Analysis of the magnetic rod interfacial stress rheometer

Interfacial rheometry of both Langmuir and Gibbs monolayers, composed of surfactants, proteins, particles and complex mixtures thereof, has gained considerable interest during the last decade. The rheological characterization of these materials at interfaces is stimulated by their use in industries ranging from food [Dickinson (1999)], pharmaceutical [Haynes and Norde (1994)] to biomedical [Wüstneck et al. (2005); Zasadzinski et al. (2005)], as well as biological applications. Interfacial rheological properties play an important role in determining the stability of high interphase systems such as emulsions and foams. For example, a relationship between the shear rheology of beta-lactoglobulin at a water-oil interface and the stability of protein stabilized oil-in-water emulsions was established [Dickinson et al. (1999)]. However, increasing the interfacial moduli does not always result in a better emulsion stability as cross-linking can lead to aggregation and a decreased stability [Damodoran and Anand (1997); Chen and Dickinson (1998)]. Human lung surfactant, a complex mixture of lipids, cholesterol, and proteins, is another example demonstrating the relevance of surface rheometry. This surfactant mixture lowers the alveolar surface tension to a few milli-Newton per meter and facilitates breathing. However, one of the lung surfactant specific proteins regulates the ratio of solid to fluid phase in the monolayer, affecting the rheological properties, thereby possibly protecting alveoli from collapse [Zasadzinski et al. (2005)].

The existence of surface viscosity was first suggested by the Belgian physicist J. Plateau [Plateau (1869)], who described the damping of a magnetic needle on a surfactant laden interface and proposed its use as a measurement technique. His experimental setup was, however, plagued by Marangoni stresses that complicate the measurement of the absolute interfacial stress [Marangoni (1871)]. Alternative techniques, including the deep channel surface viscometer [Mannheimer and Schechter (1970)] and knife-edge devices [Edwards et al. (1991); Ghaskadvi and Dennin (1998)], mainly focused on the (non-Newtonian) surface viscosity. Recently, tools have become available on standard rotational rheometers such as rotating disks [Miller et al. (1997)], rings and the bi-cone geometry [Enri et al. (2003)]. The original magnetic needle technique of Plateau has been adapted by Shahin (1986) and further improved by Brooks et al. [Brooks et al. (1999); Brooks (1999)], yielding a sensitive interfacial stress rheometer (ISR) [Brooks et al. (1999); Brooks (1999); Ding et al. (2002)]. Contrary to bulk rheometry, the fundamental sensitivity of interfacial rheometers is not only determined by the limits in applying stress and detecting deformations, but is also dependent on the coupling of both the measurement probe and monolayer with the surrounding bulk phases. In addition to the desired response of the interface, the presence of a drag force exerted by the bulk phases complicates the analysis. The measurement geometries must be such that they provide adequate sensitivity to detect stresses in the surface film, in the presence of bulk stresses in the adjacent fluids. The Boussinesq number is indicative of the relative importance of the surface and subphase contributions [Edwards et al. (1991)]:

where $ηs$ is the surface viscosity (units $Pasm$⁠) and $η$ is the bulk viscosity of the subphase, $V$ is a characteristic velocity, $LI$ and $LS$ are the characteristic length scales at which the velocity decays at the interface and in the subphase, respectively, $PI$ is the contact perimeter between the rheological probe and the interface, and $AS$ is the contact area between the probe and the subphase. The parameter $a$ has the units of length. For the disk, bi-cone or ring methods this parameter is related to the radius of the entire geometry (on the order of centimeters), for the magnetic needle it determined by the radius of the needle (on the order of millimeters and below). When $Bo≫1$⁠, the surface stresses dominate while when $Bo≪1$ the subphase stresses dominate. According to Eq. (1), a small value of $a$⁠, i.e., a minimal contact area per perimeter with the subphase, is desirable to intrinsically achieve a sensitive measurement device. From this perspective the needle rheometer has an advantage over these other techniques.

Recently, a theoretical analysis has been presented for the drag on a rod in an infinite interface in the Stokes limit [Fischer (2004a)]. Based on this analysis it was suggested that surface shear viscosity data, acquired with the needle viscometer at low Boussinesq numbers, should be reanalyzed [Fischer (2004a)], especially when the ratio between surface and bulk viscosity is smaller than the length of the needles. The conditions under which the surface drag force is linear in the velocity have indeed to be considered carefully [Alonso and Zasadzinski (2004)]. This is also the case for other surface rheometers. For both the disk and biconical interfacial viscometer for steady [Oh and Slattery (1978)] and oscillatory shear [Ray et al. (1987); Lee et al. (1991); Nagarajan et al. (1998)] the velocity profiles have been solved in the Stokes limit and data correction procedure have been suggested. In the work presented here, results from numerical calculations on model interfaces are presented for a magnetic needle rheometer in oscillatory mode, and for viscoelastic interfaces, to identify the conditions under which reliable measurements can be performed. The needle is confined in a channel, and the effect of a nonzero Reynolds number on the linearity of the velocity profiles is also found to be important. The goal of this work is hence to evaluate and also further optimize the measurement technique. The optimal design for the rheological probe is first discussed. The errors introduced by linearly subtracting the instrument contribution (inertia, compliance and drag) from the measured response will subsequently be evaluated using numerical simulations. The analysis will also be used to assess the correction procedure for conditions with a nonzero Reynolds number. Finally, conditions under which reliable measurements can be obtained are identified and the calculations are compared to experimental results.

Figure 1 shows the interfacial stress rheometer (ISR), according to the design of Brooks et al. [Brooks et al. (1999); Brooks (1999)]. A Langmuir trough (KSV minitrough from KSV Instruments, Helsinki, Finland) contains the sample, and enables control of the surface pressure, as measured with a Wilhelmy balance. When surface pressure control is not required, a smaller Petri dish may replace the Langmuir trough to reduce the sample volume. A magnetized rod with radius $a$ is supported by surface or interfacial tension at the interface and is positioned between two microscope slides arranged as a rectangular channel with a variable width $R$⁠. In the present work the distance $R$ was chosen so that the channel geometry, $λ=R∕a$⁠, is kept fixed. Two coils in Helmholtz condition are positioned around the trough and provide a potential well where the magnetic field gradient can be made zero by balancing the current though the coils. The current is controlled by two dc power supplies (Agilent Model 6644A), controlled with a function generator (Agilent model 33120A) interfaced with a PC through LABVIEW (National Instruments). The magnetic field applies a force on the magnetized rod to shear the interfacial film. It is assumed that the drag experienced by the rod predominantly arises from interfacial shear stresses developed between the glass slides and the rod. The position of the rod is detected by an inverted microscope (Nikon Eclipse TS100), which is focused on the end of the needle. The resulting image is projected onto a photodiode array thus defining the strain rate. By applying an oscillatory perturbation to the current in one of the coils, a net force can be generated on the needle. The force and position signals are digitized using data acquisition boards PCI 6023E and PCI 6034E (National Instruments). The ratio of the displacement amplitude $(z)$ to the forcing amplitude $(F)$⁠, as well as the phase difference $δ$ between these signals is determined from the frequency spectrum obtained by taking the fast Fourier transform of the two signals.

In the absence of a monolayer, it has been shown that the dynamics of the needle motion can be well described by a system [Brooks et al. (1999)]. The amplitude ratio $(z∕F)$ and phase angle $δ$ display a low-frequency plateau, a damped resonance peak and inertial damping at high frequencies. The overall frequency behavior can be described by:

and

The three parameters $k$ (a spring constant), $d$ (a drag coefficient), and $m$ (the inertial mass) are the fitting parameters used to describe the response of the device. The lowest measurable modulus at each frequency will be determined by the combination of these parameters as will be shown later. To calculate the surface modulus when a complex interface is present, the contributions stemming from the measurement system (drag from the constituent phases, surface curvature, rod inertia, etc.) need to be taken into account. The simplest way to do this is to assume that the contributions are additive and that the surface drag is linear in the velocity [Brooks et al. (1999); Alonso and Zasadzinski (2004)].

In order to optimize the sensitivity of the surface rheometer, an obvious strategy is to reduce the overall magnitude of the instrument response, by reducing the magnitude of the parameters $k$⁠, $m$⁠, and $d$⁠. The spring constant $k$ arises from the force exerted by the magnetic field. The magnitude of $k$ is determined by the curvature and magnitude (for paramagnetic rods) of the magnetic field at the center point and by the intrinsic magnetic properties of the rod. The spring constant was measured experimentally by performing needle oscillations at low frequency on the subphase without any monolayer. As can be seen from Eq. (2), the value of $1∕k$ is equal to the ratio of displacement to force $(z∕F)$ in the limit of zero frequencies. To reduce the spring constant of the rod, a geometry as in Fig. 1(G) was used, where the material is localized in the center of the rod to minimize its exposure to curvature in the magnetic field. Hollow glass capillaries, typically used in x-ray experiments (Composite Metal Services Ltd, UK) were filled only in the central section with a short piece of special grade carbon iron (0.9% C, 1% Mn, Precision Metals, Belgium) or cobalt wire (99.9% Co, Precision Metals, Belgium), magnetized in a strong magnetic field $(5T)$⁠. This type of geometry makes it possible to change the length of the probe, without changing its magnetic properties. At first glance, needles with magnetic material along their entire length (e.g., magnetic needles), would increase the force needed to move it and hence the sensitivity. However, increasing the length of the needle, without scaling up the size of the Helmholtz coils, will have the deleterious effect of increase the spring constant and result in poorer sensitivity. The diameter of the capillaries was chosen to be small. Very thin rods increase the dynamic range of the device, because at higher frequencies rod inertia (proportional to $m$⁠) decreases sensitivity. Thinner rods also increase the intrinsic sensitivity since the interfacial effects are more likely to dominate over the bulk rheology of the subphase $(Bo∼1∕a)$ for thin rods. Reducing the diameter of the rod also reduces the amount of magnetic material per unit rod length. Previously, Brooks et al. [Brooks et al. (1999); Brooks (1999)] used Teflon coated sewing needles with typical radii, $a$⁠, varying from 195 to $225μm$⁠, lengths $L$ from 25 to $50mm$ and a density of $6300kg∕m3$⁠, whereas Ding et al. (2002) used a small magnetic stirring bar of $200mg$ and a 3-cm-long, 1-mm-diam hollow Teflon tube. The diameter and mass of the five rods used in the present work are tabulated in Table I and span the range of rods used by these other works.

The use of glass needles also permits changing the surface chemistry of the needle thus controlling the wetting properties of the rod. The glass surface was rendered more hydrophobic using a silanization reaction using a 5% dimethylchlorosilane in heptane solution. The contact angle of the silanized glass was measured to be $92±4°$ using a contact angle goniometer (CAM 200, KSV Instruments Finland), whereas the Teflon needles have been reported to have a contact angle of $110°$ [Adamson and Gast (1997)]. Since the average density of the hollow glass needles is significantly lower than Teflon coated needles, experiments can also be performed on systems with a lower interfacial tension. The Bond number, Bd, represents the ratio of buoyancy forces to surface forces. The Bond number is defined as

For a rod with a radius (a) of $125μm$ and a density of $2097kg∕m3$ located at a water interface with a interfacial tension $γ=72mN∕m$⁠, the Bond number is 0.0037 while for a rod with a radius (a) of $300μm$ and a density of $4040kg∕m3$ the Bond number is 0.058. This indicates that for the needle rheometer, the surface tension force is predominantly responsible for supporting the rod. Depending on the surface pressure of the studied monolayer and the contact angle made by the needle with the interface, the needle will be supported or sink through the interface. A detailed analysis of this problem is given by Brooks (1999). For needles with similar contact angles, the simple Bond number analysis will give information on how much lower the interfacial tension can be, yet still support the lighter needles.

To evaluate in a simple manner the relative importance of the instrument contribution to any measurement, an equivalent complex “system” modulus $(Gs,SYS*)$⁠, can be defined as

where $FD$ is the amplitude of the total drag force exerted on the needle, $zoast$ is the displacement amplitude of the needle. A magnitude and phase angle can be calculated for this equivalent system modulus, as they would be measured by the instrument for an empty interface.

In a typical measurement, a certain needle is selected based on the range of forces to be exerted, and for that needle the system response $(Gs,SYS*)$ in the absence of any monolayer is measured. Subsequently the apparent complex surface modulus $(GS,APP*)$ is measured. To correct for the instrument’s response, $(Gs,SYS*)$ is usually linearly subtracted from the apparent complex surface modulus, so that a corrected complex modulus is obtained $GS,Corr*$

For this complex modulus, the corrected norm (amplitude) and corrected phase angle can now be calculated in the usual manner. Now, whereas the parameters $k$ and $m$ are intrinsic properties of the measurement setup, the drag of the needle, captured by the parameter “$d$⁠,” depends both on the geometrical aspects and the properties of the interphase and subphases. A more detailed analysis of the fluid mechanics involved is hence required to optimize the measurement geometry and to asses the intrinsic assumption that bulk and interfacial flows are decoupled.

The flow field is calculated for a rod in a flow cell as depicted in Fig. 2. A cylindrical geometry is chosen to facilitate the numerical simulation of the apparatus. Although rectangular channels are most often used in actual instruments. It has been previously experimentally verified that, at least for sufficiently deep channels, this is a good approximation [Brooks (1999)]. The rod is assumed to have an infinite length, implicitly neglecting end effects such as the possible effects coming from Marangoni flows near the needle tip. Further simplifications include assuming that the contact angle is $90°$ and that the interface is flat. The Navier-Stokes equation governs the momentum transport in the subphase, and, written in terms of the fluid displacement, $z$⁠, reduce to

with boundary conditions:

Symmetry dictates that:

The interface is treated as a mathematical surface, characterized by an “excess” complex modulus $Gs⋆=Gs′+i∙Gs″$⁠, with the elastic $(Gs′)$ and viscous $(Gs″)$ surface modulus having known (imposed) values. The stress condition yields the boundary condition at the interface as:

Time is now rescaled by frequency and the radial coordinate by the rod radius and the following dimensionless variables are defined:

The parameter $λ=R∕a$ describes the channel geometry. Equation (11) is further transformed by using the substitution $p=ln(r¯)$ and by employing a separation of variables, assuming that the displacement of each fluid element has the form

Equations (7)–(11) can then be rewritten in a form suitable for solving with a finite difference approach. The problem reduces to

with boundary conditions

and

The Reynolds number (Re) follows from rendering the Navier-Stokes equation dimensionless:

and the Boussinesq (Bo) number appears in the stress boundary condition at the interface

where $ηs*$ is the complex surface viscosity (equivalent to $Gs⋆∕(i*ω)$⁠, and equal to $ηs*=ηs″−iηs′$⁠). The Boussinesq number is a complex number for the case of a viscoelastic interface as treated here, with a positive real part determined by the interfacial loss modulus and a negative imaginary part corresponding to the interfacial storage part. Equations (15)–(19) are solved using a finite difference approach, derivatives are approximated using second order forward and backward finite differences and the actual calculations are done using $Maple™$⁠. For more details on the numerics we refer to the supporting information. The solution is calculated using a rectangular mesh of 40 by 40 nodes. It was verified that enhancing the mesh size did not yield increased accuracy. The result of the finite difference calculation gives, after rewriting everything back in dimensional form, the deformation profile. Using the displacement profile from Eqs. (15)–(19), the force exerted on the needle can now be calculated. The ratio of the total force on the needle to its displacement is given by

The first term on the right hand side of Eq. (22) corresponds to the drag on the needle stemming from the viscoelastic surface, the second term is the subphase drag while the two last terms respectively take into account the system compliance and inertia and are independent of the resulting flow. The system compliance $k$ arises from the restoring force created by the magnetic coils. Its value is the product of the curvature of the magnetic field and the intrinsic magnetic properties of the rod used.

Using the equivalent complex system modulus [Eq. (5)], the magnitude of the instrument contribution can be assessed. Clearly, the magnitude of the apparent modulus $Gs,SYS*$ will affect the lower sensitivity limit in terms of the magnitude of the measurable modulus, as a function of frequency. The maximal Reynolds number was calculated to be 0.0294 and 0.0283 for needles with respective diameters of 250 and $600μm$ (needles No. 2 and No. 5, details in Table I) on a water subphase $(T=25°C)$⁠, at frequencies ranging from 0.05 to $0.3Hz$⁠. These conditions compare well with actual experimental conditions. Figure 3 shows the norm of the system complex modulus $(Gs,SYS*)$ and the corresponding phase angle $(δs,SYS)$⁠. The calculations, given as open symbols, correspond to a typical Teflon coated sewing needle, as compared to a typical hollow glass rod with a localized magnet. Typical values for the system compliance $(k)$ vary from $0.0025N∕m$ for the sewing needle down to $0.000005N∕m$ for the hollow glass rod. The compliance of all the rods and needles used in the present work is summarized in Table I. A channel geometry with $λ=30$ is used in the calculations. Figure 3 demonstrates how $Gs,SYS*$ is significantly decreased when a thinner hollow glass rod with a smaller, more localized magnetic core (needle No. 2: weight of $3.5mg$⁠, $1.5mg$ carbon steel, diameter of $250μm$⁠) is used compared to a full Teflon coated needle rod with a mass of $38.8mg$ and a diameter of $600μm$ (needle No. 5). Reducing the amount of magnetic material also reduces the maximum force that can be exerted on the rod. Typically a rod or a needle has a measuring range of two decades above the minimum measurable modulus as will be shown later. This restriction is determined by the maximum current that can be applied to the coils without causing them to overheat.

The agreement between calculation and experiment (closed symbols in Fig. 3) is satisfactory, yet at intermediate frequencies some deviations occur which are caused by a small difference in damping. Most likely this is due to deviations from the conditions used in the calculation, i.e., the assumptions of neutral wetting, a circular channel and neglecting end effects. The damping is slightly overestimated in the calculations in the case of the silanized glass rod (No. 2, Table I) while the damping is underestimated in the case of the Teflon coated needle (No. 5, Table I). Both probes have a contact angle slightly above $90°$ so an overestimation of the damping is expected for both needles. However, the Teflon needle has a rough surface, amplifying the damping, while the glass needles have a mechanically smooth surface.

In this section, numerical calculations are used to evaluate under which conditions the deformation at the interface is nonlinear, due to coupling of bulk and interphase flow or due to a nonzero Reynolds number. This influences the manner in which the drag contribution “$d$” is taken into account. The complicance and inertial terms also contribute to the measurements [see Eq. (22)], and they can only be subtracted to within a certain numerical accuracy, depending mainly on the relative measurement errors.

A typical assumption in analyzing magnetic needle rheometry is that the displacement at the interface is linear, so that the interface is subjected to a uniform strain. To visualize this directly, the dimensionless surface displacement function $f$ as a function of the dimensionless position in the channel $(r)$ at the interface [Eq. (14)]. In Fig. 4 the surface displacement profile for a channel geometry $λ=30$ is shown for various Boussinesq numbers. The displacement profile becomes complex valued with components that are in phase (real) and out of phase (imaginary). Only at large values of Bo $(Bo>1000)$⁠, a linear profile is found with dominant real values. At low Boussinesq numbers the displacement at the surface deviates strongly from the linear profile. The deformation also becomes confined to a region close to the rod surface. An important out-of-phase component appears, which displays a complex dependence on position.

Similar profiles are obtained as $λ$⁠, the radius of the channel compared to rod diameter, is varied. For larger values of $λ$⁠, a real valued linear profile is only found at higher Boussinesq numbers. This implies that $λ$ should be as small as possible, i.e., the channel should be as narrow as possible. However, the capillary length scale $q−1$ $(=γ∕(ρ∙g))$ [Krachlevsky and Nagayama (1994)] dictates the scale at which lateral capillary forces between the walls and objects within the complex fluid interface (such as particles) become significant. For a water-air interface $q−1$ is equal to $2.7mm$⁠; for interfaces with lower interfacial tensions, the value of $q−1$ is reduced. Therefore, the width of the channel should not be made much smaller than this value. In our experiments, the thinnest glass rod (No. 1) has a radius of $85μm$⁠, and a corresponding channel with a width of $∼3mm$ has been used, yielding $λ∼35$⁠. For the thicker rods and needles, $λ$ can be reduced further. However, in the calculations lambda is kept fixed at a value of $λ=30$⁠, to isolate the effect of geometrical factors related to the rod.

The presence of the nonlinear deformation profile leads to the “system drag” contribution not being correctly accounted for under all conditions when a simple linear substraction is applied. Figure 5(a) presents results of the magnitude of the norm of the apparent complex modulus $Gs,APP*$⁠. This is the value of the modulus as it would be measured from the total drag force, without corrections. This value is plotted as a ratio to known value of the norm of the modulus $Gs*$ which was used as input in the calculations. The ratio is given as a function of the Boussinesq number, for both the glass rod (No. 2) and the magnetic needle (No. 5) as also used in Fig. 3. Figure 5(b) gives the corresponding evolution of the apparent phase angles, i.e., as they would be obtained without corrections, to the expected phase angle (for the present case of a viscous interface this is equal to $90°$⁠). Both properties are plotted as function of Bo at similar Reynolds numbers (0.0283 and 0.0294) for the needles used previously to calculate the system response. These ratios give an estimate of the relative contributions of the system response as a function of Bo. At low Re and Bo numbers, and for the corresponding small drag forces, the system compliance as well as the nonlinear deformation profile can both contribute to the measurement error. As a consequence the apparent phase angle can be underestimated as well as overestimated. For example, the apparent phase angle $δs,APP$ for the needle with a diameter of $600μm$ (No. 5) is dominated by the system compliance at low to intermediate Bo number, and a negative phase angle is obtained. As the Bo number is increased the relative importance of the compliance term decreases. For a thinner needle with localized magnetic insert (⁠$250μm$ (No. 2), the effect of complicance is much smaller. At the smallest Bo number, the effects of complicance and system drag cancel out. Figure 3 shows that the presence of the drag exerted by the bulk leads to an overestimation of the phase angle until the surface drag dominates the response at a Boussinesq number of $50000$ for the Teflon needle, the system response remains important. The phase angle is still underestimated by $5°$⁠, whereas the magnitude of the measured complex modulus is within 1% of the actual value. The error is smaller for the needle with a diameter of $250μm$ where the same sensitivity is achieved at a Boussinesq number of around 5000.

The typical procedure to account for the system response is to linearly subtract it from the experimental results [Eq. (6)] assuming a linear system response $(k,m)$ and a drag force that is linearly dependent on deformation. The ratio of the corrected modulus and phase angle to the known value of modulus and phase angle are included in Fig. 5. As all dimensionless geometric parameters $(λ=30)$ are similar for both needles, the two curves for the corrected data nearly collapse. The correction procedure leads to a dramatic improvement for the obtained values of the moduli. The error on the phase angle remains substantial at small Bo, and is due to the presence of the nonlinear deformation at the interface. The deformation profile at the interface (Fig. 4) shows that a phase lag is present at low Bo, which will result in a positive phase angle when using a linear correction. The remaining error can be quantified, e.g., at $Bo=100$ the norm of the modulus is overestimated by 21% and the phase angle $12°$⁠, at $Bo=1000$ the error is reduced to 1.5% on the magnitude of the modulus while the error on the phase angle is $1.5°$⁠.

In practice, the accuracy of the correction will also depend on the relative magnitude of $(Gs,APP*)$ and $(Gs,SYS*)$⁠, since subtracting two numbers of equal magnitude is numerically unstable. The most important error in the experiments comes from the measurement of the displacement amplitude, for the equipment used it is on the order of 5%. To suppress the propagation of the relative error when subtracting numbers of similar magnitude, an empirical “safety” factor, equal to twice the maximum relative error, on the position measurement error, is taken. This choice is somewhat arbitrary as a systematic error propagation analysis has not yet been performed, but the exact value of the safety factor does not change the analysis of the effect of the magnitude of the system response for different needles on the minimum measurable surface viscosity. In our experiments, datapoints are only accepted when $Gs,APP*∕Gs,SYS*>1.20$⁠. To demonstrate that this also depends on the measurement geometry, $Gs,APP*∕Gs,SYS*$ is plotted as a function of the Boussinesq number in Fig. 6 for the same rheological probes as in Fig. 5. For the glass rod No. 2, this condition is achieved at $Bo≥1.2$ $(Re=0.0294)$⁠, while the Boussinesq number must exceed 2000 for the needle No. 5 (Re 0.0283). The corresponding minimal apparent surface viscosity is 0.125 $10−6Pasm$ at a frequency of $0.3Hz$ for the thinnest needle and 0.6 $10−3Pasm$ at $0.05Hz$ for the thickest needle. Therefore, for most practical situations, the effect of the system response on minimum measurable apparent surface viscosity can be lowered by approximately 2 decades when using a needle with a lower system compliance. However, the coupling between subphase and interphase deformations then needs to be considered.

The previous section provides a procedure to assess the lowest surface viscosities that can be measured for viscous interfaces at low Reynolds number. In practice, it is sometimes necessary to increase the frequency of the applied force signal, either to extend the frequency window or to generate a force larger than the minimal force that can be applied by the magnetic coils. This causes the Reynolds number to increase [Eq. (20)]. Increasing the bulk Re, keeping all other factors constant, enhances the nonlinearity of the deformation at the interface, similar in effect as decreasing the Bo number. The resulting effect of Re on $Gs,Corr*$ and the phase angle $(δs,Corr)$⁠, relative to $Gs*$ and $(δs)=90°$⁠, is given in Fig. 7 for a needle with a diameter of $250μm$ (No. 2), assuming a purely viscous interface. The frequency of the force signal was chosen to be 0.05, 0.3 and $1Hz$⁠, resulting in a Reynolds number of respectively 0.0049, 0.0294 and 0.0982. The channel geometry $λ$ is kept at 30. Although Re is still relatively small, the effect on both the modulus and phase angle increases with Re at low Bo. For example, for $Re=0.0982$⁠, corresponding to a frequency of $1Hz$⁠, and $Bo=100$⁠, the modulus is overestimated by 48% and negative elasticities are predicted with a phase angle of $17.6°$⁠. For Bo larger than 5000, the effect of increasing Re can be neglected.

The effect of Reynolds number is most pronounced at relatively low Bo. At low enough Re, the Stokes equations evidently are recovered. The critical Reynolds number below which the Stokes result is obtained depends on both Bo number and needle geometry. For $λ=30$⁠, the critical Re number increases from about $10−3$ for $Bo=1$⁠, $3∙10−3$ for $Bo=10$⁠, to $5∙10−3$ for $Bo=100$⁠. It is also a strong function of $λ$⁠. For a Bo number of 10, at $λ=100$ the critical value is as low as $10−4$⁠, whereas for $λ=100$ it is on the order of $10−2$⁠. For Reynolds number above the critical value, numerical calculations for the error made when using the linear subtraction rule go roughly as $Re1∕3$ for typical values of the Re number in the experimentally accessible range.

Because the interface appears as a boundary condition in the problem of the flow field, changing its viscoelastic nature can substantially alter the deformation profiles both in the bulk and at the interface. The error made when using the linear drag substraction now becomes more difficult to intuitively assess. Therefore the case of a purely elastic interface is examined using numerical calculations as a second limiting case. In this case the channel geometry and Reynolds number are chosen to be equal to those used for the purely viscous case but the Bo number is now a negative imaginary number. In Fig. 8 the displacement profiles at the surface are shown for an elastic interface for varying Boussinesq number. These results can be qualitatively compared with the results in Fig. 4. Focusing on the profiles at $Bo*=100$ and comparing the real component of the displacement profile in both cases, the profile is found to be curved and in the opposite direction for an elastic interface when compared against a purely viscous interface. Also, the out-of-phase imaginary component goes trough a deeper minimum. This stronger nonlinear deformation profile can be expected to have a distinct, more pronounced effect, compared to the case of a purely viscous interface, on both the magnitude and the phase of the apparent modulus.

In Fig. 9(a) the ratio of the norm of the measured apparent complex modulus $(Gs,APP*)$ (circles) and the corrected apparent complex modulus $(Gs,Corr*)$ to the norm of the actual complex modulus $(Gs*)$ is plotted. Compared to the case where the interphase is purely viscous, the linear subtraction procedure does not a priori yield worse results for an elastic interface, but the dependence on the parameters becomes harder to intuitively predict. The norm of the complex modulus can be either over- or underestimated. This occurs because the effects on the deformation profile for the elastic interphase on the flow field differ from the viscous case, the momentum diffusion into the subphase causes an opposite effect on the deformation field at the interface. The error on the phase angle (compared to $δs=0°$⁠) is shown in Fig. 9(b), which shows a nonmonotonic dependence on Bo. When comparing the results for viscous and elastic interfaces in more detail, at $Bo=10$ and $Re=0.2940$ the modulus for a viscous interface is overestimated by 250% and the error on the phase angle $25.7°$⁠. For $Bo=−10∙i$ the norm of the complex modulus is overestimated by 295% and the error on the phase angle is as large as $58.2°$⁠.

The two dominant contributions to the measurement error are again stemming from the compliance $k$ and the surface drag. The effect of compliance can lead to results which give a correct phase angle; even at low Bo number for the elastic interface a value close to $0°$ is found [Fig. 9(b)], yet the apparent modulus is way too high [Fig. 9(a)]. Under these conditions, Fig. 3 shows that the system response is dominated by the elastic contribution $(k)$ due to the magnetic field. Subtracting the system reference significantly reduces the error on the modulus [triangles in Fig. 9(a)], but apparently amplifies the error on the phase angle. The linear correction procedure adequately removes the in-phase, elastic, “compliance” component of the system response. Yet the nonlinearity in the deformation profile now affects the resulting phase angle more importantly. The nonlinear deformation will again induce a phase lag in the expected response leading to a higher phase angle at low Bo.

For a viscoelastic interface the errors in the correction procedure now depend in a nontrivial way on the Boussinesq number. To establish the limits where the simple background correction procedure can be trusted, the error on the corrected modulus and phase angle for a viscoelastic system can be calculated for any given value of the Re number. It is found that the error on the norm of the complex modulus varies monotonically between the purely viscous and purely elastic interface (see, for example, Fig. 10 for $Re=0.03$ and $λ=30$⁠). The graph as in Fig. 10(a) can be used to estimate the error made by using the linear substraction profile. An error on the order of 20% is obtained for Bo number higher than 50, more or less in the entire range of phase angles. The error on the phase angle displays a more complex nonmonotonic behavior with a maximum for intermediate Boussinesq numbers and elastic interfaces. The phase angle is more sensitive to effects of instrument compliance. An error of $5°$ is only obtained for $Bo>300$⁠.

An important conclusion from our work is that at small Boussinesq number, the effect of Reynolds number and the instrumental constant have important contributions in addition to the effect of the coupling of bulk and interphase flows in the Stokes limit. Hence an approach using a integral formulation of the Stokes equations (rather than the full Navier-Stokes) and the Boussinesq-Scriven equations to correct for the coupling as proposed by Oh and Slattery (1978) as correction scheme for the experimental data [Enri et al. (2003)], has not been pursued here.

To now experimentally assess the operating window of the ISR and to test the optimization of the measurement geometries, a viscous and a viscoelastic interface was prepared. Thin films of silicon oil served as a reference Newtonian interface to test the lower measurement range of the needle rheometer. Three different types of oils were used (Brookfield Engineering Laboratories Inc., Middleboro) with bulk viscosities of, respectively, $752mPas$⁠, $10Pas$ and $100Pas$ at $25.0°C$⁠. The oils are insoluble in water, forming a thin Newtonian layer of several micrometers on top of the water. The interfacial viscosity of an insoluble thin film is the product of its thickness times the bulk viscosity. Interfacial layers with viscosities in the desired range to test the lower limits of the ISR can be prepared. Results on interfaces with an interfacial viscosity of $2.4∙10−6$⁠, $2.5∙10−5$⁠, and $1.83∙10−4Pasm$ will be presented. The layers were prepared in a Petri dish with a diameter of $100mm$⁠, using bidistilled de-ionized water as the subphase. A rectangular channel with a width of $12mm$ and a length of $70mm$ was placed in the center of the dish. A hollow glass needle (No. 3), filled with an iron insert with a diameter of $400μm$ and length of $45mm$ was used so that $λ$ was fixed at 30. The system compliance $k$ is $0.000036N∕m$ and the mass of the needle is $10.2mg$⁠. The water level was chosen so that it was equal to half the channel width.

As a viscoelastic test material, the protein lysozyme was used. Lysozyme extract from chicken egg white was purchased from Sigma Alldrich (lyophilized powder, $50000units∕mg$⁠). Potassium dihydrogen orthophosphate and sodiumhydroxide (VWR International Ltd.) were used to cross-link the protein at the interface. All reagents were used as received; $100mg$ lysozyme was first dissolved in $1ml$ bidistilled de-ionized water; $100μl$ of this solution was further diluted into a $0.1M$ phosphate buffer. The pH was regulated by adding sodiumhydroxide to the buffer. The subphase was prepared using a glass Petri dish (diameter $100mm$⁠) containing an amount of water. The amount was chosen so that the height of the water level corresponds to half of the width of the used channel. The system response of the needle on a pure water subphase was first measured. Subsequently, half of the pure water was gently replaced by the lysozyme solution so that the concentration of the phosphate buffer is $0.05M$ and a pH of 11 was obtained. The ISR was placed in a closed box saturated in humidity by placing containers with excess water in the box. Evaporation could be suppressed to a large extent and the development of the interfacial properties was followed for a period of $10h$⁠. Four different rheological probes (No. 1, 3, 4, and 5) with different diameters (170, 400, 400, $600μm$⁠) are used in channels with a width of 6, 12, and $18mm$ so $λ$ was fixed at 35, 30, and 30, respectively.

To test the practical limits of the ISR and to compare the results with the calculations on the effects of the Reynolds and Boussinesq number on the flow profile, Newtonian films with known interfacial rheological properties were prepared. The combined effect of varying the bulk viscosity and film thickness of silicon oils resulted in Boussinesq numbers varying between 12 and 915. This corresponds to the lower measuring window of the magnetic rod rheometer. Measurements were carried out for the three oil films as a function of frequency. For a purely Newtonian material $Gs″$ scales with $ω$ and the Boussinesq number is independent of frequency, whereas the Reynolds number increases with frequency. For $Bo=915$ no reliable datapoints could be measured at higher frequencies since the displacement of the needle at maximum force was too small to be resolved accurately for the needle used. The ratio between the norm of the corrected and the exact known complex modulus is plotted as a function of the ratio of the drag force on the viscous interface with the drag force observed on the system without monolayer (Fig. 11). It can be concluded that three factors contribute to the measurement error: the magnitude of the Boussinesq number, the ratio between the sample and system response as well as the Reynolds number. For a fixed force ratio the error on the modulus is expected to increase with Re, which causes the upwards curved shape for $Bo=12$ and $Bo=126$⁠. The effect of Re on the phase angle is smaller, as could be expected from the calculations presented in Fig. 3.

Comparing the experimental data of Fig. 11 with numerical calculations corresponding to the same geometric and physical conditions for two values of Re (Fig. 12), it can be concluded that the combined error on the phase angle is slightly larger $(10°)$ while the observed error on the modulus is somewhat smaller. This is especially true for $Bo=12$ and $Bo=126$ where $Gs,APP*∕Gs,SYS*$ is still small. The deviation between experiment and prediction is probably caused by the overestimation of the subphase drag (Fig. 3). At $Bo=915$ accurate measurements of both the modulus and the phase angle could be made and a good agreement between experiment and predictions was obtained. Based on the calculations and measurements, the range of surface viscosities that can be accurately measured can be determined. In Table II, the minimum and maximum measurable surface viscosity is given for the needles tabulated in Table I. The upper limit is mainly set by limits in the sensitivity of the position detector and the maximum magnetic field gradient that can be imposed without overheating the coils, and these are not intrinsic limitations of the magnetic rod device.

Lysozyme was used during its gelation at the interface. In this manner a broad range of phase angles and moduli can be explored, in order to produce a reference data set for viscoelastic materials and assess the operating limits of the ISR for viscoelastic interfaces. Lysozyme was dissolved into the aqueous subphase and the subsequent adsorption of lysozyme was monitored by measuring the surface pressure of the layer. For the different experiments, the kinetics agreed very well. However, in one experiment, somewhat slower kinetics were observed, and the time scale was adjusted by a factor of 1.15 in order to make the surface pressure kinetics for this experiment agree with the others. The surface pressure never reached a steady state value but after $1h$ the rate at which surface pressure increases, becomes small $(∼0.2mN∕m)$⁠. Values of $13–14mN∕m$ are found at this pseudo-plateau, which agrees with earlier results by Roberts et al. (2005). Figure 13 gives the evolution of the norm of the apparent complex modulus as a function of time. We take advantage of this evolving surface modulus to test the limits of the ISR, as it does not require different layers to be prepared to explore four orders in magnitude. The minimum measurable modulus with every needle is related to the plateau value at small times after addition of the lysozyme solution. For example, the thinnest needle (No. 1) with a weight of $2.2mg$ has a minimum sensitivity of order $10−6N∕m$⁠. After $40min$ the layer has sufficiently developed so that a modulus can be measured. All the other needles are less sensitive and experimental data could only be obtained after longer times, once the surface became sufficiently rigid.

Plots as in Fig. 10 can be used to verify the conditions under which accurate data can be obtained, when taking into account the system response and the effect of Reynolds number. The lysozyme layer starts out as a dominantly viscous interface with $Bo*=100$ and $Re=0.005$⁠. For these conditions the magnitude of the modulus is within 6% of the correct value, the error on the phase angle is within $2°$⁠. This condition is achieved for the thinnest needle (No. 1, $2a=170μm$⁠) at $Gs*=5∙10−6N∕m$ and sets the lower limit for our measurements. The error on all measurements of moduli at longer times with this needle is smaller than 6%. The data can now be used as a reference for data measured with other magnetic rods.

Removing the parametric dependence on time, a phase plot can be obtained where the norm of the complex surface modulus is plotted as a function of the phase angle. In this way, all possible complications due to time effects can be ruled out and the conditions under which reliable phase angles can be obtained can be calculated. The result is presented in Fig. 14. The thinnest, lightest needle (No. 1) enables measurements in the range of $Gs*=5∙10−6N∕m$ to $Gs*=5∙10−4N∕m$⁠. A hollow glass rod and a coated Teflon sewing needle with a diameter of $400μm$ (No. 3 and 4, Table I) can be used to cover the intermediate range of moduli and phase angles. At $Bo*=300$ and a phase angle of $60°$⁠, the error on the modulus is within 2% and the phase angle is within $6°$⁠, as can deduced from Fig. 10. Some deviations occur for the first three data points for needle No. 4. Despite the fact that only data that match $Gs,Corr*∕GSYS*>1.2$ are presented, subtracting the system response can induce a large uncertainty. However, when $Gs,Corr*∕GSYS*∼10$⁠, subtracting the system response has only a minimal effect and the points nicely overlay with results obtained with other needles. The datapoints obtained with the needle with a diameter of $600μm$ (No. 5) complete the data set. The arrow in Fig. 14 indicates the point at which $Bo*=1000$ when $Re=0.06$⁠. At higher $Bo*$ the error on $Gs,Corr*$ and $δs,Corr$ is sufficiently small so the data can be accepted. Indeed, the data obtained with the thickest needle nicely correspond with the data measured with thinner needles. Concluding, the ISR enables accurate measurements of the moduli from $Gs*=5∙10−6N∕m$ to $Gs*=5∙10−2N∕m$⁠, in the entire range of phase angles.

Interfacial rheometry is nontrivial because, in all measurement techniques the coupling of bulk and interfacial flows need to be accounted for. When the limits of operation are carefully explored, even using a simple linear substraction of the subphase drag. Comparing literature data using rotating disks [Miller et al. (1997)] and the bi-cone geometry [Enri et al. (2003)], the reported lowest values are 1–2 orders of magnitude higher than the lower limits for which it has been shown here that reliable data can be obtained. Knife edge devices permit low steady shear surface viscosities to be measured [Edwards et al. (1991)]. For complex fluid interfaces the rates are, however, often in the shear thinning regime. For low shear rheometry or when used in rotational rheometry, the knife edge devices have lower proven sensitivities compared to the needle device discussed here [Ghaskadvi and Dennin (1998)]. Microrheological techniques clearly have great potential in lowering the sensitivity further. However, at present reliable data seem only to have been collected for purely viscous interfaces. Prasad and Weeks have shown excellent sensitivity using both single and two-particle tracking for viscous interfaces with values as low as $10−9Pasm$ [Prasad and Weeks (2006)]. However, the value and analysis of both passive and active microrheological techniques for complex fluid interfaces is still under debate as the role of Marangoni flows and the role of the Boussinesque number have yet to be conclusively settled [Sickert and Rondelez (2003); Fischer (2004b); Sickert and Rondelez (2004); Fischer et al. (2006)]. Microrheological methods do not enable one to easily select the stresses and strains exerted onto the interface. Concluding, the magnetic rod rheometer provides rheometrical data, with control over stress and frequency, in the range of magnitude for moduli. The ISR enables accurate measurements starting at moduli of $Gs*=5∙10−6N∕m$ to $Gs*=5∙10−2N∕m$⁠, in the entire range of phase angles and strains. This is the regime where the viscoelasticity (and not only viscosity) typically starts to become an important aspect of the interfacial dynamics.

The lower limits in sensitivity of a magnetic rod interfacial rheometer have been investigated. The effects of rheometer compliance, rod inertia as well as the nonlinearity of the deformation profile on the measured moduli have been analyzed. The coupling of deformations in bulk and for viscoelastic interfaces lead to deviations in the deformation profile at low Boussinesq numbers, but also effects of nonzero Reynolds numbers are observed. The magnitude of the measurement error has been quantified as a function of the magnitude of the modulus, and the viscoelastic nature of the material at the interface—conditions under which reliable measurements can be obtained even for very low modulus interfaces. Based on the analysis of the effects of the system response, the Boussinesq and Reynolds number, modifications to the measurement probe have been proposed. By reducing the mass and and localizing the magnetic material in the center of the rod, dramatic improvements in instrument sensitivity can be obtained. Using a range of magnetic rods of varying weight and magnetic material, the magnetic rod interfacial stress rheometer can be used to measure a wide range of moduli. Especially at the lower end of the spectrum these moduli and viscosities are not accessible by other methods.

The authors would like to thank Dr. J. Vanacken (Dept. of Physics, K.U. Leuven) for suggesting the magnetic materials and access to the high field magnet. The research council of the K.U. Leuven is acknowledged for financial support through GOA-2003/06. J.V. acknowledges support from a research program of the Research Foundation-Flanders (FWO - Vlaanderen, Project No. G.00469.05) and the NoE Softcomp (EU, 6th Framework). Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockhead Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract No. DE-AC04-94AL85000.