Advanced lab course physical chemistry

Viscosity

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Summary

In this experiment, you will use a double-gap cylindrical rotational viscometer to determine the shear dependent viscosity, that is, the rheological behavior, of various aqueous samples. These cover pure water, dilute sugar solutions and semi- concentrated solutions of the biopolymer Xanthane. The experiments illustrate the application of viscometry in physical chemistry: for example, you will determine the degree of hydration of sugar (see also the PhD thesis of Albert Einstein!), or the swelling of a biopolymer in pure water. The later will also be studied in dependence of shear frequency.

Learning goals:

Theoretical background

What is viscosity?

According to a dictionary, viscosity is the inner friction or flow resistance of liquids and gases. A quantitative definition may be given in two different ways: First, viscosity on a microscopic scale is reciprocal to diffusion, as given by the Stokes-Einstein-equation, which describes the Brownian motion of nanoscopic particles in dilute dispersions:

D_\text s=\frac{kT}{6\pi\eta R_\text H}

Here D_\text s is the self-diffusion coefficient of a Brownian particle, kT the thermal energy (“driver of the diffusion“), \eta the viscosity of the medium (causing friction and hindering the diffusion) and R_\text H the hydrodynamic radius of the Brownian particle.

The second, alternative possibility is to macroscopically consider viscosity as the coefficient of momentum transport. In general, transport or 1-dimensional flux of an arbitrary physical quantity depends on a gradient, and is given as:

J=\frac{1}{A}\frac{\mathrm d N_\varphi}{\mathrm dt}=-C\frac{\mathrm d\varphi}{\mathrm dx}

Here J is the flow density, N_\varphi the amount of physical quantity \varphi, A the flow cross section, C the coefficient of transport and \frac{\mathrm d\varphi}{\mathrm dx} the spatial gradient of the flowing physical quantity (i.e. a generalized force).

Examples are:

\begin{aligned} J_N&=\frac{1}{A}\frac{\mathrm d N}{\mathrm dt}=-D_\text s\frac{\mathrm d c}{\mathrm dx}\qquad &&\text{Fick's law (flux of matter)}\\[3ex] J_Q&=\frac{1}{A}\frac{\mathrm d Q}{\mathrm dt}=-\lambda_e\frac{\mathrm d V}{\mathrm dx}\qquad &&\text{Ohm's law (flux of electrical charges)}\\[3ex] J_W&=\frac{1}{A}\frac{\mathrm d W}{\mathrm dt}=-\lambda_T\frac{\mathrm d T}{\mathrm dx}\qquad &&\text{Fourier's law (flux of heat)}\\[3ex] J_p&=\frac{1}{A}\frac{\mathrm d p}{\mathrm dt}=-\eta\frac{\mathrm d v_x}{\mathrm dx}\qquad &&\text{Newton's law (flux of momentum)} \end{aligned}

Newton’s law of viscosity therefore defines viscosity as the coefficient of momentum transport in viscous media. This is illustrated by the model of laminar flow: let us consider the motion of a solid body of surface area A on top of a thin liquid layer of thickness z. If the velocity is constant, the force to drag the body is compensated by the friction, i.e. F = R. On the other hand, according to Newton’s law of viscosity the force per surface area corresponds to the flux density of momentum transfer, and is proportional to the viscosity of the liquid and the gradient of velocity:

J_p=\frac{1}{A}\frac{\mathrm d p}{\mathrm dt}=-\frac{F}{A} =-\eta\frac{\mathrm d v_x}{\mathrm dz}

The unit of the viscosity therefore corresponds to pressure times time and is Pa s.

Model of laminar flow. The left-hand graph shows the velocity v as function of the distance z from a fixed body (placed below). The right-hand side shows schematically the moving liquid molecules.

How to measure viscosity

Capillary viscometer

The working principle of a capillary viscometer is to determine the time it takes for a defined sample volume to flow through a thin channel (capillary). To reach a constant flow velocity, frictional force and hydrostatic pressure difference creating the flow have to balance each other. This results in a parabolic velocity profile given as:

v(r)=\frac{p_1-p_2}{4\eta l}\cdot(R^2-r^2)

Here, R is the radius of the capillary, and r the distance of a cylindrical flow layer from the center of the capillary. The total volume per time \dot V can be calculated from the velocity profile v(r) by integrating, leading to the Hagen-Poisseuille-law:

\dot V=\int_0^R 2\pi r\cdot v(r)\mathrm d r=\frac{\pi (p_1 - p_2)}{8\eta l}R^4

Falling bead viscometer

A spherical bead of defined size and mass is sinking with constant velocity within a cylindrical vessel, filled with the liquid whose viscosity is to be determined. Therefore, frictional force and buoyancy have to balance the gravitational force:

6\pi\eta R v + \pi R^3 \rho g = \frac{4}{3} \pi R^3 \rho_\text K g

R is the radius of the bead, \rho the density oft he liquid, and \rho\text K the density of the bead. This relation leads to a falling bead velocity given as:

v=\frac{2R^2(\rho_\text K-\rho)g}{9\eta}

It should be noted that a falling bead viscometer is only feasible to study liquids with high viscosity, because only then the friction can balance the sedimentation force. However, with a slight modification (rolling bead viscometer), this technique can be applied to measure the viscosity of even gases.

Scheme of a falling bead viscometer.
Interactive visualization showing the trajectory of the bead in the falling bead viscometer. The viscosity \eta, the radius of the sphere R and the density difference between the sphere (\rho_\text K) and the used liquid (\rho) can be adjusted.

Double-gap rotational viscometer

The double-gap rotation viscometer uses the principle of laminar flow (see fig.1), replacing the one-dimensional flow with a cylindrical geometry for practical reasons: a hollow metal cylinder is placed within a cylindrical beaker containing the viscous liquid, and set into constant rotation. If you then measure the force or, better, torque necessary to rotate the cylinder, the viscosity can be determined. Important experimental parameters in this case are:

The linear velocity gradient then is given as:

Double-gap rotation viscometer, experimental setup and cross secion.
\frac{\mathrm d v_x}{\mathrm dz}=\frac{\omega R}{D} dv x dz =ωR D

Consequently, Newton’s law of viscosity in cylindrical flow geometry is given as:

\tau(\omega)=\eta(\omega) \cdot A \cdot \frac{\omega R}{D}=\eta(\omega) \cdot A \cdot \dot\gamma

with rotational force = torque \tau(\omega), shear stress \sigma(\omega) = \tau(\omega)/A, and shear gradient or shear rate \dot \gamma = \omega R / D.

Viscosity of solutions and dispersions

For dilute solutions and dispersions (water/sugar, water/Xanthane, in your experiment) the viscosity is given as a series expansion in solute volume fraction (Einstein) or solute mass concentration c (Staudinger), which can be truncated at the first-order:

\eta(\varphi) = \eta_0 (1 + 2.5 \varphi) \qquad \text{(Einstein)},
or:
\eta(c) = \eta_0 (1 + [\eta] c) \qquad \text{(Staudinger)},
η(c)−η0

with \eta_0 the viscosity of the pure solvent, and [\eta]=\lim_{c\rightarrow 0}\frac{\eta(c)-\eta_0}{c\eta_0} the so-called intrinsic viscosity or Staudinger index.

Comparison of these two expressions then allows us, for example, to determine the degree of hydration of sugar molecules in aqueous solution:

\eta(\varphi) = \eta_0 (1 + 2.5 \varphi) = \eta(c) = \eta_0 (1 + [\eta] c), \qquad \text{and}\quad [\eta]=2.5\frac{\varphi}{c}=\frac{2.5}{\rho}

with \rho = c/\varphi=(m_\text{tr}/V)/(V_\text s/V) the formal particle density in solution, m_\text{tr} the mass of solute (i.e. in “dry state“), V_\text s the effective volume of all solute particles in the dissolved state.

Note that V_s is the dry particle volume of the solute V_\text{tr} plus the hydration volume of water molecules V_\ce{H2O} associated with the sugar molecules in solution, and therefore:

\rho = \frac{m_\text{tr}}{V_\text{re}+V_\ce{H2O}}\quad\text{or}\quad V_\ce{H2O}=\frac{m_\text{tr}}{\rho}-V_\text{tr}=m_\text{tr}\cdot\left(\frac{1}{\rho}-\frac{V_\text{tr}}{m_\text{tr}}\right) =m_\text{tr}\cdot\left(\frac{1}{\rho}-\frac{1}{\rho_\text{tr}}\right)

with \rho_\text{tr} the density of dry solute. The ratio V_\ce{H2O}/m_\text{tr} (check dimensions carefully!) then yields the volume of water of hydration per gram sugar.

The degree of hydration is defined as the number of associated water molecules per solute molecule or particle, and can easily be determined experimentally from the intrinsic viscosity [\eta] and the corresponding formal density \rho, if the molar mass of the solvent, the density of the solvent, the molar mass of the solute and the density of the dry solute are known.

(Note: the density of water is 1 g/ml, molar mass = 18 g/mol, density of sugar in dry state = \rho_\text{tr} = \pu{1562 g/l}, molar mass of sugar M_\text{tr} = \pu{342 g/mol})

Non-Newtonian behavior

Non-Newtonian behavior or structural viscosity is the phenomenon of viscosity depending on velocity gradient or shear-rate, or viscosity depending on measurement time. In principle, one can distinguish three different types of materials:

Variation of viscosity (left) or shear stress (right) with shear-rate: Newtonian (black), shear-thinning (blue), shear-thicking (red).
Variation of viscosity (left) and shear stress (right) with the duration of the measurement: Constant (black), thixotropic (blue), rheopectic (red)

Potential causes of a change in viscosity with shear-rate can be a) a change in the spatial arrangement, or b) a change in particle size and/or morphology of the solute (see the following figure).

Shear-induced elongation and deswelling of a polymer coil in solution.

Experimental setup

The double-gap rotational viscometer used in the practical course has been described in detail before. Let us just repeat the most important experimental parameter of this setup:

Newton’s law of viscosity in cylindrical flow geometry is given as:

\tau(\omega)=\eta(\omega) \cdot A \cdot \frac{\omega R}{D}=\eta(\omega) \cdot A \cdot \dot\gamma

Experimental Tasks

All measurements last 20 seconds, each! Each sample corresponds to a mass of about 30 grams!

  1. Measure pure water at rotation speeds 25, 50, 75 und 100 U/min.
  2. Measure aqueous sugar solutions (c_\text{sugar} = 6, 8 und 10 wt%) at a suitable rotational speed (see (1)). Note: saccharose, M = \pu{342 g/mol}, \rho_\text{tr} = \pu{1562 g/l}
  3. Measure aqueous solutions of Xanthane, concentrations as following:
    1. stock solution (c = \pu{1.5 g/L})
    2. c = \pu{1 g/L}
    3. c = \pu{0.5 g/L}
    (U = 2, 10, and \pu{50 U/min}, each sample).

Important: make sure all samples are homogeneous. However, only stir gently and don’t hold the sample vial in your hand for longer than a few seconds, to avoid uncontrolled sample heating!

Data evaluation

  1. Compare your result to literature, and discuss potential deviations.
  2. Calculate the intrinsic viscosity for each concentration, and determine the corresponding degree of hydration. Discuss a potential dependence of your results on sugar concentration.
  3. Calculate the intrinsic viscosity for each concentration and shear rate. Also, determine the degree of hydration (amount of water associated per gram of solute polymer), using a dry solute density of 1 g/L. Discuss your results in terms of potential reasons for non-Newtonian behavior.

Questions for preparation

  1. What is viscosity?
  2. How can you measure viscosity (at least 3 different experimental methods)?
  3. How does the viscosity of liquids change with temperature?
  4. What is the intrinsic viscosity?
  5. What is non-Newtonian behavior? Give potential reasons, and examples.

Literature