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--- lmn:isr:theory [30/12/2013 18:52] – davide.orsi
+++ lmn:isr:theory [31/12/2013 10:13] (versione attuale) – davide.orsi
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+==== Instrument design ====
+{{lmn:isr:isr.png |ISR setup}}
+The interfacial rheometer we developed is an upgraded version of the
+instrument described by Fuller, adapted to the study of photosensitive
+polymers. A sketch and two photographs of the instrument are shown in
+the figure aside.
+The instrument is build around the Langmuir trough described [[lmn:langmuir|here]]. Two parallel glass plates, each of them $10cm$
+long, define a channel used to constrain the needle position along the
+direction parallel to the short edge of the trough. The stainless steel
+needle is typically $13mm$ long and has a diameter of $0.35mm$; its
+surface is kept clean by immersion in chloroform. The needle is
+magnetized to saturation value before each experiment by contact with a
+permanent magnet.
+Two coils in Helmholtz configuration (diameter $D=32cm$, resistance
+$R=7\Omega$) are placed around the Langmuir trough: each of them carries
+a static current $I_0 = 1 A$. The static magnetic field generated by
+$I_0$ aligns the needle along the axis of the coils, as shown in the figure below. The whole system is oriented parallel to the North-South
+direction, so the Earth field does not affect the alignment of the
+needle. A sinusoidal current is added to one of the coils, producing the
+oscillating field gradient that moves the needle. A typical sinusoidal
+wave of frequency $0.25Hz$ and amplitude $I=1mA$ generates a force $F$
+of the order of $50nN$ on the needle, resulting in a displacement of
+about $10\mu m$.
+This “two-coils” design is a simplification of the design originally
+proposed by Fuller and coworkers, which makes use of two coils in
+Helmholtz configuration to generate the static magnetic field, and two
+smaller coils in anti-Helmholtz configuration as the source of the
+oscillating magnetic gradient that moves the needle. In the following,
+we show that the use of two coils do not alter significantly the
+behavior of the oscillating magnetic field gradient, with respect to the
+one obtained with four coils.
+We name $x$ the distance from the center of the trough, along the axis
+of the two coils. Figure below, left panel reports the magnetic fields $H_1$
+and $H_2$ generated by each coil -placed respectively at $x=-8cm$ and
+$x=8cm$- and by the two coils in Helmholtz configuration. Close to
+$x=0cm$, the total field $H_{tot}$ is constant over $x$. When the
+oscillating current $I$ is added to one coil, near to $x=0cm$ the field
+$H_{tot}$ grows linearly with $x$. If the needle, aligned parallel to
+the axis of the coils, is described as a magnetic dipole of momentum
+$\mu$, the force exerted on the needle is given by
+$$\overrightarrow{F}(x) = \overrightarrow{\mu} \times \nabla \overrightarrow{H}(x)$$
+In the following, we consider only the component of the force parallel
+to the axis of the two coils. A calculation of the behavior of $F(x)$
+for our instrument is shown in the central panel of the next figure. Close to $x=0$, $F$
+is directly proportional to $x$: therefore, an elastic constant can be
+defined as
+$$k=\frac{F(x)}{x}.$$
+{{lmn:isr:isrcampi.png|ISR - magnetic fields}} {{lmn:isr:isrforza.png|ISR - forces}} {{lmn:isr:gradient_variation.png|ISR - gradient variation}}
+Given this linearity, the energy of the dipole-field interaction $U(x)$
+is parabolic in proximity of $z=0$. The needle will change its position
+accordingly to the position $x_0$ of the minimum of $U$. Since $x_0$
+depends on the value of the oscillating current $I$, we have that
+$$x_0 = A e^{i\omega t}.$$
+One may consider that, while in the four-coils configuration the value
+$k$ is constant, this do no happen in a two-coils design.
+The calculation reported in the previous figures
+shows that the dependence of $k$ over the current $I$ is negligible in the
+present experimental setup; the relative variation of $k$ reaches a
+maximum value of the order of $10^{-3}$.
+The resulting displacement $d(\omega)$ of the needle -of the order of
+$50\mu m$- is detected by a CCD camera [1] equipped with a long focal
+objective [2]. An hardware DAQ board [3] is used to control the
+instrument: using its analog outputs it is possible to drive the trough
+barriers and to apply the oscillating current to the Helmholtz coils via
+a set of power supplies. The DAQ analog inputs are used to measure the
+trough area, the surface pressure and the current on the coils as
+functions of time. Both the DAQ hardware board and the CCD camera are
+triggered by an external reference wave, oscillating at 14Hz. The
+external triggering is needed to properly measure the phase lag
+$\delta(\omega)$ between stress and strain, ensuring that the correct
+mechanical response is retrieved. A connection scheme of the
+experimental setup is sketched in the figure below: the connections relative to the trigger signal are depicted in
+red.
+  - The Imaging Source, 1024x768 pixels
+  - Mitutoyo MPLAN-APO-20X
+  - National Instruments PCI-6036e
+{{lmn:isr:schem-isr.png?600px|Electrical connections}}
+==== Calibration: oscillations on pure water ====
+In order to characterize the inertial response of the needle, we
+consider the oscillatory movement of the needle on a pure water surface.
+The total force exerted on the needle given by the composition of:
+**a)** a viscous drag force $F_c = -d \dot{x}$, where $d$ is the drag coefficient for a motion parallel to the axis of the needle. If the needle is thin - its radius is much shorter than its length ($a \ll L$)- then the drag coefficient is given by a relatively simple expression @Levine2004
+$$d= \frac{2 \pi \eta L}{\log \left( 0.43 L/l_0 \right)}$$
+The drag coefficient $d$ is then related to the water viscosity $\eta$, the needle length $L$ and a characteristic length $l_0$ over which the two dimensional fluid velocity field can vary, which can be safely assumed to be of the order of the needle’s radius $a$;
+**b)** an harmonic force originating from the interaction between the magnetic dipole of the needle and the magnetic field gradient generated by the system of coils,
+$$F_m = -k \left( x-x_0 \right) = - \frac{\partial^2 U}{\partial x^2} \left( x-x_0 \right)$$
+where k is the elastic constant previously defined and U is the energy of the dipole-field interaction. The center $x_0$ of the harmonic potential oscillates with angular frequency $\omega$ (see equation [eq:cenharm]).
+The resulting differential equation is
+$$m \ddot{x} = - d\dot{x} -kx + kAe^{i\omega t}$$
+This equation is that of a forced-dumped oscillator of mass $m$,
+spring constant $k$ and damping constant $d$ @Reynaert2008. The ratio
+$AR$ of the amplitudes of the needle and force oscillation, and the
+phase lag $\delta$ result as:
+$$AR=\frac{X}{kA}=\frac{1}{\sqrt{(k-m\omega^2)^2+(\omega d)^2}}$$
+$$\delta=\arctan \left( \frac{-\omega d}{k-m \omega^2} \right)$$
+Commonly, instead of measuring the elastic constant $k$, the simpler
+approach of measuring the oscillatory voltage $V$ applied to one of the
+two coils is preferred. Therefore, a calibration constant has to
+determined; in order to do that, the amplitude ratio $AR_V=\frac{X}{V}$
+is measured. Observing the high-frequency limit of $AR$, it can be noted
+that it is proportional to $\omega^2$ via the mass of the needle $m$.
+$$\frac{X(\omega)}{F(\omega)} = \frac{X(\omega)}{\alpha V(\omega)} \rightarrow {m \omega^2}, \; \omega \rightarrow \infty$$
+This relation -in analogy with @Brooks1999- allow the determination of
+the *calibration constant* $\alpha$:
+$$\alpha = \frac{1}{m\omega^2}\lim_{\omega \rightarrow \infty}{\frac{X}{V}}$$
+A measurement of $AR$ and $\delta$ is reported in the figure below in logarithmic and semilogarithmic scale, respectively.
+{{lmn:isr:calibrazione_acqua_ar.png|Measurement on water - Amplitude ratio}}
+{{lmn:isr:calibrazione_acqua_del.png|Measurement on water - Phase shift}}
+While the mass $m$ of the needle is measured, the oscillator parameters
+$k$ and $d$ were determined by a fit performed on the stress-stain ratio
+curve using equation [eq:isr-amplratio]; they are reported in table
+[tab:kmd]. The phase lag $\delta$ is in very good agreement with the
+model curve built from the parameters of table [tab:kmd] (figure
+[fig:isr-water]b) using equation [eq:isr-phase].
+^ $k$ | %% $1.13 \cdot 10^{-5} N/m$  %% |
+^ $m$ | %% $ 9.9 \cdot 10^{-6} Kg$   %% |
+^ $d$ | %% $1.74 \cdot 10^{-5} Kg/s$ %% |
+It is worth comparing the value of $d$ obtained from the fit of the
+experimental curves with its theoretical value calculated using equation
+[eq:drag]. If the characteristic length $l_0$ is taken equal to the
+needle’s radius $a$, we obtain $d_t=2.38 \cdot 10^{-5} Kg/s$, with
+respect to the measured value of $1.74 \cdot 10^{-5} Kg/s$: the order of
+magnitude of the drag is correctly predicted.
+==== Oscillations in presence of a film ====
+The presence of a film at the air/water interface induces an additional
+hindrance of the motion of the needle, thus reducing the amplitude of
+its oscillations and modifying the phase lag between stress and strain.
+An measurement of the response of a Langmuir film can be considered
+meaningful is it is significantly different from the measurement
+performed in absence of film, which characterize the intrinsic
+oscillating behavior of the system. If this deviation is too small, than
+the measurement is likely to include huge effects due to the drag of the
+subphase.
+As stated at the beginning of this chapter, the contribution to the drag
+due to the water subphase are negligible if the Boussinesq number $B$ is
+high: $B>>1$.
+Usually, a so-called *experimental Boussinesq number* @Brooks1999
+[@Reynaert2008]
+$$B_{exp} = \frac{AR_{sys}}{AR_{film}}$$
+is used in the literature to quantify the deviation of the response of
+the needle from the behavior observed on pure water. If
+$B_{exp} > 10-100$ in the frequency range investigated, then the
+mechanical response measured may be considered as not affected by
+hydrodynamic contributions due to the bulk of the subphase. In this
+case, the shear modulus $G$ is given by
+$$G(\omega) = \frac{W}{2L}\frac{\alpha V(\omega)}{X(\omega)} e^{i\delta}$$
+If $B \simeq 1$, the subphase drag contribution to $G$ has to be
+decoupled from the one due to the film. A simple way to decouple the
+contributions to $G$ arising from the drag of the film and from the drag
+of the subphase, when the latter is not negligible, is to imply a linear
+relation between the two quantities @Brooks1999a [@Reynaert2008], so
+that
+$$G_{film}= G_{meas}- G^*_{water} =G_{meas} - \frac{W}{2L}AR_{w} \exp{(i\delta_{w})}.$$
+The accuracy of this procedure is questionable. In this work, the
+preferred approach consists in setting a confidence threshold value
+$B_{exp} = 10$, above which the measurement can be considered as not
+influenced by subphase drag contributions.
+==== Measuring the response of a Newtonian fluid ====
+An additional control of the instrument calibration is conducted
+measuring the response of films of poly(dimethylsiloxane) of controlled
+thickness $d$ and known viscosity $\eta_ {bulk}=0.97 Pa\cdot s$. This
+oil has the mechanical response of an ideal (newtonian) fluid: it
+presents a purely viscous response with frequency-independent shear
+viscosity $\eta_s$.
+In case of predominantly viscous systems, the generalized viscosity
+$\eta_{2D}$ is obtained from $G$ through the simple relation:
+$$\eta_{2D}=\frac{G"}{\omega}.$$
+It is reasonable to assume that $\eta_{2D}$ is proportional to the
+bulk viscosity through the thickness of the film,
+$\eta_{2D}= d \cdot \eta_{bulk}$, thus ignoring confinement effects.
+{{lmn:isr:calibrazione1.png| viscous response of a Newtonian fluid}}{{lmn:isr:calibrazione2.png| Sensitivity}}
+Data are shown in figure above, left panel: the film behaves as a Newtonian
+fluid with $G"$ (circles) growing linearly with the frequency, as
+indicated by fit represented by the dashed line, while $G'$ is zero
+within the error. On the right we compare the value of the
+viscosity $\eta_{2D}$ obtained from $G''$ for each film thickness d with
+its expected value based on the bulk viscosity.
+The results are consistent with the theoretical behavior for an ideal
+viscous fluid. However, careful inspection of the data from the lowest
+thickness ($d<50\mu m$, represented by the gray area in the picture)
+shows a small deviation which can be ascribed to the drag due to the
+water subphase, which becomes comparable to that of the film when this
+becomes too thin. This effect has already been found and discussed in
+the literature @Reynaert2008. If we suppose, in a simple approximation,
+a linear superposition of these effects, a rough estimate of the
+subphase drag is obtained: $|G|_{sub} \simeq 10 \mu N/m$.
+==== Adaptation for photosensitive polymers ====
+The study of the mechanical properties of photosensitive polymers
+requires a proper adaptation of the instrument described so far. In
+particular, the instrument had to be enclosed in an aluminum box
+equipped with removable walls, to ensure that spurious light from the
+ambient does not influence the rheological properties of the film.
+Moreover, the illumination light required for the tracking of the
+needle’s position has to be chosen in a spectral region where the sample
+is characterized by low absorbance. For the photosensitive azopolymers
+investigated in this work, red light ($\lambda >630nm$) was chosen.
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