RHEOLOGY	

THEORETICAL
	Rheology (“study of flow”, Greek) is the science that studies the flow of materials and other phenomena associated  with it.  A special accent is taken to the viscous flow.
	The viscous flow can be regarded as the conductive transport of impulse (momentum).  As it is velocity what is conjugated as intensivity to extensivity impulse, flux of impulse JI must be proportional to the negative gradient of velocity v, analogously to Ohm’s law: 

										1)

	However, the conductive transport of impulse is more complex process than electric current both in physical and in mathematical sense. First, contrarily to the electric current, it is accompanied by displacement of the whole matter. 
Second, impulse is a vector, not a scalar as electric charge.  If it is displaced, each of its three components  may move in all the three directions of the space, consequently, its flux is a second order tensor that is a 3x3 matrix representing it.  As operator gradient always increases the tensorial order by one, gradient of the velocity must be a second order tensor, too. 

	The followings has to be considered yet:				
a)  The flux of the impulse can be regarded as a negative stress tensor  -T.   (The dimension of the impulse flux is force/area.  Any mechanical tension, pressure included, is entirely a flux of impulse. The negative sign comes from the fact that strain tension is regarded positive. 
b)  The two second order tensors in Eq. 1)  may be connected by a scalar, provided the material is isotropic.  In the case of an anisotropic material like a liquid crystal, a fourth order tensor connects the two second order ones.  The high number of independent entries in it (21 from the 3x3x3x3=81) excludes these materials from students’ laboratory measurements.  We confine our investigations to isotropic materials.
c)  The gradient of velocity is not the only driving force and viscous flow is not the only impact in conductive transport of impulse. 

Any side of an Ohmic relation may be regarded as driving force and the opposite one as impact. 
.
Impulse arriving at an elementary volume cell of the material causes one or more of the followings:
1. pressure increases in the cell;
2. the whole cell is (linearly) displaced;
3. the whole cell rotates;
4. the cell is elastically deformed;
5. the cell is plastically deformed, it flows;
6. the cell breaks up.
We will avoid the high impulse fluxes needed in case 6.  In a so called substantial description cases 2 and 3 disappear as coordinates move along with the cell under that method.   
In case 4 a strain tensor D (multiplied with the appropriate modulus)  has to be added to the right side of Eq. 1). Strain tensor D ist the gradient of the displacement vector so it is a second order tensor, too.  To counterbalance the addition, making use of its linearity, stress tensor T  is resolved into two tensors. One of them depends purely on Grad v, the other purely on D. Doing this, elastic deformation and plastic flow have been, at least formally, separated.
Case 5 is the viscous flow, the subject of our interest.
d)  Stress tensor T, due to conservation of angular momentum, must be symmetrical, consequently, only the symmetrical part of tensor Grad v called strain velocity tensor D˙ may be present on the right side of Eq. 1) if viscous flow is to be described.  (The  antisymmetrical part of the tensor would represent a rotation.)

				 							2)

A notice for the case when elasticity of the material is not negligible : strain tensor D is essentially symmetrical.

Any second order tensor can be resolved into the sum of a symmetrical and an antisymmetrical second order tensor. The resolution is unique.

So Eq. 2) splits in three scalar equations with an isotropic material:

				  , i ≠ j						3)

Any of them may be used to evaluate the linear coefficient viscosity η (more precisely: shear viscosity). 

The entries in the diagonal (i=j) of tensor T (denoted σ’s) are called normal stresses while the off-diagonal (i ≠ j) entries (denoted τ’s) are called shear stresses.  Strain velocity is usually denoted  γ˙. Equation 3) is valid even with normal stresses for some materials, however tensile viscosity ή replaces shear viscosity η in it.  

	If there is merely viscous flow in the material and viscosity, at the given temperature and pressure, is constant  (so called Newtonian liquid), then viscosity alone characterizes its reology, nothing else should be measured.
	In other cases the complete stress – strain velocity (τ – γ˙) graph  has to be measured. Important phenomena like thixotropy  may be studied by them.  (Thixotropy is a reversible solid-liquid phase transition caused by shear stress.) 
	Viscosimeters (or rheometers) are the devices that can measure such properties.  There are capillary, falling-ball, rotational and vibrational viscosimeters.  Each type works with a two dimentional laminar flow. 
	In a falling-ball viscosimeter a sphere of known radius and density is sinking in the liquid.  By means of Stokes equation, viscosity can be calculated from velocity of sinking. 
	In a capillary viscosimeter material flows through a cylindrical capillary of known legth and diameter. One can measure either the flow rate with a given driving force (that is pressure difference) or the pressure difference over the capillary with a given flow rate.
	In a rotational (vibrational) viscosimeter a rotating (vibrating) body is immersed in the material and the braking moment is measured at a given number of revolutions (frequency).  Contrarily to the previous two methods,  this one allows continuous measurements and studying time-dependent phenomena. 

INSTRUMENTS
a) Ostwald capillary viscosimeter
	A cylindrical tube of length L and radius R has two independent coordinates, the longitudinal X and the radial r.  If pressure difference Δp maintaines a steady laminar flow in the tube, Eq. 3) takes the following form: 

				 							4)

The shear stress in Eq. 4) can be expressed from the balance of forces in a cylinder of radius r:

										5)

After inserting the shear stress into Eq. 4) one gets a differential equation.  Ist solution is function vX(r)  whose integral for the whole cross-section of the tube  gives the flow rate, as the Poiseuille equation writes:

										6)
 
	During use of the Ostwald viscosimeter (Figure 1) the majority of the liquid of given volume (10.00cm3 in our case) is forced (with blow) into the right branch of the device until sphere B completely fills.  While letting the liquid flow back to the left branch under ist own weight one can measure the time t needed to clear sphere C. (Sphere C has an upper and a lower marker to make this measurement possible.)  As the pressure difference can be calculated from the difference of the liquid levels in the branches Δh, Eq. 6) can be rewritten as follows: 

										7)

	Eq.  7) shows more problems: 
a)  The measured time is proportional to kinematic viscosity (=viscosity/density), so density of the measured liquid must be known if viscosity is needed.
b)  The radius R has to be constant along the capillary with a very high degree of precisity as the measured time depends on its fourth power.  A mass-produced viscosimeter may hardly be expected to fulfill such a requirement.
c)   Difference of the levels Δh decreases during the measurement, strictly speaking the system is not in steady state. 
d)   Driving force Δp does not drop completely on the capillary, however in less extent, there is some Δp drop on the spheres and on the tubes of larger diameter, too.

				Figure 1
	
	Due to points  b), c) and d), this device may be used only for relative measurements. Time of flowing through sphere C must be measured with a liquid of known viscosity and density before measuring a material of unknown viscosity:

										8)

	Distilled water shold be chosen as known material if aqueous solutions are dealt with, as in our case. Its data can be taken from Table 1.

					Table 1
Temperature, ˚C	Viscosity,  mPas	Density,   kg/m3	
0.0	1,797	999,87	
5.0	1,525	999,99	
10.0	1,301	999,73	
15.0	1,138	999,13	
20.0	1,006	998,23	
25.0	0,8938	997,07	
30.0	0,7998	995,67	
35.0	0,7229	994,06	
40.0	0,6563	992,24	
60.0	0,4735	983,24	
80.0	0,3570	971,83	

b) Rotational viscosimeter with concentric cylinders
	The sample fills the space between the concentric cylinders completely and it is adhered to both  of them. As the outer cylinder is standing and the internal one is rotated with known (and controllable) angular velocity, the internal layer of the material rotates with the angular velocity of the internal cylinder while the outer layer is standing: there is a velocity gradient along radius R. (See Figure 2.) 
In steady state moment M must be independent of R:   

			9) 

 τi and  τR are  the shear stresses at the internal cylinder and at radius R, respectively, Ri is the radius of the internal cylinder and h is the length of the internal cylinder.

					Figure 2

Moment M causes a torsion spring of spring constant k to change its angle to α:

											10)

From Eq. 9) and 10) one can see that the shear stress is inversely proportional to square of the radius:

				 						11)

If the material being between the cylinders is a Newtonian liquid, strain velocity γ˙ changes with radius in the same way. 
It is advantageous  to introduce cell constant Z:

				  					12)

In strict sense, no unique value of strain velocity γ˙can be connected to a given  τi . In order to overcome this difficulty, we try to work with the possibly narrowest slit between the two cylinders.  If the difference of Re and Ri is small, the average of γ˙ ’s at Re and Ri, respectively, is a good approximation to the second order hyperbolic function in Eq. 11).  The tables belonging to the viscosimeters contain these average strain velocities. 

	There may be other problems in this measurements:
a)   End-effects: There is some braking moment on the top and mainly on the bottom of the internal cylinder.  In order to minimize it the bottom is hollowed to have an air pocket instead of the direct contact with the material.
b)  Slip on the surface of the internal cylinder causes uncertain or wrong results.  A less smooth surface is advantageous for the internal cylinder.
c)  Turbulence and heat production, at  high strain velocities and/or low viscosities.  Rheological measurements should be thermostated as rheological properties depend strongly on temperature.
d)  Elastic liquids (gels, for instance) may show the so called Weissenberg effect: a stress builds up paralelly to the axis of rotation and drives the material out of the slit.
e)  Plastic materials have flow limit (or yield limit).  The material does not flow in the slit at those radii where the shear stress is less than the flow limit.
 
	The rotational viscosimeter can be seen in Figure 3. (The thermostating jacket 4 will not be used.)

					Figure 3

The device has two torsion springs 2. Switch 1 selects between the softer (position I) and the harder (position II) ones.  Bayonet-locks 3 and 5 serve to connect the axis of the internal cylinder to the torsion spring and the external cylinder to the house of the viscosimeter.  By means of switch 11 and range selector 8 angular velocity can be controlled in 24 degrees.  Switch 12 controls the synchronous motor and switch 13 the angle measuring instrument of the viscosimeter.  There is a frequencymeter 10 too, to check the precisity of 50 Hz.

					Table 2
Cylinders	Volume of the sample, cm3	Ri, cm	Ri/Re	h, cm	
S/S1
S/N
S/S2
S/S3
H/H	25
10
30
50
17	1,957
1,957
1,875
1,61
1,045	0,98
0,98
0,94
0,81
0,81	7,19
7,57
7,2
7,19
4,69	

	There are two devices at our disposal.  Both of them have more pairs of cylinders (see their data in Table 2).  Cylinders S/S1 and S/S2 are to be used in this measurement.
	Device constants, needed at evaluation of the measurement, can be taken from Table 3.

					Table 3
Number of the device: 01980/2386	
Cylinders	Position I
Z, Pa/scale	Position II
Z, Pa/scale	
S/N
S/S1
S/S2
S/S3
H/H	0,321
0,565
0,599
0,823
2,96	3,18
5,58
5,92
8,03
28,98	
Number of the device: 0961/0961	
S/N
S/S1
S/S2
S/S3
H/H	0,303
0,563
0,578
0,773
2,79	3,01
5,61
5,71
7,78
28,57	

	The average strain velocities belonging to a given degree can be found in Table 4.

					Table 4
degree	
1a	
2a	
3a	
4a	
5a	
6a	

d/dt	
1b	
2b	
3b	
4b	
5b	
6b	

S1	
3,0	
5,4	
9,0	
16,2	
27,0	
48,6	

1,5	
2,7	
4,5	
8,1	
13,5	
24,3	

S2	
1,0	
1,8	
3,0	
5,4	
9,0	
16,2	

0,5	
0,9	
1,5	
2,7	
4,5	
8,1	

H, S3	
0,33	
0,6	
1,0	
1,8	
3,0	
5,4	

0,167	
0,3	
0,5	
0,9	
1,5	
2,7	

7a	8a	9a	10a	11a	12a	degree	
7b	8b	9b	10b	11b	12b	d/dt	
81,0	145,8	243	437,4	729,0	1312,0	S1	
40,5	72,9	121,5	218,7	364,5	656,0		
27,0	48,6	81,0	145,8	243,0	437,4	S2	
13,5	24,3	40,5	72,9	121,5	218,7		
9,0	16,2	27,0	48,6	81,0	145,8	H, S3	
4,5	8,1	13,5	24,3	40,5	72,9		

EXPERIMENTAL
a) Measuring viscosity of a copper sulfate solution
1. Pipette 10,00 cm3 distilled water into the Ostwald viscosimeter, attach a rubber tube to its left branch and let it thermostate for 10 minutes. Pay attention to the position of the viscosimeter in the thermostate: markers on sphere C shold be well visible.
2. With gentle blowing into the rubber tube, push majority of the liquid to the right branch. Be the level in the right branch over the upper marker of sphere C and be sphere in the left branch not completely empty.  Allow the liquid to flow back and, with a stopclock, measure the time elapsed while the level in the right branch proceeds from the upper marker to the lower one.
3.  Repeate the previous measurement several times. (Have at least 5 results.)
4. Clean the viscosimeter and add 10,00 cm3 22% copper sulfate solution from a burette. Thermostate it and measure times according to points 1 and 2-3, respectively.
5. The copper sulfate solution should be poured back to its flask.  Clean the viscosimeter with distilled water.
6.  Calculating with the averages of the parallel measurements, using Eq. 8), calculate the viscosity of the copper sulfate solution on the temperature of the thermostate.  Density of the copper sulfate solution on  25.0˚C is 1150 kg/m3. 

b) Studying a bentonite suspension
1. Make a 12%  bentonite suspension.  Be the suspension as smooth as possible.
2.  Fill the suspension into the external cylinder and assemble the device. Allow five minutes rest to the suspension before starting the measurement.
3.  Starting with torsion spring I, increase  the angular velocity gradually, and after reaching its maximum, decrease it gradually. If  it is needed switch to torsion spring II.  Read α at each angular velocity when steady state has been achieved.  (If the number of revolutions is steady that can be seen on bayonet-lock 3.)
4.  Clean the device. 
5.  Taking the average strain velocities from Table 4 and calculating shear stresses with the data from Table 3, plot the stress – strain velocity (τ – γ˙) graph, both branch measured with the increasing and branch measured with the decreasing γ˙‘s. The larger their difference (the hysteresis) is, the more thixotropic the suspension is. Read the flow limit.