Torque rotational viscosity

Rotational viscosity methods measure viscosity by measuring the torque required to rotate a spindle immersed in the fluid sample. 

The physical meaning of the above equation is that viscosity torque, which is the product of the rotational viscosity coefficient y and the angular speed dG/dt, is balanced by -SfISG, which is the sum of the elastic and electric torques. Using Equations (5.53), (5.54) and (5.55), it can be obtained that... 

The switching time of the director tilt angle in the electroclinic effect is independent of the electric field, and is defined only by the rotational viscosity je and the elastic modulus A. The corresponding switching times are derived from the Landau-Khalatnikov equation for the balance of the viscous and elastic torques... 

Measuring the torque on a sample of a nematic liquid crystal in a magnetic field rotating with an angular velocity smaller than the critical one represents a relatively simple method for the determination of the rotational viscosity coefficient. Below the critical angular velocity Eq. (24) is valid with 0 = F. Neither the phase lag F-0 nor the anisotropy of the magnetic susceptibility have to be known. This method will be thoroughly discussed in Chap. Ill, Sect. 2.6 of Vol. 2A of this Handbook. 

An externally applied torque on the director can only be transmitted to the surfaces of a vessel without shear, if the director rotation is homogeneous throughout the sample as assumed in Sect. 8.1.2 for the rotational viscosity. Otherwise this transmission occurs partially by shear stresses. The resulting shear flow is called backflow. As there is usually a fixed director orientation at the surfaces of the sample container, a director rotation in the bulk of the sample by application of a routing magnetic field leads to an inhomogeneous rotation of the director and to a backflow [42]. 

The viscous torque is obtained according to the procedure in Sect. 8.1.2 on the rotational viscosity. 

Larger fields and larger sample volume are possible if the sample rotates instead of the magnetic field, and electromagnets can then be used [62]. The torque on the suspension wire must then be measured in a rotating system, which requires some effort. The rotation of the sample can be monitored by means of a laser beam, which is reflected from a mirror attached to the sample vessel. The reflected laser beam is detected twice in every revolution by a photodiode. If the suspension is rotated by a computer-controlled stepper motor and the photodiode interrupts a timer on the computer, the phase lag between motor and sample vessel can be determined without and with field, and the rotational viscosity can be calculated from the difference between these phase lags. The uncertainty in the determination of the rotational viscosity with this method can be as small as 1%. 

In the smectic C phase, the rotational viscosity Yq> can be estimated by observing the polarization reversal or the electrooptic properties of the cell, as described in Sec. 2.7.6. The estimation may, for instance, be based on the approximation mentioned there, using the elastic torque [137]... 

Where f" is the viscous torque and /i is the rotational viscosity. With V being the velocity, one has ... 

Figure 6.8 shows the variation in rotational viscosity ( y ) with time t. The most remarkable feature of the result in (6.56) is that, as a function of time, the square of the amplitude of the rotational viscosity y q ) oscillates sinusoidally (Fig. 6.8). After rising to a maximum of [ (t-Tc)-kK ]Y 2, it drops back to zero. Since, (6.56) is obtained by comparing (6.55) with (6.45), the maximum value allotted to ( y ) must not exceed one, else the perturbative method will be invalid and hence, (6.56). This result is highly surprising. At times t = where n = 1, 2, 3... the FLC molecules certain to have almost infinite rotational mobility resulting into zero rotational viscosity as evident from Fig. 6.8. Therefore, the applied field should not keep for a longer period of time but it should turn off after an interval of time for maximizing the chances to produce finite rotational torque on the FLC director and as a result to obtain the system with finite rotational viscosity. Once the field is turned off, this reduced viscosity seems to produce a finite but weaker torque at the memory state and thereby, promotes multistability and resolution in the memory states as proposed earlier.

If n is mobile (no body forces present), then in cases (i) and (ii) above a viscous torque rotates n, and conversely rotation of n by body forces induces a flow. Two additi(Mial viscosity coefficients, yi and 72, which have no counterpart in isotropic liquids, are necessary to describe this situatitm. The first coefficient, yi, characterises the torque associated with rotation of n. The latter coefficient, j2, gives the contribution to the torque due to a shear velocity gradient in the nematic. The two coefficients also define the flow alignment of the director under stationary shear flow ... 

The rotational viscosity coefficient yi is the most frequently determined viscosity coefficient of liquid crystals. By a straightforward adaptation of the shear flow technique, Tsvetkov originated the idea of measuring a specific torque, TA, exerted on the sample by n rotating with a constant angular velocity (0[8] ... 

So far we have introduced four Miesowicz viscosities. Two other viscosities can be proposed by considering the following. The director n in Fig. 4.1(a), if free to move, will rotate due to a viscous torque the viscosity coefficient 71 is introduced to describe this situation and characterises the torque associated with a rotation of n. For this reason 71 is often called the rotational viscosity or twist viscosity. The coefficient 71 generally determines the rate of relaxation of the director. Also, a rotation of n due to body forces will induce a flow. The viscosity coefficient 72 characterises the contribution to the torque due to a shear velocity gradient in the nematic and is sometimes referred to as the torsion coefficient in the velocity gradient it leads to a coupling between the orientation of the director and shear flow. The two viscosities 71 and 72 have no counterpart in isotropic fluids. We therefore have a total of six viscosities four Miesowicz viscosities plus 71 and 72. It turns out, as will be seen in the problems to be discussed in later Sections, that 7i and 72 are precisely the viscosities introduced in the constitutive theory at equations (4.78) and (4.79), namely. 

In most rotational viscometers the rate of shear varies with the distance from a wall or the axis of rotation. However, in a cone—plate viscometer the rate of shear across the conical gap is essentially constant because the linear velocity and the gap between the cone and the plate both increase with increasing distance from the axis. No tedious correction calculations are required for non-Newtonian fluids. The relevant equations for viscosity, shear stress, and shear rate at small angles a of Newtonian fluids are equations 29, 30, and 31, respectively, where M is the torque, R the radius of the cone, v the linear velocity, and rthe distance from the axis. 

The Nametre Rotary B rotational viscometer measures torque in terms of the current needed to drive the d-c motor at a given speed while a material is under test. The standard sensors are coaxial cylinders or Brookfield disk-type spindles, but a cone—plate system is also available. The viscosity range for the coaxial cylinder sensors is 5 to 5 x 1(T mPa-s, and the maximum shear rate is 200. ... 

In this apparatus the polymer melt is sheared between concentric cylinders. The torque required to rotate the inner cylinder over a range of speeds is recorded so that viscosity and strain rates may be calculated. 

Direct Indicating Viscometer. This is a rotational type instrument powered by an electric motor or by a hand crank. Mud is contained in the annular space between two cylinders. The outer cylinder or rotor sleeve is driven at a constant rotational velocity its rotation in the mud produces a torque on the inner cylinder or bob. A torsion spring restrains the movement. A dial attached to the bob indicates its displacement. Instrument constants have been so adjusted that plastic viscosity, apparent viscosity, and yield point are obtained by using readings from rotor sleeve speeds of 300 and 600 rpm. 

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