Dynamic Light Scattering
Version 1.07. Last Edit 24 January 2011, Richard Kingston.
Introduction!
3
Relevant Physical Theory and Models of Scattering!
4
Data Collection!
6
Cell Cleaning.!
6
Sample preparation!
8
Routine Data Collection.!
9
Checking for scattering from the buffer alone!
13
Make measurements from a protein dilution series. And replicate !!!! 14
Correcting to standard conditions!
15
Thermal unfolding experiments using the Event Scheduler.!
15
Exporting data from Dynamics!
16
Data Analysis!
16
Measurement of Solution properties.!
17
Measuring solution density.!
18
Measuring solution viscosity!
20
Measuring solution refractive index.!
22
Appendix A: Refractive Index (831.1 nm), Dynamic Viscosity, and
Density of Pure Water, as a Function of Temperature (Atmospheric Pressure)!
26
Appendix B: Calculating the density of air.!
38
Introduction
These brief notes describe how to obtain good quality DLS data using the Wyatt
DynaPro Titan. Soon they will describe how to analyze that data too !! Unfortunately,
there are not very many reviews of Dynamic Light Scattering that are accessible to
biologists. Hence these notes will eventually be expanded to include the relevant
physical background ... as time permits.
Relevant Physical Theory and Models of Scattering
Very briefly for now ...
The DLS Instrument measures the intensity autocorrelation function (ACF) of light
scattered from a solution. This can be analyzed to yield information about the translational diffusion of the light scattering species, and hence molecular size.
If there is a single scattering species present in solution and the usual conditions are
satisfied (e.g. non-interacting, basically “spherical” particles, much smaller than the
wavelength of the light being used to probe them) then the normalized intensity ACF
G(τ) is described by ...
1
where α is an instrument constant, DT is the translational diffusion coefficient and q is
the length of the scattering vector, given by
2
where n is the refractive index of the buffer, λ is the wavelength of the incident light,
and θ is the scattering angle.
Non-linear least squares fitting of G(τ) can provide estimates of the translational diffusion coefficient. The translational diffusion coefficient DT can be related to the hydrodynamic radius RH (the radius of the hard sphere which would diffuse at the same
rate as the particle) via the Stokes-Einstein relationship ...
3
where kB is the Boltzmann constant, and η the dynamic viscosity of the buffer.
If the light scattering particles are not a single size but instead exhibit some size distribution (they are polydisperse, in the usual terminology of DLS), then things get
more difficult. You cannot recover a size distribution from the intensity autocorrelation function measured in DLS, without making some additional assumptions (the
problem is mathematically ill-conditioned). Fortunately there are still some useful
ways to proceed ...
The Method of Cumulants was introduced by Dennis Koppel (Koppel D.E. (1972) J.
Chem. Phys. 57, 4814-4820) and can be used to recover the mean Diffusion coefficient and its variance. It does not require any assumptions about the nature of the underlying particle size distribution. The Cumulants analysis preformed by the Dynamics software is a slight variant of this method. The %Polydispersity, reported by the
Dynamics package is the (estimated variance / estimated mean ) x 100. However if the
underlying distribution function is complicated (for example ... if it is not monomodal)
then the results of the Method of Cumulants may not be very informative.
Dynamics also implements a Regularization method, which attempts to recover the
particle size distributions from the ACF, imposing restraints on the smoothness of the
distributions to make the process converge. This method is not very well-documented
and it is difficult to assess its general reliability.
Data Collection
Cell Cleaning.
It is important to keep the cells used for DLS meticulously clean. These are the count
rates (photons per second) that the DynaPro Titan instrument was giving when new.
No Cell / 100% Laser Power
~2800
Empty Cell / 100% Laser Power
~7000
Water filled Cell / 100% LaserPower
~50000
These are with the original, Wyatt supplied cell. If you are getting much higher count
rates than these, the cell may need cleaning.
Mild Cleaning
1. Wash the cell with water /acetone and dry
2. Immerse the cell in a 1% solution of LA2 detergent (~50 mls in a Pyrex beaker)
3. Heat the solution to ~ 70 °C, and let the cell soak for a few minutes
4. Wash the cell with water /acetone and dry.
Acid Cleaning
1. Prepare a 3:1 (vol:vol) mixture of concentrated Sulfuric and Nitric acid (Caution dispense Sulfuric and Nitric acid in a fumehood using gloves and eye protection).
2. Put the cuvette into a small glass beaker.
3. Fill to the top with the Sulfuric/Nitric acid mixture using a glass Pasteur pipette.
4. Stopper the cuvette, and invert to mix.
5. Leave to sit for 1 hour - overnight.
6. Cautiously (!) empty the cuvette, rinse several times.
7. Finish by washing copiously with H2O using the cuvette washer.
See Manske et al. A LESS HAZARDOUS CHROMIC-ACID SUBSTITUTE FOR
CLEANING GLASSWARE. Journal of Chemical Education (1990) vol. 67 (11) pp.
A280-A282
Sample preparation
The active volume of the DLS cell is 15 µL. In practice you need 20 µL to fill the cell
without difficulty, and 40-50 µL if you’re going to centrifuge your samples and draw
liquid from the top of the spun material. The required protein concentrations depend
strongly on the size of the protein. Here’s a rough guide ...
Hydrodynamic radius
Appropriate protein
concentration
1 nm
300-1000 µM
2-3 nm
100-600 µM
10 nm
1-10 µM
For quantitative work it’s important to characterize light scattering as a function of
protein concentration ... see below.
It is critical that your sample be free of dust and other particulate matter. You should
filter your sample then subject it to a high speed spin in a benchtop centrifuge. For filtration of small volumes we generally use Microcentrifuge Spin Filters (Millipore
UltraFree-MC with a 0.1- 0.2 µM pore size). Following filtration we spin the sample
at high speed in a Benchtop centrifuge (e.g. 30 mins @ 16400 rpm in an Eppendorf
5417R centrifuge).
Sample Loading
Loading the sample into the cell is best achieved with a micropipette and a gel load
ing tip. The thin tip will fit inside the optical cell and can be gradually withdrawn as
you fill the cell with liquid. Be careful not to introduce bubbles. If you do, these can
usually be dislodged by gently (!) tapping the base of the cuvette on a solid surface.
Routine Data Collection.
Switching on
•
•
•
•
Turn on the DLS (both units) and the associated computer.
Log in and start the Dynamics Software, using the Desktop Link.
Select File -> New
Establish a connection to the DLS Instrument.
Setting parameters
On the left of the data collection window is a panel which allows you to set some parameters. Nothing in the Hardware node is configurable. What you need to check are
the instrument parameters.
Set the Temperature to the value your require. The target temperature is reported on
the bottom left of the MicroSampler, while the actual temperature is reported on the
top left. Transfer your sample into the cuvette and place it in the instrument. After
reaching the target temperature, allow 5 minutes equilibration before beginning data
collection.
You also need to pay attention to the ...
Acquisition time: For small, well-behaved, proteins acquisition times of 1-2s will be
sufficient to observe the complete decay of the intensity correlation function. 5s
should be a universally safe value. What you require is that the ACF decays to a stable
baseline before you stop measuring it. The minimum acquisition time is 1s.
The Number of Acquisitions is what it says ... the number of intensity correlation
functions that will be acquired.
The Laser power is adjustable, and you need to set it to an appropriate value. With
your sample in place, open the instrument control panel. The Counts per Second will
be continuously reported. You should adjust the laser intensity so that you are getting
no more than 800 000 counts/sec. According to Wyatt, this ensures that the photon
counter in the DynaPro Titan is responding linearly.
The other parameters are important for Wyatt’s data analysis procedures, and will influence the numbers it produces, but will not influence how the data is collected.
Collecting data
At this point you can hit start and data will be collected. Every time you hit start you
begin what Wyatt terms a “Measurement”. You will accumulate individual Intensity
autocorrelation functions, which Wyatt terms “Acquisitions”, until you hit stop. The
Wyatt software averages the Acquisitions within a single measurement - outputting a
single averaged ACF and a single set of fit model parameters. Although there’s no way
to access the individual ACFs within a measurement through the Wyatt software, they
are still present within the output “.exp” file and we can access them with lab software. So the distinction between “Measurements” and “Acquisitions” may or may not
matter, depending on your purpose.
Often it’s useful to use the Events Scheduler to automate data collection. This allows
control of the instrument using a simple scripting language. Here’s an example, which
shows how to use the Events Scheduler to collect 500 measurements, with a single
Acquisition in each Measurement.
• With the Parameters node highlighted, right click with the mouse. This will add the
“Event Schedule” node. Highlight it.
• Add a “Do” Event, and enter a value for the number of ACF’s you want to measure
(e.g. 500)
• Add a “Collect Acquisitions” Event. Enter the associated value (e.g. 1)
• Add a “Loop” event, creating a Do loop.
When you’re done it should look like this
Now when you press start, the event scheduler will take over, and you will make the
specified number of measurements 500, each containing only a single acquisition.
Like this (only hopefully your data will look a bit better !!!)
Regardless of how you collect data, once you are done you should save it (File->Save)
For convenience you can save your Event Schedule and other settings (File-> Save
Presets). You can then quickly load these before commencing each data collection
(File->Load Presets).
How Much data?
To get really good size estimates you will need more data than you might think. 5001000 intensity correlation functions would not be excessive ... particularly for small
proteins (< 20 kDa) and lower protein concentrations Note that with an acquisition
times of 1-2s this only requires 15-30 minutes of instrument time.
Checking for scattering from the buffer alone
DLS data analysis will be greatly complicated if the buffer appreciably scatters light .
You should check for light scattering by the buffer ... particularly if you are including
co-solvents to stabilize or unfold your protein, or detergents to keep it in solution. In
the ideal case, random intensity correlation functions will be obtained, resembling
those from pure water. If the buffers you are using do scatter light, then it may be possible to take care of that during data analysis ... but it will require thinking.
Make measurements from a protein dilution series. And replicate !!!
A dilution series is an important part of any quantitative DLS study. Properly, estimates of the translational diffusion coefficient - or the Hydrodynamic radius - need to
be linearly extrapolated to infinite protein dilution. Replication is also an important
part of a quantitative DLS study. This means more than simply collecting a large
number of ACFs on a single sample. The entire dilution series needs to be performed
in triplicate, preferably in randomized order. This is because physical placement of the
cuvette in the instrument, and cuvette cleaning, are both significant sources of variation in the measurements.
For a small monomeric proteins, 5-10 kDa, you’ll probably want to begin with a concentration of 500 - 2000 µM.
The nature of the dilutions is dictated by the volumes you can accurately pipette.
Make sure your pipettes are calibrated. Probably 10 µL is about the smallest you want
to go.
A series like this would be a good start ...
1. 20 µL protein solution
2. 20 µL protein solution + 10 µL sample buffer (0.67x orig concn)
3. 10 µL protein solution + 10 µL sample buffer (0.50x orig concn)
4. 10 µL protein solution + 20 µL sample buffer (0.33x orig concn)
5. 10 µL protein solution + 30 µL sample buffer (0.25x orig concn)
6. 10 µL protein solution + 70 µL sample buffer (0.13x orig concn)
The sample buffer you use for the dilutions must also be filtered and spun.
Correcting to standard conditions
When reporting diffusion coefficients, values are usually corrected to standard conditions (water as a solvent, 20 °C). Here’s how we make that correction (Noting that 20
°C =293.15 K) :
hbuffer,Tb
Dwater,Tw = Tw
D
Tb hwater,Tw buffer,Tb
hbuffer,Tb
Dwater,293.15 = 293.15
D
Tb 1.002 # 10 -3 buffer,Tb
Here Dbuffer,Tb is your measured diffusion coefficient. Tb is the temperature at which
you made your measurements (in Kelvin !) and is ηbuffer,Tb is the dynamic viscosity of
your buffer at temperature Tb. For discussion see Biophysical Chemistry Part II by
Cantor & Schimmel (eqn 10.67, page 584), or Physical Biochemistry by Van Holde
(eqn 5.20, pg 117).
Thermal unfolding experiments using the Event Scheduler.
To be completed.
Data Analysis with Local software
Exporting data from Dynamics
Our lab software now analyzes the Wyatt “.exp” file directly so there is no need to export into any other format.
Data Analysis
To be completed. (It’s all writtenM in the highest quality Fortran 90 (!!) Some familiarity with the Unix Command Line is required).
Measurement of Solution properties.
Quantitative DLS studies require that the density, viscosity and refractive index of
your buffer are all reliably known. For some buffers you can approximate using the
values for pure water (see the Appendix). But this will not always be sufficient ... particularly if you are performing a chemical titration in which some additive (e.g.
TMAO or Glycerol) markedly alters these properties.
Measuring solution density.
The easiest way to measure solution density is to use a modern digital density meter
employing the oscillating U-tube technique (e.g. those manufactured by Anton Paar).
Unfortunately we don’t have such an instrument, so for now we have to do it the oldschool way.
In simplified overview: We weigh a vessel when empty, and when filled with pure water. Since the density of water is known with great accuracy, we can calculate the volume of the vessel. If we then repeat these measurements using our solution of interest,
we can then calculate its density using the known volume of the vessel. Essentiallywe are using water as a reference fluid to calibrate the vessel. If the vessel is transferred into a water bath, and allowed to come to thermal equilibrium after filling then
we can make density measurements at different temperatures. The basic procedure is
laid out nicely in Schiel & Hage (2005). Density measurements are best done using
specialist glassware as a vessel - Guy-Lussac specific gravity bottles. However we can
make do with an ordinary volumetric flask in a pinch.
It sounds relatively simple - but for high precision measurements we need to make
buoyancy corrections to account for the weight of the air being displaced. Let’s consider the first case, after we have weighed the vessel when empty, and when filled
with water (in replicate !!!!). The mean difference between these two sets of measurements is the observed weight of water Wwater/observed which is related to the true
weight of water Wwater/true (as would be measured in a vacuum), by the following expression.
Wwater/true = Wwater/observed c 1 -
tair
tair -1
mc 1 m
tref
twater
Here, ρair, ρwater, and ρref, are the densities of air, water, and the weights used to calibrate the balance, respectively. See Battino and Williamson (1984) for discussion
ρair is ~ 1.2 kg/m3 at room temperature. The exact value can be readily calculated
given the temperature, atmospheric pressure and relative humidity (see Appendix).
ρwater is ~ 1000 kg/m3 at room temperature. The exact value can be looked up in a table (see Appendix). As for ρref, according to Sartorius, the CPA225D mass balance we
have in the lab was calibrated using weights with density 7.95 g/cm3 = 7950 kg/m3. We will
neglect any change in their density as a function of temperature.
Assuming you have obtained the true weight of water in the vessel, the volume of the
vessel can be simply calculated V = Wwater/true / ρwater .
Now finally, if we make replicate measurements on the vessel filled with buffer, and
subtract the weight of the empty vessel, we get Wbuffer/observed. As before ...
Wbuffer/true = Wbuffer/observed c 1 -
tair
tair -1
mc 1 m
tref
tbuffer
Now noting that V = Wbuffer/true / ρbuffer , if we combine these expressions and rearrange, we get the required formula for ρbuffer, involving only known quantities.
Wbuffer/observed c 1 tbuffer =
V
tair
m
tref
+ tair
Sweet ! There is a spreadsheet available to take the labor out of performing these
calculations.
References
Battino and Williamson. Single-pan balances, buoyancy, and gravity or "a mass of
confusion". Journal of Chemical Education (1984) vol. 61 (1) pp. 51-52
Schiel, J.E. and Hage, D.S. Density measurements of potassium phosphate buffer
from 4 to 45 degrees C. Talanta (2005) vol. 65 (2) pp. 495-500
Measuring solution viscosity
We use a Cannon-Ubbelohde Viscometer ... a capillary action viscometer which allows you to estimate the kinematic viscosity of a buffer by measuring the efflux time
through a capillary, under the action of gravity. You need a large volume of buffer
(~50 mls) to make these measurements. The manufacturer’s instructions are good, and
if followed, will yield reliable measurements (see following page). You need to make
3-4 measurements and average them. Because we don’t have a controlled temperature
bath for the viscometer, you’ll need to make measurements in a temperaturecontrolled room (e.g. Room 480B ... 18 °C). From the efflux time you can calculate
the kinematic viscosity as follows
kinematic viscosity (mm2/s) = efflux time (s) x viscometer constant (mm2/s2)
The constant is different for every viscometer- for ours (Number 50 B805) the value
is 0.004474 mm2/s2
Note that the dynamic viscosity (kg m-1 s -1) required for light scattering applications,
is calculated from the kinematic viscosity (m2 s-1) through multiplication by the solution density (kg m-3).
(Watch the units ... 1 mm2 s-1 = 10-6 m2 s-1).
It’s a good idea to make some measurements on water to make sure your technique is
good.
The main difficulty is in cleaning and drying the viscometer following use. This can
be achieved by draining the viscometer; rinsing with water; rinsing with acetone; and
finally pumping dry air through the viscometer to evaporate the residual acetone
(There is a dry air unit for this purpose in room 464).
Instructions for the use of
The Cannon-Ubbelohde Viscometer
See also ASTM D 445, D 446, ISO 3104 and ISO 3105
1. Clean the viscometer using suitable solvents, and by passing clean, dry,
filtered air through the instrument to remove the final traces of solvents.
Periodically, traces of organic deposits should be removed with chromic
acid or non-chromium cleaning solution.
2. If there is a possibility of lint, dust, or other solid material in the liquid
sample, filter the sample through a fritted glass filter or fine mesh screen.
3. Charge the viscometer by pouring enough sample through tube L to fill
the lower reservoir until the liquid meniscus is between the minimum and
maximum fill lines marked on the reservoir.
4. Place the viscometer into the holder and insert it into the constant
temperature bath. Vertically align the viscometer in the bath if a selfaligning holder has not been used.
5. Allow approximately 20 minutes for the sample to come to the bath
temperature.
6. Seal the branching vent tube M with a finger or stopper and apply suction
to tube N until the liquid reaches the center of bulb D. Remove suction
from tube N. Remove seal from vent tube M, and immediately seal tube
N until the sample drops away from the lower end of the capillary R into
bulb B. Then remove seal and measure the efflux time.
7. To measure the efflux time, allow the liquid sample to flow freely down
past mark E, measuring the time for the meniscus to pass from mark E to
mark F to the nearest 0.1 second or 0.01 second.
8. Calculate the kinematic viscosity of the sample by multiplying the efflux
time by the viscometer constant.
9. Without recharging the viscometer, make check determinations by
repeating steps 6 to 8.
The combined expanded1 uncertainty with 95% confidence of the calibration
measurements relative to the primary standard is as follows:
Range of
Constants
2
2
mm /s
RECOMMENDED VISCOSITY RANGES FOR THE
CANNON-UBBELOHDE VISCOMETERS
Size
25
50
75
100
150
200
300
350
400
450
500
600
650
700
Kinematic Viscosity Range
mm 2/s, (cSt)
mm 2/s 2, (cSt/s)
0.002
0.004
0.008
0.015
0.035
0.1
0.25
0.5
1.2
2.5
8
20
45
100
0.5
0.8
1.6
3
7
20
50
100
240
500
1600
4000
9000
20000
to
2
to
4
to
8
to
15
to
35
to
100
to
250
to
500
to
1200
to
2500
to
8000
to 20000
to 45000
to 100000
<0.025
0.025-0.25
0.25-2.5
2.5-25
>25
Expanded Combined
Uncertainty
0.16%
0.22%
0.29%
0.38%
0.44%
The assigned uncertainty of the primary viscosity standard at 20°C is
±0.17%. See ISO 3666.
1
An expanded uncertainty U is determined by multiplying the combined
standard uncertainty uc by a coverage factor k: U = k uc where k = 2. See
NIST Technical Note 1297, 1994 edition, Guidelines for evaluation and
Expressing the Uncertainty of NIST Measurement Results
THIS PRODUCT WAS CALIBRATED WITHIN A QUALITY SYSTEM WHICH IS REGISTERED TO ISO 9002.
CANNON INSTRUMENT CO.
P10.0118
2139 HIGH TECH ROAD
©2002
STATE COLLEGE, PA. 16804
0702
Measuring solution refractive index.
The laser in the DynaPro Titan operates at 831.1 nm. The solution refractive index at
this wavelength can be measured using the Schmidt & Haensch Digital Multiple
Wavelength Refractometer (DSR λ) located in Room 464, For Physics Types ... this is
a critical angle refractometer.
The DSR λ refractometer has 4 programmable methods. Method 1 (the default when
you start the instrument) will measure the refractive index at a fixed temperature (default 18 °C) and at wavelengths of 496.30, 588.20, 650.50, 755.25, 827.40 and
910.90 nm. It outputs the refractive index at nine user selectable wavelengths within
this range, based on a polynomial interpolation of the actual measurements. One of
these wavelengths is 831.1 nm - the wavelength of the laser in the Dynapro Titan DLS
instrument.
Condensed Operating Instructions for the DSR λ
1. Turn on the electronic control unit (back right)
2. Check that the indicator on top of the desiccant cartridge (to the left of the sample
chamber) is completely blue. If it’s pink, don’t use the instrument. You need to regenerate the zeolite beads by removing them from the cartridge, and heating them in an
oven for 1 hour at 275 °C. After repacking them and replacing the cartridge, the indicator should turn blue. See page 7 of the User Manual for details
3. After the instrument finishes initializing, the top line of the display should read
“Method METHOD1”. If it doesn’t someone has been fiddling with the instrument
defaults.
4. Open the cover to the sample chamber and pipette your solution into the sample
compartment. It should cover the prism surface. This will require > 0.3 mls of solution. The chamber holds ~ 7mls of solution when completely full.
5. Press 1 and set the temperature to that required ( default is 18 °C). The instrument
can be operated between 9 and 81 °C. If the ambient temperature in the lab gets too
warm, the manufacturer advises that the inbuilt Peltier device may struggle to maintain 9 °C - but we haven’t yet had any problems.
6. Wait until the temperature has stabilized. The actual temperature is shown at bottom
left; the set temperature is shown at bottom right. This will take a few minutes. Once
the temperature is set, wait an additional 5 minutes before making measurements.
7. Hit start to measure the solution refractive index. A table of refractive index versus
wavelength will result. At the moment, you will have to record these values manually
- there is no electronic output or interface with a computer. You can display the data in
graphical form by pressing the (←) key.
8. Repeat at least once. There will probably be variation in the last decimal place.
9. Switch off the instrument.
10. Remove the sample from the chamber. Rinse with distilled water and gently wipe
dry with a KimWipe.
Controls are good: Measurements on water
Here are the measured and “calculated” values for deionized water @ 18 °C, obtained
in Sept 2008. Without much special effort you should be able to obtain 3DP accuracy.
Actually ... we should be able to do better than this ... investigation is ongoing.
Measured
Calculated*
490.0
1.33711
1.33747
546.1
1.33469
1.33500
589.3
1.33322
1.33352
630.0
1.33204
1.33235
690.0
1.33055
1.33090
750.0
1.32927
1.32968
831.1
1.32780
1.32825
860.0
1.32734
1.32777
910.0
1.32659
1.32698
*See the Appendix for details
If you can’t reproduce at least this, the sample chamber isn’t clean, the water isn’t
deionized, or the instrument needs calibration (in rough order of likelihood).
A note on making measurements at elevated temperatures:
When making measurements on solutions above 40 °C, solution evaporation becomes
a problem, as there is a large dead air volume in the refractometer sample chamber.
Using the minimal volume required to cover the prism surface is not a good idea in
this case, as significant volume evaporation will occur. However using very large volumes is also sub optimal, as the Peltier unit will be extremely slow to shift temperature. The best solution is to set the empty sample chamber to the required temperature and to heat the solution to the required temperature in an external heat block or
water bath. Obviously the solution should be in a tightly capped vessel during this
step. Then rapidly transfer 1-2 mls of the solution into the refractometer sample
chamber. It will quickly come to equilibrium, and refractive index measurements can
be made before significant evaporation has occurred.
Appendix A: Refractive Index (831.1 nm), Dynamic Viscosity, and Density of
Pure Water, as a Function of Temperature (Atmospheric Pressure)
A1. Density of pure water at atmospheric pressure was calculated using the basic expression of Kell (1975). The coefficients in Kell’s expression were slightly revised by Bettin &
Spieweck (1990). See Batista & Paton (2007) for an English language commentary.
References:
Kell, G.S. Density, thermal expansivity, and compressibility of liquid water from 0 degrees
to 150 degrees C - Correlations and tables for atmospheric-pressure and saturation reviewed
and expressed on 1968 temperature scale. J Chem Eng Data (1975) vol. 20 (1) pp. 97-105
Batista, E. and Paton, R. The selection of water property formulae for volume and flow calibration. Metrologia (2007) vol. 44 (6) pp. 453-463.
Bettin H and Spieweck F Die Dichte des Wassers als Funktion der Temperatur nach
Einführung der Internationalen Temperaturskala von 1990. (1990) PTB Mitt. 100 195–6
Temperature (°C)
Density (kg m-3)
0.00
999.8395
1.00
999.8986
2.00
999.9399
3.00
999.9642
4.00
999.9720
5.00
999.9638
6.00
999.9401
7.00
999.9014
8.00
999.8481
9.00
999.7807
10.00
999.6994
11.00
999.6048
12.00
999.4971
13.00
999.3767
14.00
999.2439
Temperature (°C)
Density (kg m-3)
15.00
999.0991
16.00
998.9424
17.00
998.7742
18.00
998.5948
19.00
998.4043
20.00
998.2031
21.00
997.9914
22.00
997.7693
23.00
997.5372
24.00
997.2951
25.00
997.0433
26.00
996.7820
27.00
996.5114
28.00
996.2315
29.00
995.9427
30.00
995.6450
31.00
995.3386
32.00
995.0237
33.00
994.7003
34.00
994.3686
35.00
994.0288
36.00
993.6810
37.00
993.3253
38.00
992.9618
39.00
992.5906
40.00
992.2119
41.00
991.8257
42.00
991.4321
43.00
991.0313
Temperature (°C)
Density (kg m-3)
44.00
990.6234
45.00
990.2084
46.00
989.7864
47.00
989.3575
48.00
988.9219
49.00
988.4795
50.00
988.0304
51.00
987.5748
52.00
987.1128
53.00
986.6443
54.00
986.1694
55.00
985.6883
56.00
985.2010
57.00
984.7075
58.00
984.2079
59.00
983.7024
60.00
983.1908
61.00
982.6734
62.00
982.1501
63.00
981.6209
64.00
981.0861
65.00
980.5456
66.00
979.9994
67.00
979.4476
68.00
978.8902
69.00
978.3274
70.00
977.7591
71.00
977.1854
72.00
976.6063
Temperature (°C)
Density (kg m-3)
73.00
976.0219
74.00
975.4322
75.00
974.8372
76.00
974.2370
77.00
973.6316
78.00
973.0210
79.00
972.4054
80.00
971.7847
81.00
971.1589
82.00
970.5281
83.00
969.8923
84.00
969.2515
85.00
968.6058
86.00
967.9552
87.00
967.2997
88.00
966.6394
89.00
965.9742
90.00
965.3043
91.00
964.6295
92.00
963.9500
93.00
963.2658
94.00
962.5768
95.00
961.8831
96.00
961.1848
97.00
960.4818
98.00
959.7741
99.00
959.0618
100.00
958.3449
A2. Dynamic viscosity of pure water at atmospheric pressure was calculated using equation 16 of Batista and Paton (2007). The form of this expression is originally due to Kestin,
Sokolov & Wakeham (1978)
References:
Batista, E. and Paton, R. The selection of water property formulae for volume and flow calibration. Metrologia (2007) vol. 44 (6) pp. 453-463.
Kestin, J. et al. Viscosity of liquid water in the range -8 °C to 150 °C. J Phys Chem Ref Data
(1978) vol. 7 pp. 941
Temperature (°C)
Dynamic Viscosity
(Pa.s = kg.m-1.s-1)
0.00
1.79064E-03
1.00
1.73088E-03
2.00
1.67410E-03
3.00
1.62012E-03
4.00
1.56877E-03
5.00
1.51988E-03
6.00
1.47331E-03
7.00
1.42892E-03
8.00
1.38659E-03
9.00
1.34618E-03
10.00
1.30760E-03
11.00
1.27073E-03
12.00
1.23548E-03
13.00
1.20176E-03
14.00
1.16948E-03
15.00
1.13857E-03
16.00
1.10894E-03
17.00
1.08053E-03
18.00
1.05328E-03
19.00
1.02712E-03
Temperature (°C)
Dynamic Viscosity
(Pa.s = kg.m-1.s-1)
20.00
1.00200E-03
21.00
9.77859E-04
22.00
9.54649E-04
23.00
9.32322E-04
24.00
9.10835E-04
25.00
8.90145E-04
26.00
8.70212E-04
27.00
8.51000E-04
28.00
8.32473E-04
29.00
8.14598E-04
30.00
7.97344E-04
31.00
7.80682E-04
32.00
7.64583E-04
33.00
7.49023E-04
34.00
7.33976E-04
35.00
7.19419E-04
36.00
7.05330E-04
37.00
6.91688E-04
38.00
6.78473E-04
39.00
6.65667E-04
40.00
6.53252E-04
41.00
6.41211E-04
42.00
6.29529E-04
43.00
6.18191E-04
44.00
6.07183E-04
45.00
5.96490E-04
46.00
5.86101E-04
47.00
5.76004E-04
Temperature (°C)
Dynamic Viscosity
(Pa.s = kg.m-1.s-1)
48.00
5.66186E-04
49.00
5.56637E-04
50.00
5.47348E-04
51.00
5.38307E-04
52.00
5.29506E-04
53.00
5.20936E-04
54.00
5.12589E-04
55.00
5.04457E-04
56.00
4.96532E-04
57.00
4.88807E-04
58.00
4.81276E-04
59.00
4.73932E-04
60.00
4.66768E-04
61.00
4.59780E-04
62.00
4.52961E-04
63.00
4.46307E-04
64.00
4.39812E-04
65.00
4.33471E-04
66.00
4.27282E-04
67.00
4.21238E-04
68.00
4.15336E-04
69.00
4.09572E-04
70.00
4.03942E-04
71.00
3.98444E-04
72.00
3.93074E-04
73.00
3.87828E-04
74.00
3.82704E-04
75.00
3.77699E-04
Temperature (°C)
Dynamic Viscosity
(Pa.s = kg.m-1.s-1)
76.00
3.72811E-04
77.00
3.68036E-04
78.00
3.63373E-04
79.00
3.58819E-04
80.00
3.54373E-04
81.00
3.50031E-04
82.00
3.45793E-04
83.00
3.41656E-04
84.00
3.37620E-04
85.00
3.33681E-04
A2. Refractive index of pure water (831.1 nm) at atmospheric pressure was calculated
using an expression due to Schiebener et al (1990). The coefficients of this expression have
been revised subsequent to this paper - see the IAPWS release cited below. Note that the density of water appears, which appears in the expression for the refractive index, was calculated
as detailed above.
References:
P. Schiebener, J. Straub, J.M.H. Levelt Sengers and J.S. Gallagher, J. Phys. Chem. Ref. Data
19, 677, (1990).
The International Association for the Properties of Water and Steam: Release on the Refractive Index of Ordinary Water Substance as a Function of Wavelength, Temperature and Pressure (1997).
Temperature (°C)
Refractive index @
831.1 nm
0.00
1.32902
1.00
1.32902
2.00
1.32901
3.00
1.32900
4.00
1.32898
5.00
1.32895
6.00
1.32893
7.00
1.32889
8.00
1.32885
9.00
1.32881
10.00
1.32876
11.00
1.32871
12.00
1.32866
13.00
1.32860
14.00
1.32854
15.00
1.32847
16.00
1.32840
17.00
1.32832
Temperature (°C)
Refractive index @
831.1 nm
18.00
1.32825
19.00
1.32816
20.00
1.32808
21.00
1.32799
22.00
1.32790
23.00
1.32780
24.00
1.32771
25.00
1.32760
26.00
1.32750
27.00
1.32739
28.00
1.32728
29.00
1.32717
30.00
1.32705
31.00
1.32693
32.00
1.32681
33.00
1.32669
34.00
1.32656
35.00
1.32643
36.00
1.32630
37.00
1.32617
38.00
1.32603
39.00
1.32589
40.00
1.32575
41.00
1.32561
42.00
1.32546
43.00
1.32531
44.00
1.32516
45.00
1.32501
Temperature (°C)
Refractive index @
831.1 nm
46.00
1.32485
47.00
1.32469
48.00
1.32453
49.00
1.32437
50.00
1.32421
51.00
1.32404
52.00
1.32387
53.00
1.32370
54.00
1.32353
55.00
1.32336
56.00
1.32318
57.00
1.32300
58.00
1.32282
59.00
1.32264
60.00
1.32246
61.00
1.32227
62.00
1.32208
63.00
1.32189
64.00
1.32170
65.00
1.32151
66.00
1.32131
67.00
1.32112
68.00
1.32092
69.00
1.32072
70.00
1.32051
71.00
1.32031
72.00
1.32011
73.00
1.31990
Temperature (°C)
Refractive index @
831.1 nm
74.00
1.31969
75.00
1.31948
76.00
1.31927
77.00
1.31905
78.00
1.31884
79.00
1.31862
80.00
1.31840
81.00
1.31818
82.00
1.31796
83.00
1.31774
84.00
1.31751
85.00
1.31728
86.00
1.31705
87.00
1.31682
88.00
1.31659
89.00
1.31636
90.00
1.31613
91.00
1.31589
92.00
1.31565
93.00
1.31541
94.00
1.31517
95.00
1.31493
96.00
1.31469
97.00
1.31444
98.00
1.31420
99.00
1.31395
100.00
1.31370
Appendix B: Calculating the density of air.
Air has a density around 1.2 kg/m3 but the exact value depends on the amount of water it’s carrying. In order to calculate the exact value, we assume you have measured
the usual variables: Temperature (°C), Relative Humidity (%) and Atmospheric Pressure (mbar = hPa). We have a Digital Barometer which can make all three measurements.
Note the following simple unit conversions
P (kPa) = 10-1 x P (hPa)
T (K) = 273.15 + T (°C)
Next we calculate the Saturation Vapor Pressure, PSAT, of water (in kPa) from the
Temperature. There’s a huge number of different empirical equations to do this. We’ll
use the very simple expression of Bolton (1980)
17.67T
PSAT = 0.6112 exp ( T + 243.5 )
In this formula T is specified in °C. Bolton’s formula is pretty accurate up to 30°C.
Hopefully the mass balance isn’t in a room hotter than this !!!
Then we can calculate the actual Vapor Pressure, PV, of water (in kPa) from PSAT and
the Relative Humidity (RH). By definition ...
PV =
PSAT # RH
100
Now that the vapor pressure of water is known, we can calculate the vapor pressure of
dry air, PD, in kPa from the total (= atmospheric) pressure, P.
P = PV + PD
Finally we can obtain calculate the density of air, ρair, (in kg/m3) as a mixture of ideal
gases (water vapor and dry air)
tair =
PV # 10 3 PD # 10 3
RV T + RD T
where RV = 461.495 J.kg-1.K-1, is the specific gas constant for water vapor
and RD = 287.058 J.kg-1.K-1, is the specific gas constant for dry air
(are these the “best” values for the gas constants ? ... who can say)
Reference
Bolton, D., The computation of equivalent potential temperature, Monthly Weather
Review, 108, 1046-1053, 1980