscope and the scientific grade system by substituting Eqs. (19) and (20) into Eq. (21). This analysis shows that the scientific grade system, with its lower static error, can theoretically measure higher viscosity fluids. In order for the results to be comparable between the portable system and the scientific grade system, the viscosity of the fluid cannot exceed the value that is measurable on the portable system. This analysis serves as a guide for determining the limitations of viscosity measurements with the portable microscope viscometer.
4. Results and discussion
= (0.08µm)
0.973
where p and s stand for portable microscope viscometer and scientific grade microscope, respectively. It is evident that the static error decreases with increasing exposure time. This is expected since increasing exposure time allows for more light to hit the CMOS sensor, which aids in localizing particle centers. The static error also places practical limits on the measureable viscosity of a fluid using microrheology. As the viscosity of a fluid increases, the diffusion coefficient of the particle will decrease. This has the effect of slowing the particle dynamics to the point where the displacement observed by the camera could not be distinguishable from camera noise. The upper limit for fluid viscosity that can be measured using diffusing particles is [46],
(18)
where r 2 is the MSD corrected for dynamic error, n is the exponent from Eq. (7), and = t is a dimensionless quantity normalized by the exposure time. During image acquisition of a particle diffusing in a viscous fluid, the characteristic relaxation time for the diffusing particle is often less than the interval between images. If particle dynamics slow down in comparison to the camera exposure time, the particle will appear to blur on the image [44,43]. This effect leads to an overestimate for measured MSD of diffusing particles [45]. In other words, diffusing particles will appear to diffuse faster (i.e., super-diffusion) than they act are. Decreasing the exposure time helps with this issue, but this also reduces the amount of available light, which makes it difficult to track the particle. The model for MSD presented in Eq. (18) helps correct for the effect of a finite exposure time [41].
p
= (1.4 × 10 3µm)
(19)
Fig. 2. (A) The static error measured on the portable (circles) and scientific grade (triangles) microscopes as a function of exposure time. The bars represent the standard deviation calculated from at least 10 measurements. (B) The maximum viscosity measurable on the portable (solid line) and scientific grade (dashed lines) microscopes calculated using Eqs. (19)–(21).
33
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Fig. 3. The two-dimensional mean square displacement corrected for static error obtained on portable (circles) and scientific grade (triangles) microscopes. The measurements were conducted in 5 wt% (A and B) and 10 wt% (C and D) aqueous polyvinylpyrrolidone solutions. The bars represent the standard deviation calculated from at least 10 measurements.
data in Fig. 3B. The 10 wt% PVP measurement on the scientific grade microscope (Fig. 3C) exhibits increasing standard deviation with lag time, which is consistent with the 5 wt% observations from Fig. 3A. However, the error appears to be smaller at lower lag times in comparison to the portable microscope viscometer due to the low static error observed in the scientific grade microscope.
We also compare the percent difference for the values of m and n obtained using both the scientific grade microscope and portable microscope viscometer. The value for m appears to be insensitive to the platform used to perform the measurement. The largest percent difference between the two m values occurs at 10 wt%, before dynamic error is accounted for. The power factor, n, appears to be the variable that is most sensitive to the type of platform used. We find that n is consistently higher when measured on the portable system. The difference between measured n values becomes less apparent when dynamic error is accounted for. The power factor, n, was also critical in determining the possible effects of migration due to sample tilting or convection. Migration is a form of directed motion which, with diffusion, has a characteristic MSD of [49],
4.3. Measurement of viscosity The parameters, m and n, for the power law fluid viscosity model can be obtained by fitting the MSD data to either Eq. (7) or Eq. (18). The parameters for Eq. (7) are obtained using the curve fitting algorithms available in Sigma Plot. Eq. (18) is computationally more expensive to solve for, but it represents a critical correction due to camera distortion. In order to do this, we adapted the non-linear solver algorithm available on Microsoft Excel [47] to find the parameters for Eq. (18). The fits to both equations were adjusted to achieve a coefficient of determination of r2 = 0.99 or better. The parameters obtained by fitting our data to these equations are shown in Table 2 for the 5 wt% PVP solution and Table 3 for the 10 wt% PVP solution. When dynamic error is accounted for, we observe that the parameter, m, does not appear to change significantly at 5 wt%. The value of m is larger at 10 wt%, indicating that this solution is more viscous. The correlation between increasing polymer concentration and m value has been observed in other aqueous polymer solutions [48]. When we account for dynamic error at 10 wt%, we find that the change in m appears to be more pronounced when compared to 5 wt%, indicating that dynamic error is more significant at higher concentrations. For both cases, we observe a slight increase in the value for n when dynamic error is accounted for.
r 2 = 4D t + (Vt )2 where V is the average migration velocity of the particle. If migration were present in our sample, we would expect n to scale with a value of 2. However, our measurements indicate that n has a value that is consistently close to 1. This indicates that migration caused by sedimentation driven by sample tilting or convection driven by temperature or airflow changes are negligible in our system. Our analysis also illustrates that complex fluids will appear to exhibit a lower viscosity than expected, based on the power factor n, before accounting for dynamic error. This is because the distortion caused by dynamic error will cause the particle to exhibit super diffusive behavior. In this case, the particle will appear to diffuse at a faster rate. This leads to a situation where the particle will appear to be diffusing in a medium of lower viscosity. This error is particularly evident if Eq. (7) is used to find the viscosity of the fluid. Accounting for
Table 2 Fitting parameters found by fitting data from the 5 wt% PVP data to Eq. (7) and Eq. (8). The percent difference compares m and n values found by the two microscopes. 5 wt% PVP Without Dynamic Error (Eq. (7)) 2
Scientific Grade Microscope Portable Microscope Percent difference comparison
n
With Dynamic Error (Eq. (18)) n
D* (μm /s )
m (mPa-s )
n
D* (μm2/sn)
m (mPa-sn)
n
0.089 0.084 –
3.429 3.619 5.41
1.088 1.121 2.97
0.081 0.077 –
3.774 3.959 4.79
1.136 1.169 2.82
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Table 3 Fitting parameters found by fitting data from the 10 wt% PVP data to Eq. (7) and Eq. (8). The percent difference compares m and n values found by the two microscopes. 10 wt% PVP Without Dynamic Error (Eq. (7))
Scientific Grade Microscope System Portable Microscope System Percent difference comparison
With Dynamic Error (Eq. (18))
D* (μm2/sn)
m (mPa-sn)
n
D* (μm2/sn)
m (mPa-sn)
n
0.0406 0.0383 –
7.4970 7.9469 5.83
1.0958 1.2266 11.26
0.0342 0.0357 –
8.9184 8.5264 4.49
1.180 1.261 6.64
dynamic error by using Eq. (18) as a model for particle MSD leads to a reduction in measured viscosity and it allows for the values measured on the portable microfluidic viscometer to closely match those measured on a scientific grade system.
In terms of sensitivity, the portable microscope measured the complex modulus to be 0.23 mPa for the 5 wt% PVP solution and 0.36 mPa for the 10 wt% PVP solution, both measured at a shear rate of 0.066 Hz. The scientific grade microscope measured a similar complex modulus (0.24 mPa) at 5 wt%, but found the complex modulus to be 0.49 mPa at 10 wt%. For comparison, both of these results indicate that the portable microscope has a high degree of sensitivity to sub-millipascal complex moduli, whereas macroscale rheology experiments using rheometers or viscometers are limited to 1 mPa at similar shear rates [16]. As the fluid viscosity increases, we expect that the portable microscope viscometer measurements will deviate from comparable measurements made on a scientific grade microscope due to the static error limitations discussed in Section 4.1.
4.4. Complex modulus In order to connect the power law viscosity parameters to the expected mechanical response of the fluid, we substitute Eqs. (6) and (7) into Eq. (15) to derive the complex modulus for a power law fluid. This expression is,
|G | =
3m n 2 (1 + n)
(22)
where α from Eq. (16) is equal to the power factor, n, from the power law model. Eq. (22) is plotted in Fig. 4 for both the 5 wt% PVP and 10 wt% PVP solutions. To plot this curve, we use the parameters obtained from the portable microscope data after correction for dynamic error. Eq. (22) provides a simple way to calculate the complex modulus using parameters obtained from a MSD analysis of particles diffusing in a power law fluid. The value obtained for the curve representing Eq. (22) is compared to complex moduli calculated with Eqs. (13)–(16) using ensemble average MSD data for 5 wt% PVP and 10 wt% PVP solutions as inputs for MSD. The MSD data for both the scientific grade and portable microscopes are initially corrected for dynamic error using Eq. (18) before calculating the complex modulus. The results shown in Fig. 4 confirms the analysis of the data in Tables 2 and 3, indicating that both methods produce comparable values after dynamic error is accounted for. A coefficient of determination is obtained by comparing the curve representing Eq. (22) to the data from both microscopes. The coefficient of determination shows a strong correlation between Eq. (22) and the data for both solutions at all shear rate ranges between 0.066 Hz and 33 Hz.
5. Conclusion In this article, we demonstrated the development of a portable microscope viscometer for complex fluids. This platform is based on optical video microscopy measurements of MSD of colloidal particles dispersed in a model complex fluid. These measurements enable the simultaneous measurement of viscosity and rheology of the fluid. This work represents the first that microrheology has been performed in a portable microscopy format. Our results are validated by performing the same measurements on a scientific grade microscope. As part of this validation, we demonstrated the critical importance of accounting for sources of static and dynamic error for interpreting the results from our portable microscope system. The maximum viscosity measurable by this technique is limited by the static error of the system. The blurring effect caused by discrete exposure time leads to dynamic error in our samples. Our MSD results show that the power law fluid pre-factor, m, is insensitive at low polymer concentrations, but begins to deviate at higher concentrations due to the slowing of particle diffusion. The power factor, n, is insensitive to polymer concentration for the range of data explored in this work. Correcting our data for dynamic error Fig. 4. The complex modulus for (A) 5 wt% PVP and (B) 10 wt% PVP solutions is calculated using the average mean square displacement shown in Fig. 3 for both the portable (circles) and scientific grade system (triangles). For comparison, the dynamically corrected values of m and n found by the portable microscope are used as inputs into Eq. (22) for comparison. The coefficient of determination was evaluated by comparing both data sets to Eq. (22).
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increases the apparent viscosity measured by our system. This correction also has the effect of improving the agreement between the scientific grade microscope and portable microscope viscometer data. The MSD data was used to calculate the complex modulus over a frequency range of 0.066–33 Hz down to a value of approximately 0.23 mPa. We found close agreement between the complex moduli calculated using the data collected on both microscopes. A simple model of the complex modulus strongly correlates with the data when parameters from MSD measurement are used. Overall, this work demonstrates that portable microscopes are capable of the same degree of accuracy that scientific grade microscopes for microrheology at low-viscosity. As the viscosity of complex fluid increases, the portable microscope measurements are expected to deviate from comparable measurements made on a scientific grade microscope due to the static error limitations. The research presented in this article serves as a guide for quantitatively interpreting viscosity data obtained by portable microscopy platforms such as smartphones or endoscopes. Our future work aims to explore using the portable microscope as a platform for the measurement of particlesurface forces or in-situ viscosity measurements of plant-based complex fluids or biological fluids.
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