WO2013106480A1 - Measurement of rheological properties using microprobes - Google Patents

Description

MEASUREMENT OF RHEOLOGICAL PROPERTIES USING

MICROPROBES

FIELD OF THE INVENTION

The present invention relates generally to devices and techniques for the measurement of rheology. More specifically, it relates to microrheology of surfactant interfaces using microfabricated probes and to microrheology of bulk materials in very small volumes.

BACKGROUND OF THE INVENTION

Fluid interfaces are ubiquitous in industry, technology and life. Interfaces separate and organize biological systems, from organelles to cells to organs, and enable gas exchange in respiration. High-interface foams and emulsions find wide use in industry, food and personal care products: emulsified oil droplets give both taste and texture to a fine espresso. Surfactants, including traditional amphiphilic fatty acids and phospholipids and non-traditional surfactants, such as colloids, block copolymers and nanoparticles, lower the free energy of interfaces and introduce a kinetic barrier to coalescence, thereby stabilizing these high-interface systems by creating novel microstructures and phases.

The static properties of interfaces have long been studied: Langmuir isotherms relate the surface tension, γ, (or the surface pressure Π=γ(0)-γ(Γ)), to the surface concentration, Γ, of surfactant. However, the dynamic properties of interfaces— how monolayers and bilayers respond to applied forces— can be equally important, albeit less well known. The dynamic response is more challenging to study, as the bulk material generally swamps the interfacial viscoelastic response in a conventional rheometer. Probes moving within the interface feel drag from both the interface and the bulk fluid. To maximize sensitivity to the interfacial properties, the Boussinesq number,

Bo≡ drag from interface / drag from subphase = η₈Ρ_ε ηΑ_ο

must be large. Here η₈ and η are the viscosities of the interface and the bulk, and P_c and A_c are the contact perimeter and wetted surface area of the probe. Bo establishes a practical lower limit,

that can be measured with a given probe. High- aspect ratio probes such as knife-edges or magnetic needles maximize sensitivity, and even larger P_c/A_c ratios can be achieved with micron-scale colloidal spheres or ferromagnetic nanorods.

Surface active agents such as small amphiphilic molecules, block copolymers, nanoparticles, and colloidal particles modify fluid interfaces, affecting both static and dynamic properties. For example, surfactants provide better stability in emulsions and foams, while also influencing breakup, coalescence, and drainage processes. Given the ubiquity of surfactants in the food, personal care, biomedical, and petroleum industries, in addition to biological and soft matter physics, both static and rheological properties of complex interfaces have seen intense interest in both fundamental science and industrial applications in recent years. A clear example where both static and dynamic properties of surfactant monolayers matter is the lung surfactant, which must meet several demands simultaneously. It must lower the surface tension in the deep lung to nearly zero in order to reduce the power required to inflate the alveoli during respiration. At the same time, it must withstand the strong surface tension gradient between the deep lung and the bronchi. Rheological properties have been hypothesized to play a significant role in keeping lung surfactant where it belongs.

Since understanding and controlling the dynamics and rheology of fluid/fluid interfaces is crucial for a broad spectrum of multiphase materials, a variety of techniques have been developed. Classical techniques include the deep-channel viscometer, the knife-edge viscometer, viscometers that employ rotating disks, and the bicone interfacial rheometer.

As with classical rheometry of bulk materials, a standard method to measure the rheological properties of a complex interface involves the direct application of a shear stress to that interface. Any interfacial shear stress, however, is inevitably accompanied by stress from the bulk (subphase or superphase), since the interface and bulk are physically coupled. Such coupling makes it challenging to measure the rheological properties of the interface itself, as opposed to those of the bulk fluids on either side. The relative magnitude of the interfacial drag on a probe, as compared with the drag due to the bulk fluid, is given by the Boussinesq number (Bo),

Bo = TisVuPc/TiVuAc (1) Notably, the interfacial drag is established by the interfacial stress (V|_sVu) exerted along the perimeter of contact (P_c) between the probe and the interface, whereas the bulk drag arises due to the bulk viscous stress (r|Vu) exerted over the contact area (A_c) between the probe and interface. Measurements that are sensitive to the rheology of the interface, rather than the bulk, require the Boussinesq number to be large (Bo » 1). While a detailed hydrodynamic problem must be solved to relate the measured drag to surface viscosity, a characteristic scale for the surface viscosity, r\_s ^mm ~ r\A_c /P_c can be defined where Bo ~ 1. This represents an approximate scale for the minimum surface viscosity that can be clearly distinguished with a given probe from the viscosity of the bulk. We will provide a more detailed analysis, and extend its applicability to viscoelastic interfaces. The Boussinesq number for a given system depends on material parameters, which are dictated by the properties of the interface itself, and geometric factors, over which one has control. In particular, increased sensitivity can be achieved by maximizing the perimeter to area ratio of the probe, which has been accomplished in two typical ways. One is to employ high aspect ratio probes, such as needles, knife-edged rings, and centimeter-scale rotating disks. A second technique is to employ small, interfacially active probes, often driven by their own Brownian motion. In fact, the Brownian motion of fluorescent proteins or lipids, which represent perhaps the smallest probes possible, can be related to the viscoelasticity of the interface in which they are located via the Saffman-Delbruck model. Such diffusivities have been measured with both florescence recovery after photobleaching (FRAP) and single molecule tracking.

Relating rheological properties to responses of micron-scale (or smaller) probes is called microrheology. Two broad classes of microrheology have been developed: passive microrheology, in which the probe is driven by thermal fluctuations, and active microrheology, in which the probe is driven by external forces such as optical tweezers and magnetic tweezers. Active microrheology exerts an external force directly on the probe, while measuring its response. Measurements of the probe resistance encode the rheological response of the material. In the case of linearresponse measurements, one can extract the frequency-dependent, linear viscoelastic moduli by generalizing the Stokes resistance for the probe, following the correspondence principle. For nonlinear forcing, however, the quantitative interpretation is uncertain. The concept of microrheology can be extended to interfacial rheometry: both passive and active microrheology effectively measure the hydrodynamic resistance of the probe, whether the probe lies at an interface or is fully submerged in a bulk fluid. The appropriate hydrodynamic equation must be solved in order to extract the rheological properties (bulk and/or interfacial) of the material from the measured resistance.

Because passive microrheology involves the thermal fluctuations of colloidal probes, its practical use is generally limited to micron-scale or smaller probes, in soft materials whose moduli are not appreciably larger than water. Materials with G' > 1 Pa have been considered essentially inaccessible to video tracking techniques. Thus, while the minimal surface viscosity η₈ ^ιηιη measurable with a micron-scale probe is particularly small (η™ ~ 10 ⁹ N s/m with an aqueous subphase), interfaces that are even moderately more viscous (e.g. Bo ~ 1000, which would be equivalent to bulk materials with G' ~ 1 Pa), would slow the Brownian dynamics by a factor of order Bo, making passive measurements unpractically slow. The range of interfaces whose rheology can be probed with Brownian particles, then, is limited.

Well-defined, functional colloids provide critical model systems for a variety of fundamental phenomena in materials and soft matter, and enable a broad range of technological applications from optics to biotechnology. The ability to fabricate large quantities of colloids with high uniformity and specified shape and function has thus been greatly desired. For example, magnetic colloids can be externally manipulated and controlled, and have been used in applications ranging from photonic crystals, cell sorting, biosensors, drug delivery, biomedical applications, and single-molecule biophysics. Magnetic colloids that can be remotely controlled by applied fields have also been exploited in m-ink, microrheo logical probes, and a variety of self- organizing systems. In a similar fashion, ferro fluids and magneto-rheological fluids are frequently used to make useful materials because of their tuneable dynamic response. Another important direction for designing colloids involves shape anisotropy. Anisotropic particles offer additional control of light propagation for photonic crystals and resistance against efficient phagocytosis in drug delivery applications. They can also give rise to interesting and unique behaviors that are not found with spherical particles, with examples including self-assembly at fluid/fluid interfaces, novel interactions when dispersed in nematic liquid crystals, and non-Newtonian rheo logical properties.

Finally, introducing chemical anisotropy on different faces (so-called "Janus" particles) provides both technological importance and scientific interest. For example, such particles can be used for drug delivery, where one side is designed to bind specifically to a cell surface, while the other side incorporates a second functionality that binds a particular drug. Despite the separate importance of each of these functionalities, the prior art does not appear to teach particles combining all three features: magnetic, anisotropic and Janus functionality.

SUMMARY OF THE INVENTION

In one aspect, the present invention provides Janus micro-particles of defined, variable shape that exhibit either ferromagnetic properties. Among the applications where such particles open new capabilities in areas of growing interest are the rheology of complex fluid interfaces, and the reversible directed assembly of particles at interfaces. In another aspect, the invention provides techniques in which ferromagnetic Janus microdisks serve as sensitive probes for the active, interfacial microrheology of complex fluid/fluid interfaces. Surface-active molecules and particles are known to modify the surface energy of fluid interfaces; additionally, they can affect the dynamic behavior of such interfaces materials by imparting a viscosity (or viscoelasticity) to the interface itself. In one aspect, the invention provides powerful new techniques to measure the rheological properties of complex fluid interfaces. The techniques use probes with all three functionalities: ferromagnetism, amphiphilicity, and two-dimensional shape anisotropy. In particular, the microprobes 1) are small, yet visible under optical microscopy (1 to 100 μιη); 2) ferromagnetic, so as to enable external forces or torques to be applied; 3) amphiphilic, to ensure the probes physically absorb onto fluid/fluid interfaces; and 4) anisotropic, to enable the orientation of a rotating probe to be tracked optically. To satisfy these properties, the invention provides methods to fabricate ferromagnetic, Janus microdisks with two 'buttonholes' which may be circular or non-circular. A planar (2D) fabrication strategy, enabled by photo-lithography, is ideally suited for these structural properties, as it allows essentially any planar shape to be designed with various functionalities incorporated through layer-by-layer deposition. While there have been recent reports of micrometer-sized particles fabricated by photolithography, the novel introduction of magnetic properties and chemical anisotropy imparts significantly greater utility for the widespread use of these microparticles in a variety of fields.

The inventors have discovered that a photolithographic process along with evaporative metal deposition offers an extremely versatile route to the synthesis of multifunctional microparticles possessing magnetic, amphiphilic, and anisotropic structural features. The ability to control various aspects of these structural features allows micro-fabricated ferromagnetic disks to be fabricated and used as exquisite probes of the rheo logical properties of complex fluid interfaces. The modularity of this design strategy also allows the corresponding paramagnetic microtriangles to be prepared for the analysis of 2D suspensions at fluid/fluid interfaces. The total yield of this approach is limited (here to 10⁷— 10^s particles per 4-inch wafer) by the 2D surface nature of the fabrication, but can easily be scaled up, e.g., using imprint lithography techniques if needed. Nonetheless, these examples demonstrate the power of these fabrication techniques and the rich array of microstructures that can be easily prepared with control over size, aspect ratio, and shape via photolithography; magnetic properties by controlling the thickness and material of the deposited metal for ferromagnetic particles; and chemical anisotropy by selecting appropriate thiol- terminated molecules to self-assemble into monolayers on the gold surface.

Two-dimensional films of surface-active agents— from phospholipids and proteins to nanoparticles and colloids— stabilize fluid interfaces, which are essential to the science, technology and engineering of everyday life. The 2D nature of interfaces present unique challenges and opportunities: coupling between the 2D films and the bulk fluids complicates the measurement of surface dynamic properties, but allows the interfacial microstructure to be directly visualized during deformation. In one aspect, the present invention provides a novel technique that combines active microrheology with fluorescence microscopy to visualize fluid interfaces as they deform under applied stress, allowing structure and rheology to be correlated on the micron-scale in monolayer films. The inventors have discovered that even simple, single-component lipid monolayers can exhibit viscoelasticity, history dependence, a yield stress and hours-long time scales for elastic recoil and aging. Simultaneous visualization of the monolayer under stress shows that the rich dynamical response results from the cooperative dynamics and deformation of liquid-crystalline domains and their boundaries .

New active microrheology techniques according to the invention are sensitive to the surface viscoelasticity of a wide range of interfaces. Photolithography is used to fabricate micron-scale, ferromagnetic 'microbuttons', whose surface chemistry is tuned to render them amphiphilic (Fig. 3). A known torque is applied to the microbuttons by electromagnets to rotate a microbutton at the interface, and the 'buttonholes' are tracked to record the angular displacement as a function of time. The result is a micron-scale, two-dimensional (2D) Couette rheometer with P_c/A_c ~ 1/R ~ 1/(10μιη) that can measure surface viscosities of order η₈ ~ 0(1(T⁸) Nsrrf ¹ (|Bo|>l), and can simultaneously follow the evolution of the interfacial microstructure with fluorescence microscopy. The technique provides high sensitivity with simultaneous visualization. Such direct correlation is extremely challenging in traditional, three-dimensional (3D) rheology. The invention provides, in one aspect, a new technique that combines the advantages of microrheology with the versatility and dynamic range of actively-driven probes. Furthermore, it affords additional advantages with interfacial visualization and well- defined viscometric deformations. In measuring the viscoelasticity of interfaces, the translation of probes along an interface, whether thermally or actively driven, introduces several issues that are absent in the bulk. First, translation and rotation are generally coupled for particles translating along fluid interfaces, whenever the two bulk phases have different viscosities. In addition to complicating the hydrodynamic problem, this coupling could, in principle, introduce contact-line motion, which possesses a well-known singularity. Second, the flow around a translating probe would naturally tend to compress the surface in front of the probe and dilate it in the back. This, in principle, would couple the equation of state for the interface with the hydrodynamics of probe motion. Whether this occurs to an appreciable degree depends on the relaxation time for the monolayer. The relaxation time around the probe has been considered to be very fast, so that interfaces should effectively impose a two-dimensional incompressible velocity field at the interface. Finally, even incompressible two- dimensional interfacial flows around translating probes have a mixed rheological character (with various mixtures of shear and extension), which are known to significantly complicate the interpretation of nonlinear microrheology of three dimensional materials.

The present inventors have developed a technique that combines the advantages of microrheology with the versatility, dynamic range and control of macroscopic rheology. In particular, embodiments of the invention use microfabricated ferromagnetic "buttons" as probes, which are externally torqued using electromagnets, and whose orientation are tracked while they rotate. By employing rotation rather than translation, the technique avoids many of the issues of prior techniques, and introduces a variety of advantages. First, rotating disks establish a deformation field that is, in principle, pure shear, avoiding complications due to extensional, mixed and compressible flows. The interpretation of measurements using this technique is thus more straightforward, as the hydrodynamic problem is relatively simple, avoiding contact line motion and translation-rotation coupling. In addition to the advantages of rotating probes, micron-sized probes provide additional advantages. A relatively small sample area is required for experiments: areas are as small as 1 mm² are possible for Gibbs monolayers. The PJA_C ratio for a 10 μιη radius probe gives 2-3 orders of magnitude more sensitivity than macroscopic techniques. For example, with an aqueous subphase, a minimum surface viscosity scale r\_s ^mm ~ 1(T⁸ N s m^"1 can be achieved. Moreover, we can measure both linear viscoelastic surface moduli via small amplitude oscillatory stress, as well as nonlinear surface moduli via large amplitude oscillatory strain or steady rotation.

Rotating probes hold advantages for nonlinear measurements as well. Arbitrarily large strain can be imposed, and a fully-developed, Lagrangian- steady deformation field can be established. By contrast, the deformation field around translating probes is inherently unsteady in the Lagrangian sense, viewed from the reference frame of material elements. Moreover, for a given B field, a greater probe velocity is established for rotation than translation. In the Bo » 1 limit, a reference point on a rotating disk moves with linear velocity VR ~ a d9/dt ~ mB/r|_sa using a relationship d9/dt ~ mB/r|_sa², whereas the velocity of a translating probe is VT ~ (m-V|B|)/r|_s ~ mB/r|_sL, where L is the distance between the magnets. This rotational velocity VR is thus 0(L/a) larger than the translational velocity VT. Finally, these techniques have the ability to simultaneously visualize the interface as it deforms, allowing the structural deformation to be correlated directly with the rheological response. For example, a simple phospholipid monolayer exhibits a rich variety of dynamical responses, including a linear viscoelastic solid response, yielding, aging and recoiling. Direct visualization reveals these phenomena to reflect the cooperative dynamics of individual, interlocked liquid-crystalline domains.

The invention provides, in one aspect, a device and technique for the measurement of the rheology (visco-elasticity, shear-dependent viscosity and yield stress) of surfactant interfaces using microfabricated probes. These properties are relevant for the dynamics of any multiphase material and systems with fluid interfaces, including foams and emulsions (stability, coarsening, drainage), froth floatation and enhanced oil recovery, food science (stabilizing emulsions), biological systems (lung surfactant monolayers, tear film in the eyes, lipid bilayers for cells and organelles), pharmaceutical materials (protein aggregation at interfaces), suspensions and coatings (formation of surface skin layers), etc.

Another aspect of the invention enables the measurement of the linear and non-linear rheology (visco-elasticity, shear-dependent viscosity and yield stress) of bulk soft materials, while requiring microliter or smaller sample volumes. The ability to make these measurements is important for industries that work with small quantities of precious sample; for example, newly-synthesized or purified formulations that must be characterized to determine whether large-scale production is worthwhile. Examples include pharmaceutical materials (injectibility of drug formulations, early- stage protein aggregation), suspensions and coatings, high-throughput characterization in the chemical or other industries, etc.

In one embodiment, the device uses a microfabricated, permanent magnetic probe with tunable surface chemistry. The small size of the probe affords it extra sensitivity over commercial devices, the precise shape enables quantitative extraction of linear and nonlinear rheological properties; the active driving enables measurements in a wide range of interfaces (unlike passive probes, which are in principle sensitive to even weaker interfaces, but which are sensitive to a very limited range of interfaces); the small size also enables it to be used with exceedingly small sample volumes, so that measurements can be made with minute quantities of precious sample, or in parallel arrays for high-throughput characterization and screening.

Various aspects of the invention may include the following: 1) Microfabricated microbutton probes with different magnetic moments, sizes, shapes, and surface functionalities in order to probe different surfactant layers at fluid/fluid interfaces, or bulk materials with different visco-elastic characteristics. This would be analogous to providing different rheometer bobs for a rheometer. 2) Service whereby microbutton probes are custom-designed with different surface chemistry, shape, magnetic moment, size, etc. 3) An electromagnet array for microrheometry: two- or four-pole electromagnet array designed to allow a surfactant monolayer, a solution of soluble surfactant, a surfactant monolayer between immiscible liquids, or a sample volume of bulk material into the center of the poles. 4) Full interfacial microrheometry setup, including Langmuir trough for insoluble surfactants, sample holders for soluble surfactants, surface pressure sensor, electromagnet array, microbutton probes, driving and tracking software, and microscope. 5) High-throughput rheometry screening, where electromagnet arrays integrate with multi-well plates, thereby allowing automated rheological measurements of many samples, each with volumes of microliters or less.

Based on the teachings and principles of the present invention, other types of microbutton designs— e.g. circular probes with 'teeth' - may be designed and used to facilitate other types of measurements, much like vane rheometers are designed to prevent slip between the material and the rheometer surface. In addition, the range of magnetic moments and microbutton materials may be expanded to make them compatible with a greater range of interfaces (both in terms of stiffness and solvents). Also, alternate probe shape/chemistry may be used to enable its inclusion into lipid bilayers.

The techniques of the present invention allow precise, quantitative measurements of interfacial viscoelasticity that are more sensitive by a factor of 100 than existing commercial measurements; it allows linear and nonlinear measurements; it can use extremely small sample areas (i.e., on the order of 1 cm² down to 0.01 cm²) which allows measurements using just a very small amount of a precious sample, both for surfactant interfaces and for bulk materials. A key contribution of the present invention is to enable the fabrication of novel microrheological devices. According to the teachings of the present invention, a ferromagnetic and amphiphilic microbutton probe may be fabricated by a particular novel combination of using photolithography to make structures out of photoresist and evaporating magnetic metal layers onto these photoresist structures. Significantly, the ferromagnetic particles are made and lifted off the wafer, without all the other evaporated magnetic material also lifting off. Furthermore, the fabrication method is able to make these disks with "buttonholes" that enable the disk orientation to be tracked during use. The details of these two significant features are described in the attached appendices. The combination of shape anisotropy (via buttonholes), ferromagnetism (without junk nickel around that causes aggregation, etc.) and amphiphilicity (via surface chemistry) provides a new and advantageous microbutton probe for microrheology measurements. Key innovative features of the invention include the following:

1) A fabrication method that allows ferromagnetic layers to be incorporated into the probes, while allowing holes within the probes, and allowing the probes to be lifted off the wafers without the "extra" ferromagnetic junk layer being lifted off.

2) Combining ferromagnetism, shape anisotropy and surface chemical modifications to make a trackable, torqueable, interfacially active probe.

3) The design and implementation of such a probe for use in interfacial rheometry. Beyond the use of these probes to visualize the dynamics of surfactant monolayers, a key advantage of these probes is for quantitative measurements of the shear rheology (viscoelasticity and nonlinear rheology) of surfactant monolayers. This design has many advantages over other designs currently in the literature: a) It is very small, so very sensitive, b) It is perfectly round, so excites pure shear flow, c) It is perfectly round, so introduces no contact line motion, d) it is very small, so insensitive to inertia.

4) According the present technique for microrheology, the button probe is torqued into rotation, rather than forced into translation, which avoids compression/expansion in the front/rear of translating probes. The button probe may, of course, be forced into translation as well by imposing a magnetic field gradient rather than a uniform field.

5) Linear viscoelastic moduli (small oscillations at different frequencies) and nonlinear rheological measurements (large or infinite strain, at different strain rates, and "creep" (constant torque) to measure yield stresses, strain hardening or softening, shear-thickening or thinning) are all possible.

6) The ability to actively torque means a much wider range of surfactant films or complex bulk materials can be interrogated than with passive (Brownian) probes.

7) The optical nature of the rheometry measurement naturally lends itself to visualizing mesostructural dynamics within the films, if the mesostructure can be visualized.

8) The interfacial placement of the probe enables rheological measurements on opaque materials, both for surfactant films or bulk materials, using side-illumination.

Embodiments of the present invention overcome various problems in the art. For example, others have made ferromagnetic nanorods, which can be torqued and used as microrheo logical probes. This does not have the "pure shear" advantage that the present microbutton probe does, its quantitative interpretation is far more complex (and perhaps impossible, depending on exactly what torque is used), it introduces compression and extensional deformations into the monolayer, and its use of large amplitude rotations can disrupt and destroy the interfacial meso-structure responsible for the rheology of interest. Much larger needles have been developed; however, compared to the present microbutton probe they are not as sensitive (larger), may sink at concentrated surfactant monolayers (because capillary forces are not strong enough to keep heavy needles afloat), and are more susceptible to fluid and probe inertia.

In one aspect, the invention provides a rheological microprobe consisting of a microdisk which may be circular or non-circular, less than 500 microns in diameter, has a shape anisotropy created by holes through the microdisk, and is composed of a ferromagnetic material. The surface of the microdisk is amphiphilic due to surface chemical modifications. These features allow the microprobe to function as a trackable, torqueable, interfacially active rheological microprobe.

In another aspect, the invention provides a method for fabricating a rheological microprobe. The method includes using photolithography to create microdisk structures out of a photoresist layer, where each of the microdisk structures is circular or non-circular, less than 500 microns in diameter and has a shape anisotropy created by holes which may be circular or non-circular. The method also includes evaporatively depositing on the microdisk structures ferromagnetic metal layers, and depositing surface chemical modification layers on the microdisk structures to make the microdisk structures amphiphilic.

In yet another aspect, the invention provides an apparatus for interfacial microrheometry of insoluble or soluble surfactants and for the microrheometry of microliter (or smaller) sample volumes of bulk materials. The apparatus includes a sample holder suitable for containing a sample and microprobe disk (with different sample holders for insoluble surfactants, which are integrated into a Langmuir Trough, than for soluble surfactant or bulk materials, which require small volumes with a planar interface), an electromagnet array comprising at least one pair of electromagnets positioned on opposite sides of the sample holder, a microscope imager for bright-field visualization of the microprobe disk, and a microprocessor connected to the electromagnet array and microscope imager for driving the electromagnets such that a torque is exerted on the microprobe disk and for processing images from the microscope to determine an orientation of the microprobe disk.

BRIEF DESCRIPTION OF THE DRAWINGS

Fig. 1 shows a schematic of a method for fabrication of microbuttons and images thereof.

A) Photolithographic process for the microfabrication of ferromagnetic particles. SU- 8 particles are initially made photolithographically atop a sacrificial layer. An 0₂ plasma etch removes the bare sacrificial layer between the particles. A ferromagnetic nickel layer is evaporatively deposited on SU-8 particles, followed by a gold layer, and the entire wafer is immediately dunked into a solution of thiol-terminated molecules to form a self-assembled monolayer. Chemical etching and sonication removes the sacrificial layer, releasing the particles but not the metal between the particles.

C) Image of 20 μm-diameter ferromagnetic microdisks with two buttonholes, showing different surface functionalities on each side (scale bars: 20 μιη).

D) Image of magnetic 4 μιη-side triangles (scale bar: 20 μιη).

Fig. 2. Graphs showing properties of microdisks.

A) M-H curve for a batch of 20 μm-diameter microdisks with 150 nm Ni layers, fabricated following the procedure in Fig. 1A. The ferromagnetic response is clearly evident: hysteresis, coercivity, and remnant magnetization.

B) Coercivity of ferromagnetic microdisks as a function of evaporated Nickel film thickness. C) Saturation magnetization of nickel thin films as a function of thickness (circles), compared with and expectations based on equivalent volumes of bulk nickel (solid line). Fig. 3: Ferromagnetic 'microbutton' probes.

(a) Ferromagnetic Janus (two-faced) 'microbutton' probes are fabricated photolithographically from SU-8 photoresist, evaporatively coated with ferromagnetic (here nickel, but may be any ferromagnetic material that can be deposited) and gold layers, and made amphiphilic by depositing alkane-thiol or fluorocarbon-thiol monolayers on the gold.

(d) Zoomed-in view of one row of fabricated ferromagnetic micro-buttons in an array and 1-2 micron thickness of each disk; scale bar is 5 microns.

Fig. 4: A schematic drawing of an apparatus for implementing the measurement technique: a Janus ferromagnetic microbutton is placed within a surfactant layer, where two orthogonal pairs of computer-controlled electromagnets exert a defined torque (stress) on the microbutton, whose rotation (strain) is recorded with bright- field microscopy, along with simultaneous fluorescence video microscopy of the monolayer.

Fig. 5: Schematic flow chart of a measurement technique: A computer using a LabVIEW program generates a programmed voltage sequence, which is either amplified through a linear amplifier (Oscillatory or steady rotation mode) or a DC power supply (Creep mode) to drive high electric currents (~1 A) through electromagnet coils. The resulting field B(t) exerts a torque on a ferromagnetic microbutton probe, which exerts a shear stress on the interface. A CCD camera records the rotational response of the microbutton to the applied torque in order to determine the orientation by tracking two buttonholes in real-time using a LabVIEW program. Fluorescent video camera records simultaneous images of an interface as it is being sheared. Fig. 6 illustrates an apparatus according to an embodiment of the invention.

(A) A custom-built Langmuir trough with two symmetrically compressing barriers to control the surface concentration of insoluble surfactants. A Wilhelmy plate is used to measure the surface pressure.

(B) Two pairs of aligned electromagnets are orthogonally placed to generate a uniform field in any direction in the plane.

(C) A aluminum cone with two side slits to suppress stray convective flows. These thin slits allow surfactants to freely move in and out of the cone, while pinning the air- water interface along the rim.

Fig. 7. Sample holder for soluble surfactants and liquid-liquid interfaces. An aluminum cone with 5 -mm circular hole is inserted between two pairs of electromagnets and used to form a planar interface, either for liquid-air systems (soluble surfactants and bulk solutions) or liquid-liquid systems (e.g. oil-water).

Fig. 8: A Schematic diagram of a custom-built microscope that is capable of simultaneously visualizing florescent molecules at the interfaces and bright-field visualization of microbuttons. FC(Fluorescence camera), DM(Dichroic mirror), EM(Emission filter), EX(excitation filter), BFC(Bright field camera), M(mirror), CL(Collimator), OBJ(Objectives), BFL(Bright field lamp), TL(Tube lens), ML(Mercury lamp). A commercial microscope with similar capabilities could also be used. Fig. 9: Output from a program written in Lab VIEW to track two buttonholes on the microbutton in real-time.

(A) The original bright field image, which is convoluted with black square to locate the center of the microbutton.

(B) The center of the microbutton found in (A) is shown with a white dot. (C) The image is cropped to the shape of the black square in (A), thus including two holes.

(D) The image in (C) is convoluted with a small white square to locate the centroid of the button-holes.

(E) Hole centers are marked with black dots.

(F) The orientation of a microbutton at the air/glycerol interface is tracked for 80 seconds to watch fluctuations. The glycerol is viscous enough for Brownian motion to be ignored. Fig. 10 illustrates the orientational displacement of a microbutton following a suddenly applied magnetic field. We measure the microbutton orientation 9(t) in glycerol, a purely viscous liquid with a known viscosity, upon applying a known DC field (B) to align the magnetic moment direction to the magnetic field direction. The solid line represents theory 9(t) = 2tan^_1[exp(-mBt^)], and the squares are measurements from which |m| is determined. The orientation of the microbutton exponentially approaches the direction of the magnetic field.

Fig. 11 illustrates aspects of rheological measurement according to an embodiment of the invention, showing the theoretical computation used to determine surface rheological quantities from the measured rotational resistance of the microbutton..

(a) Definition sketch: a disk of radius a and height 2h sits at the interface of a viscous fluid with viscosity η, covered by an infinitely thin interfacial layer with surface viscosity η₈(ω). A no-slip boundary condition is enforced at r = 10a to mimic the u → 0 as r→∞ far-field condition.

(b) Rotational resistance ζ, normalized by the rotational resistance of an infinitely thin disk in a bulk viscous fluid, plotted against the non-dimensional Boussinesq number Bo = r|_s/(ar|), for disks of different height/radius ratios h/a. At low Bo, ζ_Γ becomes independent of Bo (and therefore is insensitive to η₈). At high Bo, however, disks of all aspect ratio have the same rotational resistance ζ_Γ = 47ir|a³Bo. In our experiments, h/a = 0.1 and the complex resistance ζ_Γ7(4πτ ³) ~ Bo^* for |Bo | > 1.

Fig. 12 illustrates the relative error made in approximating the measured (complex) rotational resistance ζ_Γ to correspond to a complex surface viscosity η₈ = ζτ /½a . Plotting (ζ_Γ - ½a η₈ )/4na r\_s against |Bo | for various degrees of viscoelasticity (phase angle δ = 0 corresponds to a viscous monolayer, and δ = π/2 corresponds to a purely elastic monolayer) reveals this simplest approximation to be within a factor of two for |Bo^*| > 1, and within 30% for |Bo^*| > 4.

Fig. 13 illustrates history-dependent linear viscoelasticity.

(a) Linear visco-elasticity of as-prepared liquid-condensed dipalmitoylphosphati- dylcholine (LC-DPPC) monolayers at air/water interfaces ( Π= 9±1 mN m ¹ ). Surface shear moduli indicate a dominant low-frequency elastic response, G_s'(oo)>G_s"(oo) down to 0.1Hz, crossing over to a primarily viscous response,

G_S"(CD)>G_s'(CD) above 5 Hz.

(b) Direct visualization of the DPPC monolayer reveals the elastic response to result from a network of interlocked LC domains that cannot deform without deforming neighboring domains (scale bar, 20 μιη).

(c) After forcing a large-amplitude microbutton rotation, LC-DPPC appears viscous G_s"(oo)>G_s'(oo) at all frequencies,

(d) Visualization reveals the creation of a 'slip line' (white arrow) within the monolayer, which enables non-cooperative deformation under shear (scale bar, 20 μιη).

Fig. 14 illustrates how a fractured monolayer heals and highlights the ability of ferromagnetic microbutton probes to measure aging and recovery of rheological properties.

(a) Initially, domains around a microbutton form a rough boundary with neighboring domains. (b) Forcing a large-amplitude rotation smooths the boundary to a nearly circular slip line, reduces the linear viscoelastic response 20-to 100-fold (f) and eliminates the elastic response.

(c, d) Line tension progressively roughens the slip line, with an accompanying increase in surface shear modulus (f).

(e) Overlaying initial (red) and final (blue, t = 60 s) images reveals the domain structure has healed almost completely, highlighting a remarkably persistent memory (scale bar, 20 μιη).

(f) Graph showing reduction of linear viscoelastic response 20-to 100-fold.

Fig. 15 illustrates the ability of ferromagnetic microbutton probes to measure surface yielding and yield stresses.

(a) shows steady rotation at 5 Hz,

(b) shows rotation at 10 Hz (b).

(c) shows rotation at 30 Hz. In (a)-(c), the monolayer is divided into two regions: an inner region that flows with the microbutton, deforms continuously and appears domain-free; and an outer region with domains that do not deform significantly. The radius Ry of the yielded region (white arrows) is set by the surface yield stress, T_s ^y. A torque balance gives T_app = 2KR_Y ²T_s ^y, consistent with (d).

(d) Graph illustrating the slope which gives τ₈ ^γ~10^~2 μΝ m^-1, ~G_s'(oo)/10 (Fig. 13) as expected by analogy with three-dimensional emulsions (scale bar, 20 μιη).

Fig. 16. Ferromagnetic microbuttons as probes of the rheology of complex interfaces between immiscible liquids, highlighting also strain sweep capabilities.

A) An image of a Janus, ferromagnetic microbutton (10 μιη radius) placed at the interface between pure water and decane. A monolayer of 1 μm-diameter polystyrene particles is then spread at the water-decane interface, and mutual electrostatic repulsion between colloids imparts local crystalline order. The viscoelastic moduli of the colloidal monolayer can be determined from the microbutton's rotation in response to an externally imposed oscillatory torque (B- C).

B) Graph showing that the linear viscoelastic modulus of the colloidal monolayer is primarily elastic (circles are G_s' and squares are G_s"), with a low frequency plateau.

C) Graph showing that increasing the amplitude of the oscillatory strain reveals a linear elastic response at low strains, with softening and yielding of the monolayer above a critical strain, which direct visualization reveals to correspond to particle hopping between lattice sites.

Fig. 17: Ferromagnetic microbutton probes measure linear viscoelastic moduli of bulk soft materials that quantitatively match traditional macroscopic rheometry.

The viscoelastic moduli of bulk solutions of xanthan gum are measured using microrheologically, using the ferromagnetic microbutton probes described here (empty symbols) and a traditional cone -plate rheometer (solid symbols). The measurements using the microbutton are in excellent agreement with those by a traditional cone-plate rheometer.

Fig. 18: Evolving visco-elastic properties of the interface of a Bovine Serum Albumin solution, which adsorbs from solution onto the interface and aggregates to form a surface layer whose rheology stiffens with time. This shows

(A) Time-dependent surface pressure measurements of a bovine serum albumin solution. The surface pressure saturates at about 20 minutes, but the rheological properties of the interface continues to stiffen, as shown in (B).

(B) Viscoelasticity of the interface of a bovine serum albumin solution as it ages, showing early viscous behavior, followed by a predominantly elastic response. Interfacial "nano-needle" measurements (Dhar et al 2010, Phys. Rev. Lett 104, 016001 , circles) are shown for comparison. Notably, the microbutton does not destroy the interfacial structure, revealing a significant elastic component, whereas the large-amplitude rotations required for the interfacial "nano-needle" destroyed the elastic structure of the film during measurements, and no elasticity is reported. This highlights the non-disruptive nature of gentle oscillatory motions of the microbutton,

Fig. 19:

(A) Isotherm of palmitic acid (PA) exhibits three different phases: gas and liquid expanded phases for extremely low pressures, liquid condensed phase (tilted) for low pressures (up to ~ 24 mN/m), and solid phase (untilted) for high pressures. (B) Rheological properties of PA as a function of surface pressures, comparing with measurements using macroscopic scale needles. For a liquid condensed phase, our microbutton measures η₈ even for very low surface pressure Π, whereas macroscopic needles cannot measure such low η₈. The viscoelasticity changes discontinuously at the tilt-untilt phase transition. For pressures above 24 mN/m (solid phase), the microbutton is capable of measuring the elasticity, whereas a steadily translating needle can not.

Fig. 20 shows creep compliance measurement of a DPPC monolayer. Using real-time measurements of the orientation of the microbutton, two pairs of electromagnets are used to apply a magnetic field that rotates with the disk, to be perpendicular to the magnetic moment of the disk and therefore to impose a constant torque.

(A) shows a creep recovery measurement. The creeping strain (rotation) of a microbutton in a DPPC monolayer is measured in response to a constant imposed torque; at 300 s the torque is turned off and the microbutton counter-rotates due to the elastic nature of the monolayer.

(B) shows a yield-stress measurement enabled by the creep compliance mode of operation. The effective instantaneous viscosity is measured as a function of time for different (constant) applied torques. When the torque corresponds to a stress below the interfacial yield stress, the instantaneous surface viscosity diverges when the microbutton rotation ceases. Above a critical torque, the instantaneous surface viscosity drops to a very low value and the monolayer yields. Fig. 21 illustrates data related to the characterization of the electromagnets.

(A) Magnetic field amplitude, as a function of DC current through electromagnet coils, measured using a Hall probe.

(B) The current through the electromagnetic coils following a step increase in voltage, indicating 1.5 ms inductance time.

(C) A power spectral density of electric current, established under a random applied voltage, and fitted with A/(l + τω²) resulting in the inductance time τ=1.8 ms.

(D) A 5 Hz oscillatory magnetic field is directly measured using Hall probe while recording applied current as a function of time, and fitted with sinusoidal waves. Time lag between the current and the magnetic field is 1.3 ms consistent with the inductance time measured in (B) and (C) independently.

Fig. 22 is a graph illustrating LED light intensity measured by applying 10 Hz oscillatory currents as a function of time to find the time lag between video microscopy and electric current. To find maximum intensity, we fit the intensity of the light, emitted by LED, with the Gaussian, and fit the current with a sine wave function. We find a 1.6 ms time lag between them.

DETAILED DESCRIPTION

The following description provides details of embodiments of microprobes according to embodiments of the invention, together with details regarding methods for making and using such probes. We begin with a discussion of the probes and methods for fabricating them, followed by the theoretical principles, apparatus, and methods related to the use of these probes for microrheological measurements.

Microprobe Fabrication

The ferromagnetic microbutton probes play the central role in the microrheology techniques of the present invention. The active interfacial microrheology technique of the present invention makes use of probes that are 1) small, yet visible under optical microscopy (10 to 100 μιη); 2) ferromagnetic, so as to enable external forces or torques to be applied; 3) amphiphilic, to ensure the probes physically absorb onto fluid/fluid interfaces; and that 4) have tracers to enable optical tracking of the orientation of the probe. We synthesize these probes using 2D photolithographic techniques, enabling multiple functionalities to be incorporated by a layer-by-layer addition onto any desired shapes.

Work by the present inventors, US. Patent Application 61/584475 filed 1/9/2012, which is incorporated herein by reference, describes the detailed fabrication procedure. Briefly, a sacrificial layer (Omnicoat, Microchem) and photoresist (SU-8, Microchem) are spin-coated on the wafer, and then exposed to UV through a patterned photomask to crosslink the photoresist in the desired microbutton shape. Ferromagnetic functionality is then imparted by evaporative ly depositing a thin (10- 300 nm) magnetic layer, typically nickel or cobalt, but any magnetic material that can be evaporatively deposited will be possible. A 10 nm gold layer is then directly deposited onto the magnetic layer, which allows us to modify their hydrophilicity of the top surface with a self-assembled monolayer of thiol-terminated ligands. (We typically use thiol-terminated fluorocarbons). In this way, we can tune surface chemistry to ensure an amphiphilic character for any particular interface.

According to one embodiment, photolithography is used to fabricate micron-scale, ferromagnetic, amphiphilic 'microbutton' probes. Briefly, a 4 inch diameter silicon wafer is cleaned with piranha solution, and a 200 nm sacrificial layer (Omnicoat, Microchem) is spin-coated onto the wafer at 1,000 r.p.m. for 30 s, followed by 1-m thick photoresist (SU8-2001) at 3,000 r.p.m. for 30 s. After baking the photoresist at 95 °C for 1 min, ultraviolet light is exposed through a Chrome mask using a 5X stepper (GCA Autostep 6300 i-line). After developing the photoresist, the wafer is exposed to 0₂ plasma for 2 min to remove the sacrificial layer. A 150-nm nickel layer is then evaporatively deposited onto the photoresist, followed by a 10-nm gold layer. The wafer is then soaked in lH,lH,2H,2H-perfluorooctanethiol (Sigma) for 8 h to promote the formation of a self-assembled monolayer on the gold surface. Finally, gentle sonication in water releases the microbuttons by dissolving the sacrificial layer. The schematic photolithographic process for production of multifunctional microparticles according to one embodiment of the invention is shown in Fig. 1. Initially, a 200 nm sacrificial layer (Omnicoat, Microchem) is spin-coated on a 4-inch silicon wafer, followed by a one -micrometer layer of photoresist (SU-8, Microchem). The bilayer structure is then baked at 95 °C for 1 min, and photoresist exposed to UV light through a patterned chrome photomask for 3 sec. After developing the photoresist, the wafer is exposed to an oxygen plasma at 0.19 Torr for 2 min, which removes the exposed sacrificial layer but not the sacrificial layer buried under the photoresist structures. (This etch step is critical, as it ensures that the metal layers that are subsequently deposited will remain on the wafer, even while the desired multilayer microparticles are lifted off. By preventing the microparticles from aggregating with magnetic metal fragments, it eliminates the need for an additional process to separate the microparticles from the magnetic metal fragments.) Ferromagnetic functionality is then imparted to the microstructures by depositing a magnetic layer, typically 10-300 nm of nickel, cobalt or iron, but can be any ferromagnetic material that can be deposited. A 10 nm gold layer is then directly deposited on the magnetic layer which gives rise to a Janus character and allows for facile functionalization with self-assembled monolayers (SAMs) of a wide variety of thiol-terminated molecules. Immediately following Au deposition, the wafers were submerged in a 1 mM solution of lH,lH,2H,2H-perfluorooctanethiol (Sigma) in ethanol for 8 hours. We then release the microdisks from the wafer by gently sonicating in deionized water. The magnetic character of the microparticles greatly facilitates subsequent purification and collection. EMPIRICAL STUDIES OF MICROPROBES

To study the magnetic properties of our microprobe structures more quantitatively, a wafer of fabricated microdisks was diced after metal deposition into 3 mm by 3 mm sections, each of which contains approximately 10⁴ ferromagnetic microdisks. The in- plane magnetic properties of microdisks were then measured using a SQUID (MPMS 5XL, Quantum design). Fig. 2(a) shows the ferromagnetic properties of a representative batch of 20 um-diameter microdisks incorporating a 150 nm ferromagnetic layer of nickel. It should be noted that the ferromagnetic properties of the microparticles are retained, with hysteresis and saturation of magnetization being observed.

The microrheological application described above requires a fixed permanent magnetic moment that does not reorient under externally applied fields. The coercivity represents the magnetic field required to demagnetize materials via reorientation of the magnetic moments of the domains. In general, bulk nickel has extremely low coercivity (~1 Oe). The large internal stresses generated during evaporative deposition of Ni films, however, increase coercivity by up to two orders of magnitude (Fig. 2(b)). The inventors have experimentally studied how the in-plane saturation magnetization of the fabricated thin Ni films changes with thickness. Unlike bulk materials, whose saturation magnetization (Ms) depends linearly on their volume, the saturation magnetization of thin films shows different behavior (Fig. 2(c)). To compare thin film measurements with predictions based on bulk properties, we estimate Ms based on the volume of the thin film. The saturation magnetization Ms of sufficiently thin Nickel films is higher than one would expect based on bulk properties, since magnetic domains prefer their moments to be oriented in the plane of the thin film, rather than randomly as in bulk. However, out-of-plane magnetic moments of the domains become favorable as the film thickness increases to minimize the magnetostatic energy, thus giving rise to smaller in-plane magnetization than bulk. APPARATUS

An advantage of monolayers of insoluble surfactant at fluid interfaces is that the surfactant concentration can be controlled by using barriers to change the surface area. The large interfacial areas typical in Langmuir troughs, however, allow strong convective flows at the interface. Such convective flow rapidly remove any probes from the ~ 200 x 200 μιη² field of view of the microscope, making measurements impossible. Accordingly, one embodiment of the invention provides a Langmuir trough (Fig. 6A) to control surfactant concentration while also incorporating several mechanisms to reduce this flow. First, we position a small aluminum conical cylinder in the center that pins the interface at the 5 mm diameter rim of the cone (Fig. 6C). Narrow slits (- 0.5 mm) are notched into cylindrical walls to allow surfactant to flow in and out, with the cylinder wall effectively suppressing convective flows. Second, the subphase is kept to a depth of less than 5 mm to further suppress subphase flows. Third, the viewing area of the trough is itself isolated from the larger reservoirs and barriers by two narrow 5 mm width channels.

Gibbs monolayers of soluble surfactant whose surface pressure Π and concentration Γ are controlled with bulk concentration, and bulk materials do not require a Langmuir trough. For such materials, we employ a sample holder with 5 mm diameter circular hole, centered between 4 electromagnets (Fig. 7). This sample holder is also effective for forming monolayers at liquid-liquid interfaces, like oil-water interfaces with particles or surfactants adsorbed.

Electromagnet systems and control

Two pairs of aligned, independently controlled electromagnets are used to generate magnetic fields of specified magnetic fields in any planar direction. Two independent signals are generated for each set of electromagnets, using two channels of a digital analog converter (National Instrument, PCI-6933). The signals are passed through a linear audio amplifier (Sony, HDMI 259), which amplifies the voltage by a factor of two, and secures up to 4A in current. This current is passed through the electromagnet coils, wrapped around 5 mm diameter pure iron (> 99 %) core, generating a uniform magnetic field between the two electromagnets. The current passing through each set of electromagnet coils is recorded with two data acquisition boards (National Instruments, USB-6009). To relate the current to the magnetic field, we measure the magnetic field as a function of current using Hall probe (F.W. Bell Inc.). As seen in Fig. 21a, the magnetic field increases linearly with electric current for all currents imposed. This linear relation is used to relate the actual magnetic field from measured electric current. Having determined the relationship between the static magnetic field and applied current, we determined the response time of the electromagnets. Fig. 21b shows the response of the electric current of the electromagnets following a step change of applied voltage. The inductance time of the electromagnetic circuit is seen to be 1.52 ms, so that frequencies below ~ 100 Hz can be accurately imposed. As a complementary technique to confirm this response time, we applied a random signal and measure the power spectral density. The expected power spectral density for an LR circuit is expected to be psd = (1 + τ²ω²)^~\ Fitting our measured PSD to this form reveals, the time lag τ is 1.78 ms, in good agreement with the LR time. Finally, we directly measure the oscillatory magnetic field with a Hall probe, and compared with simultaneous measurements of the applied current, as shown in Fig. 2 Id. Fitting the results of each to a sine wave reveal a time lag of 2.01 ms, again in good agreement with the two other measurements.

Synchronization of strain measurements (camera) and stress measurements (electromagnets)

In measuring the linear viscoelastic moduli under oscillatory stress, it is important to accurately measure the phase lag δ. Even small errors in δ provide errant results, particularly when δ approaches either zero or π/2, corresponding to the limits of nearly elastic or nearly viscous materials, respectively. Although the applied electric current has been shown to be synchronized with the established magnetic field to within 2 ms as described above, we nevertheless determine the synchronization of video microscopy with applied field. To do so, we connect a light emitting diode (LED) in series with the electromagnet coils, so that the brightness viewed in video microscopy corresponds to current through coils. Since an LED has a very short response time (< 1 ms), its intensity varies in phase with the electric currents. We place the LED under the microscope, and measure the intensity of the emitted light as a function of time while applying oscillatory currents. Fig. 22 shows 10 Hz oscillatory electric currents measured as described above, along with the intensity of light emitted by LED. The phase difference between current and light intensity is < 2 ms, which gives the maximum possible phase lag is < 4 ms. We thus expect credible measurements up to 10 Hz, with error < π/20 (rad) in phase angle.

SIMULTANEOUS INTERFACIAL VISUALIZATION DURING MEASUREMENTS

A key feature of our technique is that we can visualize the interface while it deforms. To do so, we built a microscope (Fig. 8) that employs a dichroic mirror, where light at the emission frequency of a fluorophore is passed to a fluorescent camera, and the frequency of bright field light for visualizing the buttonholes is reflected to a CCD camera. For fluorescence visualization, light from a mercury source is passed through an excitation filter, and reflected by a dichroic mirror, and illuminate the interface under investigation. Emitted fluorescence light passes through the first dichroic mirror, is reflected by a mirror, and passes through the second dichroic mirror to be imaged on a fluorescence camera (Andor Ixon). A second light source, whose wavelength does not overlap with the emission spectra of fluorescent dyes, is sent through a color filter. This bright-field light passes the interface, and reflected at the second dichroic mirror to be imaged on a CCD camera (BFC). The dichroic filters are interchangeable if needed to change fluorophores.

Image Analysis

To determine the microbutton orientation in real time from bright field image, we wrote a LabVIEW program to track the centroids of the buttonholes, adapting techniques used for colloidal particle tracking. Fig. 9 shows the procedure for the tracking algorithm. Starting with a digital image, wherein each pixel has a brightness between 0 and 255, we begin by locating the darkest pixel (assumed to be on the microbutton). We then crop a square twice as big as the disk diameter, centered at the darkest pixel to restrict the search area. To find the center of the disk, we convolute a black square whose side is 0.75 times the disk diameter with this cropped image. The maximum of this convolution represents the disk center. To locate a buttonhole, we then convolute a white square, whose size is 3/4 the buttonhole diameter, with the cropped image, the maximum of which represents the centroid of a buttonhole. To locate the other buttonhole, we "black out" the buttonhole we have just identified with a black square, and repeat the process. To confirm our tracking accuracy, we track orientation of a single stationary microbutton for 80 seconds, giving a value 1.1 rad, resulting in angular accuracy ± 0.005 radian (Fig. 9F).

Determining magnetic moment of microbutton probes

To determine the magnetic moment of the microbutton probes, we measure their response to applied fields of known magnitude, in purely viscous Newtonian liquids of known viscosity. Fig. 10 shows the response of a representative microbutton in glycerol to a suddenly applied constant magnetic field. We place a microbutton at glycerol/air interface, and measure the angular displacement of the microbutton as a function of time following the known DC magnetic field. For purely viscous materials, a response of the probe to constant field is governed by

where ζ = 4πτ ³ for an infinitely thin disk half-submerged in a viscous fluid, η is fluid viscosity, and a is disk radius. This equation can be solved exactly, giving θ(ί) = 2tan^~1[exp(-mBt^)]. With known viscosity η and magnetic field B, the magnetic moment can be determined by fitting with the solution. We measured 29 different microbuttons each with 150 nm nickel layers and found 4.0±0.9 emu. Methods and apparatus for rheological measurements

According to one embodiment, a Janus microbutton is placed within a surfactant layer, where two orthogonal pairs of electromagnets are used to exert a torque on it (Fig. 4). The applied magnetic field (and thus torque, or stress) is imposed, and the rotational displacement (and thus strain) is measured, by tracking the angle made by two "buttonholes" on the probe. The relationship between torque and rotation gives the rotational resistance, from which rheological properties can be computed.

Fig. 5 shows a general flow diagram for the experimental procedure. The magnitude (|B|) and the direction φ of the magnetic field are imposed externally, and the orientation Θ of the probe is determined. The direction of the magnetic moment m of the microbutton must be known prior to measurements. We thus initialize all measurements by using an external field to align m, or by using two pairs of electromagnets to impose the magnetic field in a specified direction relative to the direction of the magnetic moment..

We have developed three modes of operation: an oscillatory stress, a controlled stress, and a controlled strain mode. 1) Oscillatory mode. A small amplitude, oscillatory stress enables the measurements of

frequency-dependent linear viscoelastic surface moduli G_s ^*(co). First, we set an oscillatory frequency and voltage amplitude to achieve a desired stress. This signal is passed through a high-current linear amplifier to drive a current through the coils wrapped around an aligned pair of electromagnets to establish an oscillatory magnetic field, oQ^WJt. We apply the magnetic field perpendicular to the magnetic moment m of the microbutton, to exert an oscillatory torque,

T₀e^icot = m x B e^f0" ~ mB₀e^iia', (3) on the probe. Using bright-field microscopy, we track the angle ^^ωί+δ) made by the two "buttonholes" on the probe in real time as it rotates in response to the oscillatory torque. The frequency-dependent rotational resistance ζτ (ω) is obtained by

2) Controlled stress (creep) mode. Integrating a real-time feedback control enables a constant torque to be applied, which gives an interfacial creep compliance measurement. Specifically, tracking the orientation 9(t) in real time, allows the angle (p(t) of the applied field to be modulated according to cp(t) = 9(t) + π/2, so that the applied torque is constant T = mB₀. Using this method, we have measured the yield stress of a liquid condensed phase of a phospholipid monolayer. (Fig. 20)

3) Controlled strain mode. (Fig. 15) Lastly, we can force steady rotation at a given frequency by applying a large rotating magnetic field Bo with angle cp(t) = Ωί. Measuring the orientation Θ of the magnetic moments, then, gives the angular lag δ between the applied field φ and the magnetic moment Θ. The steady rotational resistance is then given by ζτ (Ω) = (|ηι|Β₀8ίηδ)/Ω. Since δ is difficult to measure precisely, this is a less precise method compared to the controlled stress mode.

Theoretical

RELATING ROTATIONAL DRAG TO RHEOLOGICAL PROPERTIES

In our experiments, we measure the rotational drag coefficient of the microbutton probe, which relates the angular displacement e^mt+s to the applied torque ToQ^mt according to

The rotational drag coefficient ζ_Γ(ω) is generally complex and frequency-dependent, and depends upon the visco(elastic) moduli of the surface and bulk phases. Just as the Stokes flow around a translating sphere must be solved to determine the steady drag of a small sphere (F_D = 67ir|aU), the Stokes flow around a rotating disk in a surfactant monolayer must be solved to relate ζ_Γ(ω) to η₈ (ω) and η, which are in turn related to the complex surface shear modulus G_s (ω) = G_s' (ω) + G_s"(co) according to η₈ (ω) =

To do so, we solve for the subphase velocity field around a disk of radius a and height h undergoing oscillatory rotations at the interface, and treat the (viscoelastic) interface as imposing a boundary condition on the subphase. This allows us to determine the torque on the disk, which then gives ζ_Γ. The interface (surface viscosity η₈) is suspended on the subphase (viscosity η) and the disk is rotating with angular velocity Ω.

When Bo»l, the rotational resistance becomes completely dominated by the interfacial rheology, and the subphase effectively decouples from the interface. A good approximation for the drag coefficient can therefore be obtained by solving the 2D Stokes equations, -Vp_s + r|_sV²u_s= 0 and V-u_S=0. Symmetry requires the flow be purely in the θ-direction with no aximuthal pressure gradient, giving simply V_r ²u_e= 0, with boundary conditions u_e = ΩΆ at r = a and u_e = 0 as r→∞. The resulting velocity field is given by u_e = Qa²/r, so that the torque on the disc is given by T

=2Kar|_s(a3_r(u_Q/r))=4Ka²r|_sQ revealing the rotational resistance to be ζ_Γ = 4na²r\_s.

NUMERICAL CALCULATION OF THE DRAG COEFFICIENT OF THE DISK WITHIN SURFACE-ACTIVE MOLECULES

In our experiments, a microbutton has finite thickness, giving an aspect ratio h/a = 0.1. In this subsection, we compute how the aspect ratio of the probes affects their rotational drag as a function of Bo. Rotational oscillations of the disk establish a swirling flow. The Stokes and continuity equations for the subphase are then given by

-Vp+r|V²u=p du/dt and V u=0 (6) In the absence of inertia— generally when pcoa²/r| « 1, meaning ω « (η/pa²)¹⁷² ~ 10 kHz frequencies— symmetry requires the flow be purely in the θ-direction with no pressure gradient, giving simply

r|V²ue =0 (7) with boundary conditions

U₀ =ΩΓ for 0<r<a on the disk (8) U₀=O as r,z→∞ (9) A two-dimensional viscous layer at the interface (with surface viscosity η₈), experiences a stress in response to in-plane shear r|_sV_RU₀. The surface Laplacian r|_sV² _RU₀ then exerts a stress on the subphase, giving a boundary condition on the subphase

r|_sV_RU₀ +η¾ιΐ_θ =0 for a<r<∞ at z=0. (10) Scaling lengths by disk radius a and velocities by Ωa yields a non-dimensional form for

the equations and boundary conditions (8-10)

η ν² ύ_θ = 0 (11) with boundary conditions

ύθ =? for 0<f<l on the disk (12) iie=0 as f,z→∞ (13) BoV² _ru₉ +δ_ζύβ =0 for 1 <f<∞ at z=0, (14) where the Boussinesq number

Bo = r|_s/r|a (15) appears naturally. We solve the Stokes equations (11) subject to boundary conditions (12-14) using boundary integral techniques, imposing a no-slip i¾ = 0 condition at f = 10 to enforce the far-field condition (13) approximately. We compute the viscous stresses

τ = ¾ΰθ (16) τ = Βο¾ ύ_θ (17) from the subphase and interface, respectively. The torque on the disk is then given by T = 2πηα³Ω [J_c τ(τ, z)dl(r) + Bo τ₈ ], (18) from which the rotational resistance

ζΓ =Τ/Ω (19) can be computed. Normalizing by the rotational resistance of an infinitesimally thin disk, fully immersed in a viscous fluid/air interface (ζτ = 4nr\a³), we compute a scaled resistance. One may then graph ζ_Γ as a function of Bo, for different disk thicknesses h = h/a. Notably, when Bo » 1 , the rotational resistance is dominated by the interfacial contribution, which does not depend on h and grows linearly with Bo. When Bo « 1, on the other hand, ζ_Γ becomes independent of Bo and depends upon h. A measured ζ_Γ which is significantly greater than 4nr\a³ thus indicates Bo » 1, enabling the corresponding Bo (and thus η₈) to be determined.

The geometry of the disk (i.e., the aspect ratio h/a) thus only affects ζ_Γ only for low- to intermediate Bo. Once Bo is high enough, the aspect ratio becomes largely irrelevant, since the drag from the interface— along the contact perimeter— is so much higher than the drag from the subphase. When Bo » 1, the subphase essentially decouples from the interface, with a velocity field given by

u₉(r;Bo»l^a²/r , (20) so that the torque on the disk is given by

T = 2nar\_s (a d/dr(y^/r) )|_r=a = 4ia² η₈Ω, (21) for a rotational resistance

ζ_Γ(Βο » 1) ~ ½a²r|_s, (22) or

ζ_Γ(Βο » l)/47ir|a³ = Bo (23) It is desirable to perform experiments in this limit, as the measurements are maximally sensitive to the interfacial rheology. While the preceding calculations were performed for purely viscous interfaces, the complex rotational resistance ζ_Γ (ω) for visco-elastic interfaces (with complex viscosity η (ω)) can be computed within the same framework. In that case, Bo becomes complex in (14). One thus finds ζκ (with real and imaginary parts) to depend upon Bo , also with real and imaginary parts. Measuring the amplitude and phase angle of ζκ , one can then determine Bo . In the limit |Bo | » 1, the analysis leading to (23) holds, with Bo in place of Bo. When |Bo| ~ 0(1), however, the relation between ζ_Γ and Bo is not simple.

RHEOLOGICAL MEASUREMENTS: VERSATILITY OF THE TECHNIQUES

Having described the design, strategy, calibration and operation of our technique, we now turn to a series of measurements to highlight key capabilities and applications. Figs. 13-15, 20 highlight experiments on insoluble phospholipid surfactants, spread as a monolayer at the water-air interface. They reveal that the monolayer dynamics of even a single-component monolayer of dipalmitoyl-phosphatidylcholine (DPPC), one of the primary lipids in lung surfactant and ubiquitous in cell membranes, can be far richer than ever expected. DPPC monolayers exhibit a disordered liquid-expanded (LE) phase at low surface pressures that transforms into a liquid-condensed (LC) phase with long-range orientational and short-range positional order (hexatic) at Π= 7 mN rrf ¹ at 20 °C. All experiments were performed at n=9±lmNrrf ¹ in the LC phase, just above the coexistence plateau. A small-amplitude, oscillatory magnetic field, Be^i<wi, applied perpendicular to the magnetic moment, m, of a microbutton suspended at the interface (Fig. 4), exerts an oscillatory torque T₀e^i<wi that drives a small oscillatory rotation θ₀6'(^{ωί+ 5}(^ω)). Measuring the buttonhole orientation along with the applied torque determines the probe's rotational resistance ζ(ω)=Τ₀6^{"ί 5(ω)}/ίωθ₀. When Bo»l, the real and imaginary parts of ζ(ω) are proportional to the surface elastic and viscous shear moduli, G_s'(oo) and G_s"(oo)=oor|_s(oo), respectively. Surprisingly, the DPPC monolayer had a primarily elastic response (G_s'~150 nN rrf¹) down to 0.1 Hz, indicating that the monolayer stored elastic energy, without appreciable relaxation, over 10-s time scales (Fig. 13a- b). Above 4Hz, however, G_s'(oo) and G_s"(oo) crossed, and the monolayer response was primarily viscous (r|_s~70nNsnT ^J).

Forcing a few large-amplitude (180°) rotations, however, changed the measured response dramatically (Fig. 13c-d). The DPPC monolayer exhibited a viscous (albeit frequency-thinning) response at all frequencies measured, without the elastic storage seen in a 'freshly prepared' monolayer. This dramatic history-dependence highlights the care required in interpreting dynamic measurements of monolayers. Our ability to visualize the monolayer while deforming it allowed us to unambiguously identify the microstructure changes responsible for this unusual behavior. The bright lines (high fluorescent dye concentration) in Fig. 13(b,d) show the boundaries of the irregularly shaped, interlocking ~10 ιη LC domains, which cannot slide past each other without deforming. DPPC monolayers in the LC phase respond to a weak applied stress with small elastic deformations of the domains, rather than rearrangement of the domains or the domain boundaries. Large-amplitude deformations, however, drive the monolayer out of its equilibrium microstructure (Fig. 13d). The domains deform enough that the boundary forms a continuous, almost circular slip line that effectively fractures the material. The microbutton and domains within the slip line rotate freely, with minimal deformation of the domains inside or outside of the slip line, eliminating the elastic response (Fig. 13c,d).

On returning the micro-button to its original orientation and removing the driving torque, the circular slip line roughens (Fig. 14), the smoothed grains heal to their interpenetrating configuration (compare Fig. 14a, e), and the original viscoelastic response (Fig. 14f) is recovered. When the slip line is formed (Fig. 14b), both G_s' and G_s" drop 20- to 100-fold (Fig. 14f), and the monolayer appears primarily viscous (G_S">G_S'). However, after shear stops, both G_s' and G_s" increase with time: the elastic response is recovered (G_S'>G_S") after 40 s, when the domain structures again interpenetrate (Fig. 14c). Even the detailed features of the grain boundaries and domains are largely restored after 150 s (Fig. 14e).

Significant domain deformation should require a stress of order G_s' as well: concentrated emulsions yield at applied stresses of -G'/IO. Monolayers respond to a steady torque (Fig. 15) by establishing two distinct regions: an outer region in which the interface does not flow, and an inner region that appears largely structure-free and freely flowing. The boundary radius, Ry , between the two is determined by the surface yield stress, τ₈ ^γ: the interface flows for r<R_Y where x_s c_s ^Y, but deforms elastically for r>Ry where τ₈<τ₈ ^γ. An interfacial torque balance relates applied torque to surface shear as T_app = 2KR_Y ²T_s ^y. Fig. 15d shows that T_app increases linearly with Ry², whose slope gives a surface yield stress τ₈ ^γ~0.01 μΝη ¹ of order G_s'/10, as expected. Fig. 20 shows an alternate method for measuring yield stress, using a constant-torque (creep) mode that does not require epifluorescent or Brewster-Angle Microscopy visualization of the surfactant monolayer.

By visualizing the structure of an interface while it is being deformed, our new technique provides an unprecedented ability to correlate structural deformations with rheological response. More generally, our technique can interrogate the dynamical response of a wide variety of fluid/fluid interfaces, of scientific, biological, industrial and technological relevance. For example, lipids, proteins and fatty acids can be added to systematically construct model monolayers of biological relevance, such as the lung surfactant monolayer.

Fig. 16 highlights an example measurement of the rheology of a particulate monolayer at the interface between two immiscible liquids. A ferromagnetic, Janus microbutton probe is made to rotate in an oscillatory fashion within a monolayer of polystyrene colloids. Both the microbutton probe and the colloids are adsorbed at the planar interface between clean, immiscible solvents (pure water and decane). Colloids within the monolayer experience a long-range electrostatic repulsion mediated through the decane, which gives rise to hexagonal crystalline order within the monolayer. Fig. 16B shows the frequency-dependent surface visco-elastic shear moduli of the colloidal monolayer itself— a low-frequency elastic plateau is observed, along with increasing visco-elasticity at frequencies above the diffusive relaxation time of colloids within their well. Additionally, Fig. 16C shows the weakening— and eventual yielding— of the colloidal monolayer for increasing applied strain, which can be directly correlated with the onset of lattice hopping by particles.

Viscoelastic bulk material: Xanthan gum

In addition to interfacial rheology, our technique also can be used to measure bulk materials. Typical rheology experiments require a few milliliters for measurements while this microrheological technique requires of the order 1 μΐ or even less of sample. To demonstrate the capability of bulk measurements, we measure linear viscoelastic moduli of 0.2 wt% xanthan gum solution in water, using both a standard rheometer and our microbutton (Fig. 17).

For microrheology experiments, we prepare a 0.3 wt% xanthan gum solution simply by dissolving xanthan gum powder in deionized water after stirring with a magnetic bar for 30 minutes, place a small volume of the solution into the sample holder, and deposit ferromagnetic microbutton probes on the interface. After waiting for 5 minutes, we measure the linear viscoelasticity as a function of frequencies shown in Fig. 17. We note that we are deliberative ly operating this experiment in the Bo « 1 limit.

For "macro" measurements, we use a cone and plate rheometer (ARES, TA instruments). We place a 0.2 wt% solution between the cone and plate. We first sweep strains to determine the range for which the measurements remain in a linear regime; for Xanthan gum, strains in excess of 100% are still linear. We then sweep frequencies while fixing the strain at 100%. For both macro- and micro- measurements, the viscous moduli is dominant for low frequencies, and the elastic moduli takes over at 2 Hz, in excellent agreement with microbutton measurements for both G' and G".

Viscoelastic Gibbs monolayer

We now turn to Gibbs monolayers of soluble surfactant, which are relatively simple to prepare by simply controlling bulk concentration. The interfacial concentration equilibrates according to an absorption/desorption isotherm specific to the surfactant. We demonstrate with Bovine serum albumin (BSA) solutions.

Bovine Serum Albumin (BSA)

Our calibration procedure (Fig. 10) follows a known procedure for measuring the large amplitude rotational response of nanoneedles for a film of Bovine Serum Albumin(BSA) at the air/water interface. In that procedure, it was found that the viscosity of BSA increases over time by orders of magnitude even after the surface concentration has saturated without any indication of elasticity. However, such large amplitude measurements cannot measure elasticity reliably: large amplitude deformations can disrupt and destroy the mesoscopic structures responsible for elasticity. To compare such results with measurements by our technique, we prepare 0.02 mg/ml BSA solution, and measure the surface pressure and the viscoelasticity as functions of time. Our measurements agree, but provide more information since our technique can measure the elasticity. The BSA film is initially viscous, and becomes increasingly elastic over time.

Viscoelastic Langmuir monolayer

Another kind of a monolayer in addition to Gibbs monolayers is a Langmuir monolayer, which is not soluble in water. It spreads and stays at the interface when the surfactant solution is spread using organic solvents. As a model Langmuir monolayer at the air/water interface, we study monolayers of palmitic acid (PA), which is a water insoluble fatty acid with a hydrophobic tail consisting of 16 saturated hydrocarbons, and carboxyl acid head group. We prepare a 1 mg/ml solution of PA in chloroform, which we spread using a microsyringe on a clean air/water interface in a Langmuir trough. The surface concentration Π is controlled by moving teflon barriers to change the area of the monolayer, and the surface pressure is measured as a function of the concentration using Wilhelmy plate (area/molecule).

Fig. 19 shows the equilibrium phase behavior of PA at 20 ± 2 °C. PA exhibits three phases at room temperature as pressure increases: a gas for Π ~ 0, a condensed liquid (tilted) L₂" phase for Π < 24 mN/m, and solid phase (untilted) for Π >24 mN/m. We measure G_s' and G_s" of the PA monolayer at 20 ± 2 °C as a function of surface pressures. For low pressures (L₂" phase), PA shows a primarily viscous response at 1 Hz. Furthermore, G_s" increases exponentially with surface pressure, as expected from the free area model. This is the two-dimensional analog of the (three-dimensional) free volume model, which postulates that the viscosity of a liquid increases exponentially with the inverse of the free area available to each molecule.

We compare our results with measurements using a macroscopic needle surface viscometer. As shown in Fig. 19B, our technique is more sensitive at low Π, since r|_s ^min = 10 ⁸ Ns/m is lower for microbuttons, compared with r\_s ^mm ~ 10 ⁶ Ns/m for needles. For intermediate surface pressures Π ~ 20 mN/m, our measured surface viscosity η₈ =G_s"/co agrees well with measurements using the needle viscometer. Moreover, we measure the discontinuous appearance of an elastic modulus just above the L₂"-solid phase transition. Steadily translating needles cannot measure the elastic moduli, although oscillatory needles could. Our recent experiments on a phospholipid shows that discontinuities in the value or slope of the moduli can indicate phase transitions which would otherwise be difficult to determine from static measurements.

Claims

1. A rheo logical microprobe consisting of a microdisk;

wherein the microdisk is less than 500 microns in diameter and has a shape anisotropy created by holes through the microdisk;

wherein the microdisk contains a ferromagnetic material;

wherein the surface of the microdisk is amphiphilic due to surface chemical

modifications;

whereby the microprobe is a trackable, torqueable, interfacially active probe.

2. The microprobe of claim 1 wherein the microdisk is circular.

3. The microprobe of claim 1 wherein the microdisk is non-circular.

4. The microprobe of claim 1 wherein the holes are circular.

5. The microprobe of claim 1 wherein the microdisk is less than 500 microns in diameter.

6. The microprobe of claim 5 wherein the microdisk is less than 200 microns in diameter.

7. The microprobe of claim 6 wherein the microdisk is less than 50 microns in diameter.

8. A method for fabricating a rheo logical microprobe, the method comprising: a) using photolithography to create microdisk structures out of a photoresist layer; wherein each of the microdisk structures is less than 500 microns in diameter, has holes and a shape anisotropy;

b) evaporatively depositing on the microdisk structures ferromagnetic and gold layers; and

c) depositing surface chemical modification layers on the microdisk structures to

make the microdisk structures amphiphilic.

9. The method of claim 8 wherein the microdisk structures are circular.

10. The method of claim 8 wherein the microdisk structures are non-circular.

11. The method of claim 8 wherein the holes are circular.

12. The method of claim 8 wherein each of the microdisk structures is less than 500 microns in diameter.

13. The microprobe of claim 12 wherein the microdisk structures is less than 200 microns in diameter.

14. The microprobe of claim 13 wherein the microdisk structures is less than 50 microns in diameter.

15. An apparatus for interfacial microrheometry of insoluble surfactants

comprising:

a trough suitable for containing an insoluble surfactant monolayer and microprobe disk;

a sample holder for containing soluble surfactants and liquid/liquid interfaces with surfactant;

a sample holder for containing small volumes of bulk solution;

an electromagnet array comprising at least one pair of electromagnets positioned on opposite sides of the sample holder;

a microscope imager for bright-field visualization of the microprobe disk;

a microprocessor connected to the electromagnet array and microscope imager for driving the electromagnets such that a torque is exerted on the microprobe disk and for processing images from the microscope to determine an orientation of the microprobe disk.