The task of calculating and interpreting the correlation functions can be
broken down into three logical parts:
Graphical output of results is catered for, and is one of the options
controlled by tailinput. The programming behind this is complicated, but was
written to ease transport of the programs. To plot a graph, a program creates
otoko format files containing the data, and a set of shell scripts. The shell
scripts run otoko and force it to plot the data contained in the files just
created. The name used for dummy otoko files is O99000.TMP, or something
similar. The shell scripts created by the programs are invoked by corfunc, and
delete themselves once plotting is over. They use the otoko ".pnt" command that
requires the user to click with the middle mouse button to clear the graph -
any of the standard plotting commands would leave no delay between plotting the
graph and clearing it, when run from a shell script. This may seem an
unnecessarily complicated way of producing plots, but it can be used on any
system that supports the Xwindows version of otoko, and in general works
seemlessly. Note however that should any changes be made to otoko, the programs
might not be able to produce graphics. In this case, the graphics option can
either be toggled off, or some minor alterations made to the Fortran code to
enable the programs to use some other graph plotting program.
Before discussing each of the analysis programs in turn, we will consider the
first few inputs made to tailinput. The user is prompted for the name of the
directory containing the otoko files to be analysed. This can be any UNIX path
name such as /img8/ajr/aug94, or ../jul94, or you can simply press return if
the current directory contains your data. The user is the prompted for a file
name. All file names should be in the form X??000.!!!, where ?? is a file
number and !!! a file extension. Using any other name format will result in
error messages like
at some point in the analysis. Whenever a fatal error occurs all the programs
stop. The user is also prompted for a Q axis filename, before tailinput checks
all the data. If the file supplied contains realtime data, the user is then
prompted for the number of the first frame to be analysed, the last frame to be
analysed, and the increment between frames. If the user, when prompted, chooses
the absolute intensities option, the programs assume the intensities are in
reciprocal centimetres. Using absolute units is recommended, as more
information will be available when interpreting the correlation functions. Note
that pressing the return key when prompted by tailinput selects the default for
that input. Tailinput keeps a record of defaults, speeding up user input.
Tailinput is case sensitive!
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Task.
Tailfit fits either a Porod3 or sigmoid4 profile to the tail of the intensity
data. It then passes the fit parameters to tailjoin for extrapolation of the
data to q=infinity. The sigmoid profile should generally be used. It uses an
intensity profile of the form:
where B is a Bonart thermal background, K is the Porod constant, and σ
describes the electron density profile at the interface between crystalline and
amorphous sections.
Sigmoid tail interface width.
The parameter σ is in Angstroms, and is a measure of the width of the
distribution that gives rise to the sigmoid electron density profile. Fitting
with a sigmoid profile is a non-linear problem, and requires a robust fitting
algorithm: a Levenburg-Marquart method5 is used. An initial guess is made at
the fit parameters by performing least squares fits to certain graphs and
measuring gradients.
The Porod tail profile has the form:
where B is a Bonart thermal background and K is the Porod constant. Using a
Porod profile usually gives poor quality correlation functions, and is included
for completeness. However, if a Porod profile is used, a greater number of
structural parameters can be extracted, including interface surface to volume
ratios and the Porod characteristic chord length. Fitting becomes a linear
problem, and so can be less problematic than fitting to a sigmoid profile. The
parameters K and B are measured from a standard Porod-type plot of I(q) vs. 1 /
q4.
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User input.
The most important parameters controlled by the user are those concerning which
data are used for the tail fitting. The user is prompted by tailinput for a
channel number at which to start fitting, and a channel number at the end of
the tail. If realtime data is being analysed, the user can input different
channel numbers for each frame by typing "n" at the prompt:
Same channel limits on each frame for tailfit [y/n] [y] -->
Choosing limits can be difficult, and some experimentation may be required. As
a general rule, the start channel should have a q value about twice that at
which any peak occurred in the intensity data. The end channel should be as
large as possible, but should avoid any detector noise. More will be said about
tail fit channel limits below. Tailfit can also search through a range of start
channels to optimise the fit. The user is prompted:
Enter start channel optimisation range (type n for no opt.) [0] -->
Typing "n", pressing return, or entering zero will result in no optimisation.
If the user entered, say, a value of five, tailfit would work through the
channels from five to the left of the start channel specified by the user, to
five channels to the right, searching for the best fit. This option is perhaps
best ignored as the optimum starting point is invariably as far as possible to
the right of the start channel specified by the user.
Tailinput also prompts the user for a number of iterations. Usually the default
value of 100 is fine, but if start channel optimisation is being performed, or
if warning messages appear during tail fitting, the number should be increased.
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Possible errors.
Any of the programs may produce errors concerning dummy files, often called
status files or transfer files by the programs themselves. If one of these
error messages occurs it will be fatal and the programs will stop. They should
only occur if a dummy file has been deleted by the user during the analysis
session, or if there are problems with the file system, for example ownership
problems or a server crashing. The only action is to run corfunc again from the
start.
However, several non-fatal errors can occur in tailfit and these should be
looked out for. If a single frame is being analysed using a sigmoid profile,
you may see a message like:
Warning! positive gradient during estimation of sigma.
This refers to the graph used to gain an initial estimate of s. When this
message appears, the initial guess at s will be poor, and this may affect the
non-linear fitting. If the same error occurs during fitting to realtime data,
no error message will appear. Following this error message, if the fits are
poor, the analysis should be performed again with different tail channel
limits.
Another message that may occur is:
Warning: ran out of iterations before convergence...
If, for a static image, this gives a poor quality fit, the analysis should be
run again with a larger number of iterations specified in tailinput. For
realtime data, if the message appears for several frames, again the number of
iterations should be increased. The number of iterations required should never
be higher than 500: if it is, something is wrong.
The most serious message that can occur is:
Fitting problem: singular alpha matrix...
This will only occur during sigmoid profile fitting. The error message refers
to a problem with the Levenburg-Marquart algorithm that isn't properly
understood. When this error occurs, the program automatically sets s to zero,
and performs a linear least squares fit for the background and Porod constant.
Changing fit channel limits or removing frames from realtime data may get
around the problem.
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Files created
(assuming intensity file A01000.XSH and q-axis file X00000.QAX).
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For all files:
|
|
Tailinfo.dat
|
Text file containing fit parameters.
|
|
For realtime data:
|
|
A01000.BAK
|
Otoko file containing thermal background vs. frame.
|
|
A01000.POR
|
Otoko file containing Porod constant vs. frame.
|
|
A01000.SIG
|
Otoko file containing sigma value vs. frame.
|
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A01000.FAX
|
Otoko file containing frame no.s for use as an axis with the files above. |
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Graphics output.
For fitting to static images an overplot of the experimental tail and
calculated tail is plotted. For realtime data, graphs of each of the fit
parameters vs. frame number will be plotted.
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Notes.
Since the experimental data is extrapolated to very high q, the majority of the
data used in the Fourier transform comes from the tail fit. It's therefore very
important to check that the tail fit is good. The tail affects points on the
correlation function at low values of R (real space coordinate) to the greatest
extent, but these points are the most important in the extraction of ideal
lamellar morphology parameters. Hence the results from an analysis session
depend greatly on the tail fit.
In particular, watch out for values of s below 1. When realtime data is
analysed, s will often be approximately zero for a large proportion of frames.
This is an indication that the tail fitting is failing, and that your channel
limits could be changed. Moving the start channel towards the beamstop usually
increases s, but moving the limit in too far while keeping the end limit at
high q will result in very poor fits.
Choosing channel limits is a payoff between using as many points as possible to
ensure a good fit, but wanting to keep as many points as possible from your
experimental data in the extrapolated data that is passed to the Fourier
transform. A noisy tail on the experimental data will result in poor tail fits,
possibly making correlation function analysis impossible.
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Task.
Tailjoin extrapolates the experimental data back to q=0 using a Guinier model.
It then creates otoko files containing intensities from q=0 to beyond q=0.6
using the parameters found by tailfit. Due to the nature of the tail fitting,
the join between experimental data and data in the calculated tail usually
involves a step that could cause ripples in the correlation function. Hence
this join is smoothed.
The intensity profile used by the Guinier model has the form:
where B is negative. The Guinier model assumes the small angle scattering
arises from particles and the parameter B is related to the radius of gyration
of those particles. This obviously has dubious applicability to polymer
systems. However, the correlation function is affected by the Guinier
back-extrapolation to the greatest extent at large values of R, and so the
back-extrapolation only has a small effect on the analysis. The Guinier profile
is fitted to the first few genuine scattering points after the beamstop. If
your experimental data does not contain an upturn in intensity at low q, back
extrapolation may fail. As an alternative to the Guinier profile, a Vonk
profile can also be used:
Little work has been performed using the Vonk model.
The channel at which tail fitting started also marks the point at which
calculated data takes over from experimental data in the extrapolated data
file. This region is smoothed using a Savitzky-Golay5 smoothing algorithm that
smoothes the join without greatly altering higher moments of the data. Note
that no smoothing is employed for the back-extrapolation as so few points are
involved. The join between data sets will give rise to ripples in the
correlation functions with a period matching the D-spacing at which the join
occurs. These aren't usually visible.
The point in q to which extrapolation is performed affects the correlation
functions, particularly if it is too small. The value of q=0.6 used by the
programs was decided on after experimentation, and gives smooth correlation
functions without loss of speed. The lower the truncation point is, the rougher
the correlation functions, while the higher the truncation point is, the slower
the transform. A truncation point of q=0.6 corresponds to fluctuations in the
correlation functions of about 10 . These are usually not observable.
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User input.
User input for tailjoin occurs in tailinput. Firstly the back-extrapolation
model must be specified: a Guinier model is best. Then tailinput searches for
the first genuine data point after the beamstop by comparing intensities with
that at q=0 and by waiting for the sudden decrease in intensity after the
beamstop. The process doesn't always work, so at the prompt:
Enter channel at start of genuine data [73] -->
press return only if the default value is sensible. If tailjoin is not passed a
sensible channel number the program will crash.
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Possible errors.
The most frequent errors are caused by an incorrect value for the start of
genuine data being passed to tailjoin or the absence of an upturn in intensity
as you approach the beamstop. A message like:
Guinier back-extrapolation failed!
or
Back-extrapolae channel optimisation has failed: fatal...
indicates problems with the first genuine data point. The user should inspect
the intensities near the beamstop before running corfunc again. The fatal error
message:
Problem with channel limits...
will only occur if one of the start or end channel limits for the tail fit is
very close to the beginning or end of the experimental data. A serious fatal
error occurs if the experimental data contains negative intensities close to
the point at which the experimental data is joined to the extrapolated tail:
Error: negative intensities in experimental data: fatal...
If this occurs the user should add some arbitrarily large constant to the
experimental data using otoko, and re-perform the tail fitting. Tailfit will
automatically subtract any constant background added to the data, and no
problems will occur in tailjoin. Finally, if the experimental work was
performed at very long camera lengths, more than 2048 points may be needed to
extrapolate the data to beyond q=0.6. If this occurs a warning message is
displayed and the program stops. A re-program would be necessary to get around
this problem.
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Files created
(assuming intensity file A01000.XSH and q-axis file X00000.QAX).
|
For all data:
|
|
A01000.FUL
|
Otoko file containing extrapolated intensities.
|
|
X00000.FLX
|
Otoko file containing extrapolated q values used as a q-axis for the above.
|
|
For realtime data:
|
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A01000.RAD
|
Otoko file containing the Guinier radius of gyration vs. frame no. |
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Graphics output.
For static data, the full extrapolated dataset is displayed, followed by a plot
of the smoothed intensities around the join area. For realtime data, a plot of
the Guinier radius of gyration obtained from the back-extrapolation is
displayed. This can largely be ignored.
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Notes.
Check correlation functions for fluctuations with periods corresponding to the
truncation point of q=0.6 (ie. period of 10) and to the point at which
extrapolated data meets experimental data. The latter effect will be most
noticeable when the SAXS curve is featureless, for example during a melt. When
inspecting the smoothed joined, don't be alarmed by strange bulges in the data.
The point of smoothing is to avoid large steps in the data, as these give rise
to the sharpest fluctuations in the correlation functions. On the other hand,
always make sure these bulges haven't introduced ripples.
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Task.
Extract interprets Γ1 in terms of an ideal lamellar morphology,
and extracts structural parameters. The program also displays the results of
calculating the moments of the experimental SAXS curve, and Porod results. If
tail fitting was performed using a Porod instead of a sigmoid profile, a more
detailed Porod analysis is performed.
The program first decides whether a lamellar interpretation can be applied. It
searches for the first local minimum with a negative Γ1 coordinate
and the first local maximum with a positive Γ1
coordinate. If these cannot be found, extraction of the lamella structural
parameters is abandoned (for that particular frame). If these features are
found extraction can be performed, and will employ these two features. Note
these criteria very carefully: local minima above the abscissa will be ignored
and not used in the calculation of structural parameters. Similarly, local
maxima below the abscissa will not be used in the calculation of structural
parameters. Indeed, the interpretation of any one dimensional correlation
function deviating from the ideal lamellar model is not yet properly
understood.
A diagram of the one dimensional correlation function from an ideal lamellar
morphology is given above. It consists of a gradually decaying oscillation,
with an initial linear section at low values of R. Structural parameters are
derived from the positions of the first local minimum and local maximum, and
the position and gradient of the linear section.
|
Parameter
|
Symbol
|
Measurement
|
|
Long period
| Lp
|
As in diagram
|
|
Bulk crystallinity
|
φ
|
Γmin / (Γmin + G*)
|
|
Av. hard block thickness
| Lc
|
As in diagram
|
|
Av. soft block thickness
| La
| Lp - Lc
|
|
Local crystallinity
| φl
| Lc / Lp
|
|
Average core thickness
| D0
|
As in diagram
|
|
Average interface thickness
| Dtr
|
As in diagram
|
|
Polydispersity
|
|
Γmin / Γmax
|
|
Electron density contrast
|
Δρ
|
Q / φ (1-φ)
|
|
Specific inner surface
|
|
2f / Lc
|
|
Non-ideality
|
| (Lp - Lp*)2 / Lp2 |
Most of these parameters are given in reference 1. The polydispersity
measurement was suggested by Guy Eeckhaut. The problem of determining Dtr
and D0 has not been straightforward. Algorithms have been developed
with consistency as a priority, so that even if the structural interpretation
of these parameters is dubious, the line they specify gives consistent values
to the bulk crystallinity, hard block thickness and so on. This has not been an
easy task, made harder by the fact that poor tail fits radically alter the
appearance of Γ1 at low R. Early algorithms involved polynomial
interpolation and algebraic differentiation to determine the first derivative
of Γ1 in the region of the linear section. The linear section
was then taken to be some region surrounding the point with greatest gradient.
However, when low s values arise from tail fitting, Γ1
often has its steepest gradient at very low R in the form of an upturn next to
the ordinate. This distracted the derivative-based algorithms away from the
genuine linear section.
The present algorithm has been slightly more successful, but still does not work
very well. It considers every possible value of Dtr and D0,
and for each pair of values constructs a calculated Γ1 profile.
This is done by performing a linear least squares fit to the points between Dtr
and D0 and then connecting this linear region back to the ordinate
and to the first local minimum with a polynomial. The position of the linear
section defines these polynomials uniquely, as we know two points on each
polynomial, and the gradient at those points. A goodness of fit parameter is
then assigned to the calculated profile. The algorithm finds the values of Dtr
and D0
that give rise to the best calculated profile.
Once extraction of the lamellar structural parameters is complete, Porod
analysis is performed. If the lamellar interpretation of Γ1
failed, a crystallinity provided by the user in tailinput is used to calculate
the electron density contrast, otherwise this was given in the lamellar
parameter extraction. If a Porod profile was used for tail fitting, a detailed
Porod analysis is performed. Those frames for which lamellar interpretation
succeeded are analysed using the bulk crystallinity measurement from the
lamella interpretation. Those frames for which lamellar interpretation failed
are analysed using the user's estimate of crystallinity. The parameters
calculated by the Porod analysis are given below.
The calculated profile fitted to Γ1 for extraction of Dtr
and D0.
|
Parameter
|
Symbol
|
Measurement
|
|
Porod const.
|
K
|
Performed in tailfit
|
|
Surface to volume ratio
|
|
π K φ (1 - φ) / Q
|
|
Characteristic chord length
| lp
|
4Q / π K
|
|
Crystalline chord length
|
| lp
/ (1 - φ)
|
|
Amorphous chord length
|
| lp / φ |
where Q is the second moment and φ is a crystallinity measurement (either
obtained from the correlation function, or user-supplied).
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User input.
Extract is an exception to the rule that all user input occurs in tailinput.
When tailinput is run, however, the user is prompted for an estimate of
crystallinity which is used in the Porod analysis should lamellar
interpretation of the correlation function fail. Extract specifies for each
frame whether the crystallinity it uses in calculations is user supplied or
provided by the correlation function.
Tailinput also prompts the user:
Do you want user control of extraction process [y/n] [n] -->
This refers to the selection of the linear section of the correlation function.
The first time you analyse a dataset you will probably want extract to select
the linear section automatically. However, if the measurements of Dtr
and D0 made by extract are inconsistent (ie. vary widely within a
realtime dataset) you may want to re-run the analysis by supplying values for Dtr
and D0 yourself. If user control is selected during tailinput,
extract will prompt the user for Dtr and D0
for every frame. A system of defaults makes this a reasonably rapid process.
Having performed an analysis with automatic selection of the linear section,
you may want to re-perform the extraction process with the user selecting the
linear section, but without having to tailfit and transform the data over
again. This can be done by editing the file corfunc.dat and changing the line
that reads "user" to read "auto". Then run extract directly (ie. not from
within corfunc.sh). Corfunc.dat resides in the directory that contains the
correlation function analysis programs.
-
Possible errors.
Unlikely to occur.
-
Files created
(assuming intensity file A01000.XSH and q-axis file X00000.QAX).
|
For all datasets:
|
|
A01000.OUT
|
Text file containing a copy of the information echoed to the screen. |
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Graphics output
None.
-
Notes.
The user should always check the values of Dtr and D0 measured
by extract. If they vary widely within a realtime dataset, or inspection of
Γ1 suggests different values, extraction should be re-performed
with user supplied values based on the appearance of Γ1. Checks
should also be made on the appearance of Γ1 at low values of R,
particularly when tail fitting has given s as zero. Finally, remember that the
parameters calculated by extract are based on the assumption of an ideal
lamellar morphology. If Γ1 differs from the type of correlation
function expected for such a morphology (for example if the first local maximum
lies below the abscissa) parameters calculated by extract may be meaningless.
