1
00:00:10,180 --> 00:00:12,830
STUDENT: Materials are often
classified as crystalline,

2
00:00:12,830 --> 00:00:14,920
semi-crystalline, or amorphous.

3
00:00:14,920 --> 00:00:17,110
Most polymers are
semi-crystalline.

4
00:00:17,110 --> 00:00:18,790
This means a fraction
of the material

5
00:00:18,790 --> 00:00:22,132
is amorphous and lacks order in
arrangements of polymer chains,

6
00:00:22,132 --> 00:00:23,590
while the remainder
of the material

7
00:00:23,590 --> 00:00:26,070
exhibits parallel,
well-ordered chains that

8
00:00:26,070 --> 00:00:28,540
define crystalline regions.

9
00:00:28,540 --> 00:00:30,850
There are several methods
for determining the degree

10
00:00:30,850 --> 00:00:32,880
of crystallinity in a polymer.

11
00:00:32,880 --> 00:00:35,380
X-ray diffraction can be
used to determine degree

12
00:00:35,380 --> 00:00:38,650
of crystallinity, because
intensity of X-ray diffraction

13
00:00:38,650 --> 00:00:42,120
reflection is proportional
to the material density.

14
00:00:42,120 --> 00:00:44,260
An X-ray diffraction
pattern displays

15
00:00:44,260 --> 00:00:46,390
both sharp Bragg's
peaks, arising

16
00:00:46,390 --> 00:00:49,650
from the crystalline portion,
and a broad diffraction peak

17
00:00:49,650 --> 00:00:51,880
caused by scattering from
the amorphous portion

18
00:00:51,880 --> 00:00:53,260
of the polymer.

19
00:00:53,260 --> 00:00:55,390
The degree of crystallinity
can be determined

20
00:00:55,390 --> 00:00:58,840
using this formula, where the
total integrated intensities,

21
00:00:58,840 --> 00:01:01,810
all of the sharp peaks, are
divided by the integrated

22
00:01:01,810 --> 00:01:04,030
intensities of all
peaks, including

23
00:01:04,030 --> 00:01:06,100
that of the amorphous peak.

24
00:01:06,100 --> 00:01:10,210
Many laboratories have access to
XRD peak fitting software that

25
00:01:10,210 --> 00:01:13,030
can be used to calculate
degree of crystallinity.

26
00:01:13,030 --> 00:01:15,760
But this software can
often be expensive.

27
00:01:15,760 --> 00:01:18,040
I set out to create a
peak fitting software

28
00:01:18,040 --> 00:01:19,930
to calculate degree
of crystallinity

29
00:01:19,930 --> 00:01:23,170
for semi-crystalline
polymers using Mathematica.

30
00:01:23,170 --> 00:01:25,030
The basic idea for
this program is

31
00:01:25,030 --> 00:01:28,030
that the user will define
a series of Gaussian peaks

32
00:01:28,030 --> 00:01:29,770
using this equation.

33
00:01:29,770 --> 00:01:34,510
The width, the height, and
the exposition of these peaks

34
00:01:34,510 --> 00:01:37,090
can be manipulated.

35
00:01:37,090 --> 00:01:39,970
This code allows the user
to adjust the appearance

36
00:01:39,970 --> 00:01:42,160
and position of multiple
peaks, and match it

37
00:01:42,160 --> 00:01:44,440
to an existing XRD pattern.

38
00:01:44,440 --> 00:01:46,660
First, the code allows
the user to adjust

39
00:01:46,660 --> 00:01:50,680
the position and the
scale of the XRD pattern.

40
00:01:50,680 --> 00:01:53,170
The user is then able
to adjust the positions

41
00:01:53,170 --> 00:01:55,615
of both the crystalline
peaks and amorphous peaks.

42
00:02:00,840 --> 00:02:03,330
Once the user obtains a fit
that they're happy with,

43
00:02:03,330 --> 00:02:05,610
they can calculate the
degree of crystallinity

44
00:02:05,610 --> 00:02:07,590
by determining the
integrated area

45
00:02:07,590 --> 00:02:10,800
under the crystalline peaks,
divided by the integrated

46
00:02:10,800 --> 00:02:12,465
area under all of the peaks.

47
00:02:15,090 --> 00:02:18,240
In this particular polymer,
the determined degree

48
00:02:18,240 --> 00:02:21,420
of crystallinity was 41%.

49
00:02:21,420 --> 00:02:24,050
To test the accuracy
of my code, I

50
00:02:24,050 --> 00:02:26,640
fit XRD patterns for
polyethylene varieties

51
00:02:26,640 --> 00:02:29,670
with known values for
degree of crystallinity.

52
00:02:29,670 --> 00:02:32,820
The polymers for which I
chose to match XRD patterns

53
00:02:32,820 --> 00:02:36,330
were low density polyethylene,
high density polyethylene,

54
00:02:36,330 --> 00:02:39,390
and ultra high molecular
weight polyethylene.

55
00:02:39,390 --> 00:02:42,690
The expected value for degree
of crystallinity for low density

56
00:02:42,690 --> 00:02:46,620
polyethylene is 44%
to 51% crystallinity.

57
00:02:46,620 --> 00:02:49,110
The value obtained using
Mathematica peak fitting

58
00:02:49,110 --> 00:02:53,580
was 46% crystallinity, so this
is within the accepted range.

59
00:02:53,580 --> 00:02:55,980
Similarly, the expected
value for degree

60
00:02:55,980 --> 00:02:58,740
of crystallinity for
high density polyethylene

61
00:02:58,740 --> 00:03:01,740
is 48% to 69% crystallinity.

62
00:03:01,740 --> 00:03:05,040
And Mathematica peak fitting
obtained a value of 56%

63
00:03:05,040 --> 00:03:06,450
crystallinity.

64
00:03:06,450 --> 00:03:09,300
Finally, ultra high
molecular weight polyethylene

65
00:03:09,300 --> 00:03:14,520
has an expected value of
75% to 77% crystallinity.

66
00:03:14,520 --> 00:03:19,340
Mathematica peak fitting showed
a value of 76% crystallinity.

67
00:03:19,340 --> 00:03:21,360
In all three polymer
case studies,

68
00:03:21,360 --> 00:03:23,940
this code obtained a degree
of crystallinity value

69
00:03:23,940 --> 00:03:27,840
within the acceptable range,
indicating manual Mathematica

70
00:03:27,840 --> 00:03:30,420
peak fitting of XRD
plots is a reasonable way

71
00:03:30,420 --> 00:03:34,380
to determine a material's
degree of crystallinity.

72
00:03:34,380 --> 00:03:36,210
The code I just
described showed a way

73
00:03:36,210 --> 00:03:38,670
to import an XRD
pattern as an image

74
00:03:38,670 --> 00:03:40,710
and manually fit the
peaks to calculate

75
00:03:40,710 --> 00:03:41,709
degree of crystallinity.

76
00:03:41,709 --> 00:03:44,910
However, I can do a better
job of determining degree

77
00:03:44,910 --> 00:03:48,450
of crystallinity if I let
the user's peak selections be

78
00:03:48,450 --> 00:03:50,520
a starting guess,
and let Mathematica

79
00:03:50,520 --> 00:03:54,420
find the best set of Gaussian
curves to fit the data.

80
00:03:54,420 --> 00:03:56,250
One of the challenges
with this method

81
00:03:56,250 --> 00:04:00,210
is I have to take data that's
currently saved as a JPG image

82
00:04:00,210 --> 00:04:02,010
and convert it into
something Mathematica

83
00:04:02,010 --> 00:04:04,120
can use to find a best fit.

84
00:04:04,120 --> 00:04:07,860
So the way I do this is I
import my JPG image here.

85
00:04:07,860 --> 00:04:09,690
And then I use
that, and I convert

86
00:04:09,690 --> 00:04:11,880
it to be either black or white.

87
00:04:11,880 --> 00:04:16,019
Then I'm able to determine the
locations of the black pixels

88
00:04:16,019 --> 00:04:17,449
and isolate them.

89
00:04:17,449 --> 00:04:18,990
However, my lines
really thick, which

90
00:04:18,990 --> 00:04:22,520
means that at each x position,
I have multiple y positions.

91
00:04:22,520 --> 00:04:24,960
So then I'm able to
use the following code

92
00:04:24,960 --> 00:04:27,210
to choose whether I
want to have the mean y

93
00:04:27,210 --> 00:04:30,270
position at each x value,
or one of the highest y

94
00:04:30,270 --> 00:04:32,310
positions at each x value.

95
00:04:32,310 --> 00:04:34,170
After my data is
in this form, I'm

96
00:04:34,170 --> 00:04:38,190
able to use it for future
curve-fitting applications.

97
00:04:38,190 --> 00:04:40,380
So after I have data,
I'm able to write

98
00:04:40,380 --> 00:04:43,500
code that allows the user
to input guesses for where

99
00:04:43,500 --> 00:04:46,140
they think the peaks
should be located.

100
00:04:46,140 --> 00:04:48,210
And then Mathematica
uses a function

101
00:04:48,210 --> 00:04:51,090
called Find Fit, that's
going to find the best

102
00:04:51,090 --> 00:04:55,495
approximation for my data given
the guesses and the model I

103
00:04:55,495 --> 00:04:56,370
give it to work with.

104
00:04:56,370 --> 00:04:58,860
So this model is a
model that includes

105
00:04:58,860 --> 00:05:01,320
three different Gaussian peaks.

106
00:05:01,320 --> 00:05:03,900
So the full code is available
to you in my notebook.

107
00:05:03,900 --> 00:05:05,970
But here's the final product.

108
00:05:05,970 --> 00:05:07,770
Like in the previous
curve-fitting program

109
00:05:07,770 --> 00:05:11,490
I showed you, you're able to
adjust the positions and shapes

110
00:05:11,490 --> 00:05:14,955
of the crystalline peaks and
the amorphous peaks as follows.

111
00:05:18,570 --> 00:05:20,870
Once you think you have a
fit that's fairly reasonable,

112
00:05:20,870 --> 00:05:23,250
you can click Find
Nearest Solution,

113
00:05:23,250 --> 00:05:25,050
and Mathematica is
going to help you find

114
00:05:25,050 --> 00:05:27,270
the best fit for this data.

115
00:05:27,270 --> 00:05:29,640
Then, using these
Mathematica found values,

116
00:05:29,640 --> 00:05:34,720
you can calculate degree of
crystallinity for the curve.

117
00:05:34,720 --> 00:05:37,080
So in conclusion,
degree of crystallinity

118
00:05:37,080 --> 00:05:39,270
can be determined
from XRD patterns.

119
00:05:39,270 --> 00:05:41,520
And today I demonstrated
two possible ways

120
00:05:41,520 --> 00:05:43,364
of doing this in Mathematica.

121
00:05:43,364 --> 00:05:45,030
There's a fair amount
of background code

122
00:05:45,030 --> 00:05:47,100
that goes into the
outputs you saw today.

123
00:05:47,100 --> 00:05:49,500
And all this code is available
to you in my Mathematica

124
00:05:49,500 --> 00:05:50,550
notebook.

125
00:05:50,550 --> 00:05:51,990
Hopefully you'll
find it very easy

126
00:05:51,990 --> 00:05:55,640
to modify to meet your
unique application needs.