WEBVTT 00:00:00.000 --> 00:00:02.452 align:middle line:90% [SQUEAKING] 00:00:02.452 --> 00:00:03.916 align:middle line:90% [RUSTLING] 00:00:03.916 --> 00:00:05.380 align:middle line:90% [CLICKING] 00:00:05.380 --> 00:00:16.058 align:middle line:90% 00:00:16.058 --> 00:00:18.600 align:middle line:84% JUEJUN HU: All right, so I will start by quickly just talking 00:00:18.600 --> 00:00:22.300 align:middle line:90% a little bit about the p set. 00:00:22.300 --> 00:00:27.200 align:middle line:84% So for the first problem, that's relatively straightforward. 00:00:27.200 --> 00:00:30.280 align:middle line:84% What you would do is just do a weighted average 00:00:30.280 --> 00:00:35.720 align:middle line:84% of the number, coordination number, 00:00:35.720 --> 00:00:38.700 align:middle line:84% essentially, with respect to the atomic fraction. 00:00:38.700 --> 00:00:43.460 align:middle line:84% So what you would do is basically, as I am listing here, 00:00:43.460 --> 00:00:44.960 align:middle line:84% you start by computing, essentially, 00:00:44.960 --> 00:00:47.760 align:middle line:84% atomic fraction of each of the components 00:00:47.760 --> 00:00:51.040 align:middle line:84% and then, essentially, calculate, 00:00:51.040 --> 00:00:53.020 align:middle line:84% what is the average coordination number? 00:00:53.020 --> 00:00:55.880 align:middle line:84% In this case, it goes directly to 2.4, 00:00:55.880 --> 00:00:57.880 align:middle line:84% which is what I have commented as something 00:00:57.880 --> 00:01:00.910 align:middle line:84% that's quite remarkable, that we are able to-- 00:01:00.910 --> 00:01:02.570 align:middle line:84% sort of by trial and error, people 00:01:02.570 --> 00:01:04.653 align:middle line:84% were able to come up through this composition that 00:01:04.653 --> 00:01:08.890 align:middle line:84% coincides with the most stable glass composition there. 00:01:08.890 --> 00:01:10.530 align:middle line:90% So that's the first one. 00:01:10.530 --> 00:01:16.130 align:middle line:84% Now, for the second question here, this 00:01:16.130 --> 00:01:18.150 align:middle line:90% is about glass batching. 00:01:18.150 --> 00:01:20.610 align:middle line:84% So batching essentially refers to a process 00:01:20.610 --> 00:01:24.210 align:middle line:84% where you essentially mix different ingredients of glasses 00:01:24.210 --> 00:01:26.770 align:middle line:84% with the right amount and then send them into a furnace 00:01:26.770 --> 00:01:28.950 align:middle line:90% to melt the glass. 00:01:28.950 --> 00:01:31.690 align:middle line:84% So I just want to comment a little bit more 00:01:31.690 --> 00:01:32.990 align:middle line:90% about the batching process. 00:01:32.990 --> 00:01:35.690 align:middle line:84% So in the class, I have already shown you a few YouTube videos 00:01:35.690 --> 00:01:38.250 align:middle line:84% that depicts how the batching process takes place, 00:01:38.250 --> 00:01:40.930 align:middle line:90% in a large glass plant. 00:01:40.930 --> 00:01:43.530 align:middle line:84% So that's for high-volume industrial manufacturing 00:01:43.530 --> 00:01:44.410 align:middle line:90% process. 00:01:44.410 --> 00:01:46.450 align:middle line:84% Now, if you are looking for, essentially, 00:01:46.450 --> 00:01:49.570 align:middle line:84% making small amount of glasses, for example, in the lab, 00:01:49.570 --> 00:01:53.530 align:middle line:84% or maybe you are trying to do it for some specialty glasses that 00:01:53.530 --> 00:01:55.330 align:middle line:84% supply only in small volume, then 00:01:55.330 --> 00:01:57.425 align:middle line:84% the process is a little bit different. 00:01:57.425 --> 00:01:59.050 align:middle line:84% What I'm showing here, this is actually 00:01:59.050 --> 00:02:02.650 align:middle line:84% an example of how you actually make small amount of glasses 00:02:02.650 --> 00:02:03.810 align:middle line:90% in the lab. 00:02:03.810 --> 00:02:05.610 align:middle line:84% Basically, what you do for oxide glasses 00:02:05.610 --> 00:02:07.410 align:middle line:84% is you just mix all the ingredients, 00:02:07.410 --> 00:02:11.009 align:middle line:84% put them into a crucible, heat them up to high temperature, 00:02:11.009 --> 00:02:14.290 align:middle line:84% and then you can then cast them or do whatever you like. 00:02:14.290 --> 00:02:16.970 align:middle line:84% So for oxide glasses, because they 00:02:16.970 --> 00:02:18.650 align:middle line:84% are usually stable in ambient air 00:02:18.650 --> 00:02:20.410 align:middle line:84% because they are already oxidized, 00:02:20.410 --> 00:02:24.250 align:middle line:84% so you can do this, essentially, in an exposed environment. 00:02:24.250 --> 00:02:29.170 align:middle line:84% Now, if you're trying to process other kind of non-oxide glasses 00:02:29.170 --> 00:02:31.610 align:middle line:84% that potentially can be oxidized, 00:02:31.610 --> 00:02:33.290 align:middle line:84% then what you will have to do is you 00:02:33.290 --> 00:02:35.890 align:middle line:84% use a setup like what I'm illustrating here. 00:02:35.890 --> 00:02:38.770 align:middle line:84% You start it out by essentially mixing all the ingredients 00:02:38.770 --> 00:02:41.690 align:middle line:84% and then seal them in a cross tube. 00:02:41.690 --> 00:02:45.450 align:middle line:84% You put in a little bit of either inert gas 00:02:45.450 --> 00:02:47.570 align:middle line:84% or leave it in vacuum, just so that you don't 00:02:47.570 --> 00:02:49.910 align:middle line:90% have any risk of oxidation. 00:02:49.910 --> 00:02:54.170 align:middle line:84% Then what you do is you put that cross tube 00:02:54.170 --> 00:02:57.420 align:middle line:84% into this so-called rocking furnace. 00:02:57.420 --> 00:03:00.120 align:middle line:84% So rocking furnace is basically like a furnace, 00:03:00.120 --> 00:03:02.660 align:middle line:84% but it constantly vibrates over time. 00:03:02.660 --> 00:03:05.060 align:middle line:84% And the purpose of that is to essentially homogenize 00:03:05.060 --> 00:03:07.740 align:middle line:84% the glass composition to ensure that there is no-- 00:03:07.740 --> 00:03:10.060 align:middle line:84% any kind of compositional segregation. 00:03:10.060 --> 00:03:12.340 align:middle line:84% So this melting process usually takes anywhere 00:03:12.340 --> 00:03:15.140 align:middle line:84% from 12 to 24 hours, just to ensure everything's 00:03:15.140 --> 00:03:16.740 align:middle line:90% perfectly homogenized. 00:03:16.740 --> 00:03:19.460 align:middle line:84% And then in the end, you bring it out, anneal it. 00:03:19.460 --> 00:03:22.820 align:middle line:84% So that will remove any kind of internal stress. 00:03:22.820 --> 00:03:27.300 align:middle line:84% All right, so that's sort of more like a small-scale lab 00:03:27.300 --> 00:03:29.740 align:middle line:90% glass production process. 00:03:29.740 --> 00:03:32.740 align:middle line:84% So now in terms of batching about calculating 00:03:32.740 --> 00:03:36.677 align:middle line:84% the composition ratios, there are a few general rules 00:03:36.677 --> 00:03:38.260 align:middle line:84% that I want to mention here, which are 00:03:38.260 --> 00:03:40.900 align:middle line:90% quoted from [INAUDIBLE] book. 00:03:40.900 --> 00:03:43.420 align:middle line:84% Basically, it says that because of the high temperature 00:03:43.420 --> 00:03:45.500 align:middle line:90% during the glass batch-- 00:03:45.500 --> 00:03:48.420 align:middle line:84% processing, in the glass process, 00:03:48.420 --> 00:03:51.900 align:middle line:84% all the hydrated crystals do lose their water completely. 00:03:51.900 --> 00:03:53.468 align:middle line:90% Organics burn off completely. 00:03:53.468 --> 00:03:55.260 align:middle line:84% Again, same thing, due to high temperature. 00:03:55.260 --> 00:03:59.620 align:middle line:84% Carbonates, sulfates, sulfites, nitrates, nitrites, et cetera, 00:03:59.620 --> 00:04:02.680 align:middle line:84% they all change to the corresponding metal oxide. 00:04:02.680 --> 00:04:04.060 align:middle line:90% The cation is kept. 00:04:04.060 --> 00:04:08.140 align:middle line:84% But carbon, sulfur, or nitrogen, because their oxide 00:04:08.140 --> 00:04:09.880 align:middle line:84% is essentially in a gaseous form, 00:04:09.880 --> 00:04:12.540 align:middle line:84% they will all get released into them during the glass 00:04:12.540 --> 00:04:14.660 align:middle line:90% fabrication process. 00:04:14.660 --> 00:04:16.860 align:middle line:84% Fluoride, chloride bromides, iodines, 00:04:16.860 --> 00:04:19.540 align:middle line:84% these are also, essentially, volatile. 00:04:19.540 --> 00:04:22.079 align:middle line:84% So most of the halide elements essentially goes away. 00:04:22.079 --> 00:04:24.420 align:middle line:84% What's left behind is the cation, the oxide 00:04:24.420 --> 00:04:26.380 align:middle line:90% corresponding to cation. 00:04:26.380 --> 00:04:29.420 align:middle line:84% And, finally, pure element added convert 00:04:29.420 --> 00:04:30.520 align:middle line:90% to equivalent oxide form. 00:04:30.520 --> 00:04:33.020 align:middle line:84% They all get oxidized in this strongly oxidizing 00:04:33.020 --> 00:04:34.380 align:middle line:90% high-temperature environment. 00:04:34.380 --> 00:04:38.340 align:middle line:84% And all the metal ions, aluminum, silicate, phosphate, 00:04:38.340 --> 00:04:40.200 align:middle line:90% the anion essentially get kept. 00:04:40.200 --> 00:04:42.820 align:middle line:84% The refractory three oxides is what stays behind. 00:04:42.820 --> 00:04:45.700 align:middle line:84% So simply put, refractory oxides and metal oxides 00:04:45.700 --> 00:04:47.400 align:middle line:84% remain in the final glass composition. 00:04:47.400 --> 00:04:51.520 align:middle line:84% Everything else essentially, if they are volatile or gaseous, 00:04:51.520 --> 00:04:54.190 align:middle line:84% they all get released into the environment 00:04:54.190 --> 00:04:56.190 align:middle line:84% during the glass fabrication process because 00:04:56.190 --> 00:04:58.770 align:middle line:84% of the high temperature and strong oxidizing environment. 00:04:58.770 --> 00:05:05.350 align:middle line:84% So then to actually calculate the glass composition, what 00:05:05.350 --> 00:05:09.310 align:middle line:84% I prefer to do is using a table, like what I'm showing here. 00:05:09.310 --> 00:05:12.710 align:middle line:84% But, basically, the idea is you start out by calculating, 00:05:12.710 --> 00:05:13.550 align:middle line:90% what is the-- 00:05:13.550 --> 00:05:16.150 align:middle line:84% starting from the, essentially, composition, calculate, 00:05:16.150 --> 00:05:17.610 align:middle line:90% what is the wave fraction? 00:05:17.610 --> 00:05:21.710 align:middle line:84% You then convert it to the amount of cation 00:05:21.710 --> 00:05:23.150 align:middle line:90% because cation is what kept-- 00:05:23.150 --> 00:05:26.950 align:middle line:84% essentially conserved in the glass fabrication process. 00:05:26.950 --> 00:05:29.830 align:middle line:84% And then you calculate, what is the amount of cation 00:05:29.830 --> 00:05:32.247 align:middle line:84% you need within one kilogram of glass? 00:05:32.247 --> 00:05:33.830 align:middle line:84% And then based on that, you calculate, 00:05:33.830 --> 00:05:36.470 align:middle line:84% for each ingredient, what is the amount of ingredient that 00:05:36.470 --> 00:05:39.150 align:middle line:84% contains that corresponding amount of the cation? 00:05:39.150 --> 00:05:41.630 align:middle line:90% So that's the idea. 00:05:41.630 --> 00:05:44.390 align:middle line:84% Now, I also want to show you another, a little bit 00:05:44.390 --> 00:05:47.110 align:middle line:90% more complicated example here. 00:05:47.110 --> 00:05:50.390 align:middle line:84% In this case, we are trying to make some borosilicate glass. 00:05:50.390 --> 00:05:53.630 align:middle line:84% And then the ingredient we are trying to use here 00:05:53.630 --> 00:06:00.010 align:middle line:84% is sodium fluorosilicate, SiO2, sand, as well as Borax. 00:06:00.010 --> 00:06:01.950 align:middle line:84% Now, this example is a bit more unique 00:06:01.950 --> 00:06:06.470 align:middle line:84% because the cations come from not just one single ingredient. 00:06:06.470 --> 00:06:07.850 align:middle line:90% They come from multiple sources. 00:06:07.850 --> 00:06:10.190 align:middle line:84% So if you think about, for example, boron, 00:06:10.190 --> 00:06:11.590 align:middle line:90% it comes from both-- 00:06:11.590 --> 00:06:13.950 align:middle line:84% sorry, if you think about, for example, silicon, 00:06:13.950 --> 00:06:18.090 align:middle line:84% it comes from both sand, as well from sodium fluoride silicate. 00:06:18.090 --> 00:06:19.730 align:middle line:90% Same thing happens with sodium. 00:06:19.730 --> 00:06:21.810 align:middle line:84% Both Borax and sodium fluorosilicate, 00:06:21.810 --> 00:06:23.630 align:middle line:90% they both contain sodium. 00:06:23.630 --> 00:06:27.550 align:middle line:84% So if you come up with a question like this, asking 00:06:27.550 --> 00:06:31.150 align:middle line:84% about, essentially, what is the composition of the ingredients? 00:06:31.150 --> 00:06:33.090 align:middle line:90% What you will do, same thing. 00:06:33.090 --> 00:06:34.870 align:middle line:84% You start with the table, but you 00:06:34.870 --> 00:06:37.150 align:middle line:84% have to be careful about sequence here. 00:06:37.150 --> 00:06:38.910 align:middle line:84% You start it out by trying to calculate 00:06:38.910 --> 00:06:40.310 align:middle line:84% the amount of ingredient you have 00:06:40.310 --> 00:06:46.870 align:middle line:84% to add for the cation that come from-- 00:06:46.870 --> 00:06:51.160 align:middle line:84% for the ingredient that contains only the supplies 00:06:51.160 --> 00:06:54.560 align:middle line:84% that only supplies the particular type of a cation. 00:06:54.560 --> 00:06:57.200 align:middle line:84% So in this case, more specifically, 00:06:57.200 --> 00:07:00.800 align:middle line:90% boron comes from only Borax. 00:07:00.800 --> 00:07:02.440 align:middle line:84% So if you know the amount of boron 00:07:02.440 --> 00:07:04.520 align:middle line:84% you need in the final glass, you can 00:07:04.520 --> 00:07:08.200 align:middle line:84% use that to actually determine how much Borax you will need. 00:07:08.200 --> 00:07:11.560 align:middle line:84% So that's what you determine first because boron only 00:07:11.560 --> 00:07:13.260 align:middle line:90% comes from Borax. 00:07:13.260 --> 00:07:15.480 align:middle line:84% Then what you do in the process, once you determine 00:07:15.480 --> 00:07:17.360 align:middle line:84% the amount of Borax, you can also 00:07:17.360 --> 00:07:20.360 align:middle line:84% compute, what's the amount of sodium that Borax also 00:07:20.360 --> 00:07:22.120 align:middle line:90% introduced into the mix? 00:07:22.120 --> 00:07:26.440 align:middle line:84% You then subtract that sodium from the total amount of sodium 00:07:26.440 --> 00:07:27.240 align:middle line:90% you need. 00:07:27.240 --> 00:07:30.120 align:middle line:84% Then the remaining will be the sodium 00:07:30.120 --> 00:07:33.720 align:middle line:84% that has to come from sodium fluorosilicate. 00:07:33.720 --> 00:07:37.680 align:middle line:84% That sodium fluorosilicate also supplies some of the silicon 00:07:37.680 --> 00:07:38.180 align:middle line:90% there. 00:07:38.180 --> 00:07:40.680 align:middle line:84% So, similarly, you subtract the silicon coming 00:07:40.680 --> 00:07:42.040 align:middle line:90% from sodium borosilicate. 00:07:42.040 --> 00:07:46.600 align:middle line:84% And the remaining amount silicon has to come from SiO2, quartz. 00:07:46.600 --> 00:07:49.640 align:middle line:90% 00:07:49.640 --> 00:07:53.480 align:middle line:84% So that's basically a process you can use to essentially infer 00:07:53.480 --> 00:07:56.120 align:middle line:84% the glass batch amount in this more 00:07:56.120 --> 00:07:59.480 align:middle line:90% complex glass-making process. 00:07:59.480 --> 00:08:02.360 align:middle line:84% Now, Zachariasen's rules, this will also be also 00:08:02.360 --> 00:08:03.880 align:middle line:90% be pretty straightforward. 00:08:03.880 --> 00:08:08.140 align:middle line:84% For barium oxide, it has a sodium chloride structure. 00:08:08.140 --> 00:08:12.480 align:middle line:84% Then you can quickly identify the coordination number 00:08:12.480 --> 00:08:14.040 align:middle line:90% around oxygen and barium. 00:08:14.040 --> 00:08:15.580 align:middle line:90% In this case, both are six. 00:08:15.580 --> 00:08:18.680 align:middle line:84% So that's why they violate the first rule and second rule. 00:08:18.680 --> 00:08:23.140 align:middle line:84% About the polyhedron around the cation, in this case, 00:08:23.140 --> 00:08:25.980 align:middle line:84% it is this kind of an octahedral. 00:08:25.980 --> 00:08:27.480 align:middle line:84% And these octahedrals, they actually 00:08:27.480 --> 00:08:30.760 align:middle line:84% share edges for the neighboring octahedrals. 00:08:30.760 --> 00:08:33.360 align:middle line:84% So same thing, they also violate the third rule. 00:08:33.360 --> 00:08:35.745 align:middle line:84% Finally, three corners are indeed shared. 00:08:35.745 --> 00:08:37.120 align:middle line:84% Otherwise, you cannot find this-- 00:08:37.120 --> 00:08:38.720 align:middle line:84% form this three-dimensional structure. 00:08:38.720 --> 00:08:40.419 align:middle line:90% That is obeyed. 00:08:40.419 --> 00:08:42.179 align:middle line:84% But the four-- three rules are violated. 00:08:42.179 --> 00:08:45.465 align:middle line:84% So that's why this is not a good glass formula. 00:08:45.465 --> 00:08:47.340 align:middle line:84% I just want to show you another example here. 00:08:47.340 --> 00:08:50.290 align:middle line:84% In this case, this is actually a sodium oxide. 00:08:50.290 --> 00:08:52.450 align:middle line:84% Sodium oxide has an antifluorite structure, 00:08:52.450 --> 00:08:57.170 align:middle line:84% like what I'm illustrating here, where the oxygen essentially 00:08:57.170 --> 00:09:00.210 align:middle line:84% occupies this blue lattice point here. 00:09:00.210 --> 00:09:05.770 align:middle line:84% And sodium is what-- represented as red atoms here, actually. 00:09:05.770 --> 00:09:09.070 align:middle line:84% No, actually, the purple one should be sodium. 00:09:09.070 --> 00:09:10.510 align:middle line:90% It has eight in a unit cell. 00:09:10.510 --> 00:09:14.910 align:middle line:84% And then the red one should be oxygen. It has less. 00:09:14.910 --> 00:09:16.170 align:middle line:90% This one. 00:09:16.170 --> 00:09:18.450 align:middle line:90% This one is three. 00:09:18.450 --> 00:09:18.950 align:middle line:90% So four. 00:09:18.950 --> 00:09:21.930 align:middle line:84% Yeah, so red lattice point should be oxygen. Purple 00:09:21.930 --> 00:09:23.870 align:middle line:90% should be sodium. 00:09:23.870 --> 00:09:27.290 align:middle line:84% It has more sodium sites than oxygen sites. 00:09:27.290 --> 00:09:31.730 align:middle line:84% So now look at the coordination number here. 00:09:31.730 --> 00:09:35.250 align:middle line:84% Oxygen, oxygen is eight because you look at one of these oxygen 00:09:35.250 --> 00:09:35.950 align:middle line:90% atoms here. 00:09:35.950 --> 00:09:39.250 align:middle line:84% It connects to four sodium atoms at the bottom. 00:09:39.250 --> 00:09:41.930 align:middle line:84% By symmetry, there's also another four on top. 00:09:41.930 --> 00:09:43.490 align:middle line:90% So it's eight. 00:09:43.490 --> 00:09:47.410 align:middle line:84% The coordination number for sodium in this case is four. 00:09:47.410 --> 00:09:53.430 align:middle line:84% So in terms of the cation polyhedra, in this case, 00:09:53.430 --> 00:09:58.810 align:middle line:84% the sodium ion sits within this tetrahedra here. 00:09:58.810 --> 00:10:04.170 align:middle line:84% So the tetrahedra, they actually share edges. 00:10:04.170 --> 00:10:06.690 align:middle line:90% And three corners are shared. 00:10:06.690 --> 00:10:10.070 align:middle line:84% So in this case, similarly, the first three rules of-- 00:10:10.070 --> 00:10:12.070 align:middle line:84% three Zachariasen's rules are actually violated. 00:10:12.070 --> 00:10:15.090 align:middle line:84% So it's also a poor glass formula. 00:10:15.090 --> 00:10:17.070 align:middle line:84% Now, to solve these kind of problems, 00:10:17.070 --> 00:10:21.050 align:middle line:84% I also have a link to this kind of website here. 00:10:21.050 --> 00:10:24.130 align:middle line:84% This is actually a very handy website. 00:10:24.130 --> 00:10:26.650 align:middle line:84% It not only allows you to show the 3D structure, 00:10:26.650 --> 00:10:30.930 align:middle line:84% but you can also actually illustrate, in this case, the-- 00:10:30.930 --> 00:10:34.770 align:middle line:84% for example, this is the [INAUDIBLE] cation-- 00:10:34.770 --> 00:10:37.650 align:middle line:84% you can see this is the cation polyhedra. 00:10:37.650 --> 00:10:40.730 align:middle line:90% This is the anion polyhedra. 00:10:40.730 --> 00:10:42.910 align:middle line:84% And it shows that these are actually sharing edges. 00:10:42.910 --> 00:10:44.940 align:middle line:84% For example, this one is actually sharing-- 00:10:44.940 --> 00:10:46.640 align:middle line:90% these are sharing edges. 00:10:46.640 --> 00:10:49.420 align:middle line:84% So this will be a fairly convenient website 00:10:49.420 --> 00:10:51.340 align:middle line:84% if you want to see any of the structures 00:10:51.340 --> 00:10:54.500 align:middle line:84% and visualize how-- whether or not they obey the Zachariasen 00:10:54.500 --> 00:10:57.420 align:middle line:90% rules. 00:10:57.420 --> 00:11:04.220 align:middle line:84% All right, next question about the 2D rigidity percolation 00:11:04.220 --> 00:11:08.600 align:middle line:84% threshold or isostatic condition. 00:11:08.600 --> 00:11:09.600 align:middle line:90% All of you get it right. 00:11:09.600 --> 00:11:11.100 align:middle line:84% I'm not going to talk much about it. 00:11:11.100 --> 00:11:12.440 align:middle line:90% The answer also makes sense. 00:11:12.440 --> 00:11:15.260 align:middle line:84% If it's 2, you can form a long polymeric chain. 00:11:15.260 --> 00:11:18.620 align:middle line:84% In 2D, because you are not allowed to rotate out of plane, 00:11:18.620 --> 00:11:21.580 align:middle line:84% so if you fix the bond angle, and also, 00:11:21.580 --> 00:11:24.380 align:middle line:84% you know the bond length, then this completely determines 00:11:24.380 --> 00:11:27.200 align:middle line:84% the configuration of the big chain there. 00:11:27.200 --> 00:11:28.438 align:middle line:90% That's why it's isostatic. 00:11:28.438 --> 00:11:30.980 align:middle line:84% In 3D, it is not because you can actually rotate the chemical 00:11:30.980 --> 00:11:32.920 align:middle line:84% bond with-- back to each other out of plane, 00:11:32.920 --> 00:11:37.160 align:middle line:84% so that this is still under constraint. 00:11:37.160 --> 00:11:39.220 align:middle line:90% Now, last, this one. 00:11:39.220 --> 00:11:43.060 align:middle line:90% This is the more tricky one. 00:11:43.060 --> 00:11:46.580 align:middle line:84% So the question is asking about whether or not 00:11:46.580 --> 00:11:50.100 align:middle line:84% you can make, essentially, a tunable lens out of these phase 00:11:50.100 --> 00:11:52.500 align:middle line:84% change materials, tunable bulk lens. 00:11:52.500 --> 00:11:57.770 align:middle line:84% The answer is that, no, because even though optically, indeed, 00:11:57.770 --> 00:12:00.020 align:middle line:84% this material can be switched between these two states 00:12:00.020 --> 00:12:03.500 align:middle line:84% to give you different optical performance, 00:12:03.500 --> 00:12:05.540 align:middle line:84% you will not be able to switch this material 00:12:05.540 --> 00:12:09.540 align:middle line:90% reversibly in a bulk form. 00:12:09.540 --> 00:12:13.040 align:middle line:84% So this goes back to the fact that this phase change material, 00:12:13.040 --> 00:12:15.740 align:middle line:84% they have very fast crystallization kinetics. 00:12:15.740 --> 00:12:18.720 align:middle line:84% That's why they can be used as fast memories to recall data. 00:12:18.720 --> 00:12:23.820 align:middle line:84% But what that also means is that it takes a very short time 00:12:23.820 --> 00:12:25.820 align:middle line:90% for them to crystallize. 00:12:25.820 --> 00:12:28.460 align:middle line:84% So I have actually looked into some literature here. 00:12:28.460 --> 00:12:30.700 align:middle line:84% So the typical crystallization time for GST 00:12:30.700 --> 00:12:32.780 align:middle line:84% is of the order of 15 nanoseconds. 00:12:32.780 --> 00:12:35.380 align:middle line:90% That's what people measured. 00:12:35.380 --> 00:12:37.200 align:middle line:84% With this very fast crystallization, 00:12:37.200 --> 00:12:39.620 align:middle line:84% it means that, in order to reamorphize it, 00:12:39.620 --> 00:12:41.760 align:middle line:84% you also have to cool down very, very fast. 00:12:41.760 --> 00:12:43.510 align:middle line:84% You have to cool down within a time period 00:12:43.510 --> 00:12:45.390 align:middle line:90% of less than 50 nanoseconds. 00:12:45.390 --> 00:12:48.590 align:middle line:84% Otherwise, you cannot bypass crystallization. 00:12:48.590 --> 00:12:51.630 align:middle line:84% Now, the problem with having a bulk piece of glass 00:12:51.630 --> 00:12:53.950 align:middle line:84% is that it always takes time for the heat 00:12:53.950 --> 00:12:56.030 align:middle line:84% to essentially get dissipated from the interior 00:12:56.030 --> 00:12:58.350 align:middle line:90% to the surface. 00:12:58.350 --> 00:13:01.290 align:middle line:84% So it means that for this bulk piece of glass, 00:13:01.290 --> 00:13:05.550 align:middle line:84% you can only amortize it on the only very top surface layer, not 00:13:05.550 --> 00:13:06.790 align:middle line:90% the inside. 00:13:06.790 --> 00:13:10.310 align:middle line:84% So more quantitatively, I did some calculations here. 00:13:10.310 --> 00:13:12.150 align:middle line:84% I look up the thermal conductivity 00:13:12.150 --> 00:13:14.090 align:middle line:84% with the volume, volumetric heat capacity. 00:13:14.090 --> 00:13:16.930 align:middle line:84% You can now actually calculate, what is the thermal diffusivity? 00:13:16.930 --> 00:13:20.730 align:middle line:84% Then you know the time of crystallization. 00:13:20.730 --> 00:13:23.190 align:middle line:84% So that allows you to calculate the heat diffusion length 00:13:23.190 --> 00:13:24.710 align:middle line:90% during that period of time. 00:13:24.710 --> 00:13:29.190 align:middle line:90% So conclusion, 76 nanometer. 00:13:29.190 --> 00:13:33.670 align:middle line:84% So what this says is that the thickness of this GST film 00:13:33.670 --> 00:13:36.670 align:middle line:84% has to be of the order of a hundred nanometer or less 00:13:36.670 --> 00:13:40.390 align:middle line:84% if you want to be able to reversibly switch it. 00:13:40.390 --> 00:13:42.290 align:middle line:84% So that's why bulk GST, you cannot switch it. 00:13:42.290 --> 00:13:44.470 align:middle line:84% In fact, even thick, thick, thicker GST films, 00:13:44.470 --> 00:13:46.590 align:middle line:84% like few hundred nanometers thick, most likely, 00:13:46.590 --> 00:13:48.690 align:middle line:84% you cannot completely reamorphize it as well. 00:13:48.690 --> 00:13:52.810 align:middle line:84% It will stay in the crystalline state, the inside. 00:13:52.810 --> 00:13:54.870 align:middle line:84% Don't worry too much about making a mistake. 00:13:54.870 --> 00:13:58.362 align:middle line:84% I saw people making this mistake all the time. 00:13:58.362 --> 00:14:00.070 align:middle line:84% There are always people trying to propose 00:14:00.070 --> 00:14:03.070 align:middle line:84% all kinds of novel optical designs out of the GST material. 00:14:03.070 --> 00:14:04.790 align:middle line:84% And if you actually read that paper, 00:14:04.790 --> 00:14:06.870 align:middle line:84% they tend to ignore this constraint there. 00:14:06.870 --> 00:14:10.250 align:middle line:84% The structure can be microns in size or whatever. 00:14:10.250 --> 00:14:11.335 align:middle line:90% It doesn't work. 00:14:11.335 --> 00:14:13.210 align:middle line:84% If they really try to experimentally make it, 00:14:13.210 --> 00:14:16.170 align:middle line:84% they'll find that they will not be able to switch it reversibly. 00:14:16.170 --> 00:14:19.790 align:middle line:90% 00:14:19.790 --> 00:14:26.870 align:middle line:84% So now, last question, this is about comparing 00:14:26.870 --> 00:14:29.150 align:middle line:84% the property of glass films prepared using 00:14:29.150 --> 00:14:31.190 align:middle line:90% different techniques there. 00:14:31.190 --> 00:14:34.270 align:middle line:84% The bottom line is, if you are trying to do it through a vacuum 00:14:34.270 --> 00:14:36.752 align:middle line:84% deposition, you have much higher effective cooling 00:14:36.752 --> 00:14:38.960 align:middle line:84% rate because what you are actually doing is you have, 00:14:38.960 --> 00:14:42.440 align:middle line:84% essentially, these glass molecules in the vapor phase 00:14:42.440 --> 00:14:44.240 align:middle line:84% or glass atoms in the vapor phase. 00:14:44.240 --> 00:14:46.960 align:middle line:84% They come into contact with a cold substrate 00:14:46.960 --> 00:14:49.000 align:middle line:84% and then condense into a solid form. 00:14:49.000 --> 00:14:51.040 align:middle line:84% So, effectively, this cooling rate 00:14:51.040 --> 00:14:54.480 align:middle line:84% is very, very fast because of the very large heat capacity 00:14:54.480 --> 00:14:57.640 align:middle line:84% of the substrate compared to, essentially, the gas phase 00:14:57.640 --> 00:15:01.440 align:middle line:84% molecules, which is much smaller amount. 00:15:01.440 --> 00:15:03.400 align:middle line:84% So that's why vapor deposition is 00:15:03.400 --> 00:15:07.240 align:middle line:84% equivalent to having a very, very high cooling rate. 00:15:07.240 --> 00:15:09.880 align:middle line:84% If you have high cooling rate for most glass materials, 00:15:09.880 --> 00:15:13.040 align:middle line:84% essentially, that correspond to essentially higher volume. 00:15:13.040 --> 00:15:16.020 align:middle line:84% That also correspond to having a higher enthalpy 00:15:16.020 --> 00:15:17.760 align:middle line:84% because, again, if you look at this plot 00:15:17.760 --> 00:15:21.280 align:middle line:84% here, faster cooling rate means that this glass line deviates 00:15:21.280 --> 00:15:23.640 align:middle line:84% from the supercooled liquid line at an earlier point. 00:15:23.640 --> 00:15:26.760 align:middle line:84% So it's in a less equilibrated state, 00:15:26.760 --> 00:15:30.800 align:middle line:84% and that's why it corresponds to higher residual enthalpy. 00:15:30.800 --> 00:15:35.760 align:middle line:84% So you would tend to expect that any vapor that deposited films, 00:15:35.760 --> 00:15:38.480 align:middle line:84% they are able to reach a state that's much more further 00:15:38.480 --> 00:15:41.680 align:middle line:84% away from equilibrium compared to if you do a [INAUDIBLE], 00:15:41.680 --> 00:15:44.040 align:middle line:84% quenching, or whatever, any other technique. 00:15:44.040 --> 00:15:47.040 align:middle line:84% That's also the reason why vacuum deposition can 00:15:47.040 --> 00:15:49.920 align:middle line:84% be used to generate amorphous films that, otherwise, it's 00:15:49.920 --> 00:15:51.880 align:middle line:90% very difficult to obtain. 00:15:51.880 --> 00:15:54.120 align:middle line:84% Vacuum deposition is a technique that people first 00:15:54.120 --> 00:15:56.600 align:middle line:84% use to actually demonstrate amorphous metal. 00:15:56.600 --> 00:16:00.040 align:middle line:84% You can also prepare a lot of transition metal oxides 00:16:00.040 --> 00:16:03.140 align:middle line:84% in amorphous form quite easily using vapor deposition, 00:16:03.140 --> 00:16:05.680 align:middle line:84% which otherwise you cannot make using bulk glass preparation 00:16:05.680 --> 00:16:08.430 align:middle line:90% techniques. 00:16:08.430 --> 00:16:10.680 align:middle line:84% Any more questions about the p set before we switch on 00:16:10.680 --> 00:16:13.020 align:middle line:90% to today's main content? 00:16:13.020 --> 00:16:19.560 align:middle line:90% 00:16:19.560 --> 00:16:24.040 align:middle line:84% So now we're going to return to the topic of relaxation. 00:16:24.040 --> 00:16:25.640 align:middle line:84% So in the previous lecture, we have 00:16:25.640 --> 00:16:28.440 align:middle line:84% talked about two examples of relaxation, 00:16:28.440 --> 00:16:30.520 align:middle line:84% where we talk about stress-strain relaxation 00:16:30.520 --> 00:16:32.360 align:middle line:90% in viscoelastic solids. 00:16:32.360 --> 00:16:34.480 align:middle line:84% We also discussed about free volume relaxation 00:16:34.480 --> 00:16:35.720 align:middle line:90% in glass materials. 00:16:35.720 --> 00:16:38.210 align:middle line:84% So what I want to do here is to introduce it 00:16:38.210 --> 00:16:42.170 align:middle line:84% in a more generic theory, basically called Tool's theory. 00:16:42.170 --> 00:16:46.010 align:middle line:84% What it describes is it gives you, essentially, 00:16:46.010 --> 00:16:52.370 align:middle line:84% a way to characterize the property change of glass 00:16:52.370 --> 00:16:55.150 align:middle line:84% materials in the glass transition process, 00:16:55.150 --> 00:16:56.330 align:middle line:90% not just for volume. 00:16:56.330 --> 00:16:58.430 align:middle line:84% That's what's characterized by free volume theory. 00:16:58.430 --> 00:17:01.870 align:middle line:84% But, essentially, any other thermodynamic parameters, 00:17:01.870 --> 00:17:04.770 align:middle line:84% how they actually changes during the glass transition, that's 00:17:04.770 --> 00:17:08.130 align:middle line:84% what Tool's theory allows you to quantitatively predict. 00:17:08.130 --> 00:17:13.050 align:middle line:84% So the big assumption about Tool's theory 00:17:13.050 --> 00:17:15.984 align:middle line:84% is about this parameter, fictive temperature. 00:17:15.984 --> 00:17:18.609 align:middle line:84% So we have already talked about fictive temperature in the very 00:17:18.609 --> 00:17:20.349 align:middle line:90% first lecture of the class. 00:17:20.349 --> 00:17:23.329 align:middle line:84% We said that the definition of fictive temperature 00:17:23.329 --> 00:17:29.410 align:middle line:84% is that, if you look at this cooling line, cooling 00:17:29.410 --> 00:17:34.425 align:middle line:84% plot of glass here, you extend the glass state line. 00:17:34.425 --> 00:17:37.050 align:middle line:84% The point where they intersect with the supercooled liquid line 00:17:37.050 --> 00:17:39.370 align:middle line:84% now defines the fictive temperature 00:17:39.370 --> 00:17:43.370 align:middle line:84% for the corresponding glassy state. 00:17:43.370 --> 00:17:47.330 align:middle line:84% The more quantitative or maybe more descriptive 00:17:47.330 --> 00:17:48.790 align:middle line:84% definition of fictive temperature, 00:17:48.790 --> 00:17:51.490 align:middle line:84% it says that the temperature on a supercooled liquid curve, 00:17:51.490 --> 00:17:53.170 align:middle line:84% at which the glass will find itself 00:17:53.170 --> 00:17:56.450 align:middle line:84% in equilibrium with the supercooled liquid state, 00:17:56.450 --> 00:17:59.592 align:middle line:90% if brought suddenly to it. 00:17:59.592 --> 00:18:01.550 align:middle line:84% Now, that may be a bit difficult to understand. 00:18:01.550 --> 00:18:03.788 align:middle line:84% So let me explain what it actually means. 00:18:03.788 --> 00:18:05.830 align:middle line:84% Starting out by looking at here, this point here, 00:18:05.830 --> 00:18:10.230 align:middle line:84% if brought suddenly to it, so what we're trying to do, 00:18:10.230 --> 00:18:13.210 align:middle line:84% we're trying to find a state, a supercooled liquid state, that's 00:18:13.210 --> 00:18:18.450 align:middle line:84% in equilibrium with the glass, the specific glass structure. 00:18:18.450 --> 00:18:20.850 align:middle line:84% So to determine if two states are in equilibrium 00:18:20.850 --> 00:18:23.410 align:middle line:84% with each other, what you can do is, 00:18:23.410 --> 00:18:25.710 align:middle line:84% essentially, you put them in contact with each other, 00:18:25.710 --> 00:18:27.670 align:middle line:84% see if there's any transition between the two. 00:18:27.670 --> 00:18:30.730 align:middle line:84% If they essentially remain-- no transition takes place, 00:18:30.730 --> 00:18:32.730 align:middle line:84% it means that they're essentially in equilibrium 00:18:32.730 --> 00:18:34.740 align:middle line:90% with each other. 00:18:34.740 --> 00:18:36.620 align:middle line:84% However, if you want to bring a glass 00:18:36.620 --> 00:18:39.860 align:middle line:84% state to the supercooled liquid line, 00:18:39.860 --> 00:18:42.720 align:middle line:84% if you do it in a normal experimental process, 00:18:42.720 --> 00:18:48.220 align:middle line:84% the problem is that this process always involves relaxation. 00:18:48.220 --> 00:18:51.100 align:middle line:84% So structure is going to change while you are trying to bring it 00:18:51.100 --> 00:18:53.780 align:middle line:84% to the supercooled liquid line, the corresponding temperature. 00:18:53.780 --> 00:18:56.280 align:middle line:84% So this relaxation then changes the structure, 00:18:56.280 --> 00:18:58.580 align:middle line:84% which essentially defeat the original assumption 00:18:58.580 --> 00:19:01.140 align:middle line:84% that we are trying to find some parameters that characterize 00:19:01.140 --> 00:19:02.580 align:middle line:90% the glass structure. 00:19:02.580 --> 00:19:05.220 align:middle line:84% So if brought suddenly to it, it essentially 00:19:05.220 --> 00:19:09.300 align:middle line:84% means that this is an ideal process. 00:19:09.300 --> 00:19:12.460 align:middle line:84% If you can do this quickly so that you 00:19:12.460 --> 00:19:15.340 align:middle line:84% avoid any structural relaxation, essentially, you 00:19:15.340 --> 00:19:18.140 align:middle line:84% maintain the glass structure, while you bring it into contact 00:19:18.140 --> 00:19:19.240 align:middle line:90% with supercooled liquid. 00:19:19.240 --> 00:19:20.500 align:middle line:90% So that's what it means. 00:19:20.500 --> 00:19:24.097 align:middle line:84% The structure of the glass stays the same 00:19:24.097 --> 00:19:25.680 align:middle line:84% when you are changing its temperature. 00:19:25.680 --> 00:19:28.540 align:middle line:90% That's the assumption there. 00:19:28.540 --> 00:19:31.100 align:middle line:84% If you do this fast enough, it do not 00:19:31.100 --> 00:19:34.380 align:middle line:84% change the microscopic structure. 00:19:34.380 --> 00:19:36.780 align:middle line:84% So then what happens if you are changing the temperature 00:19:36.780 --> 00:19:40.620 align:middle line:90% of a glass instantaneously? 00:19:40.620 --> 00:19:43.460 align:middle line:84% If you change the temperature of glass instantaneously, then 00:19:43.460 --> 00:19:44.200 align:middle line:90% what happens? 00:19:44.200 --> 00:19:46.100 align:middle line:84% There is no structural relaxation. 00:19:46.100 --> 00:19:48.500 align:middle line:84% Then the glass structure will essentially 00:19:48.500 --> 00:19:53.585 align:middle line:84% move along this glass line here, these three line. 00:19:53.585 --> 00:19:55.460 align:middle line:84% Moving along these three line, the only thing 00:19:55.460 --> 00:19:58.140 align:middle line:84% that's really changing here is the volume changes slightly 00:19:58.140 --> 00:19:59.900 align:middle line:90% because of thermal expansion. 00:19:59.900 --> 00:20:02.680 align:middle line:84% Now, thermal expansion, we talk about the atomic origin. 00:20:02.680 --> 00:20:05.220 align:middle line:84% The reason the material can actually expand as you heat it 00:20:05.220 --> 00:20:09.060 align:middle line:84% up is because, essentially, the equilibrium position of atoms 00:20:09.060 --> 00:20:12.320 align:middle line:84% changes a little bit in the potential well. 00:20:12.320 --> 00:20:15.580 align:middle line:84% That's defined by attractive and repulsive forces 00:20:15.580 --> 00:20:17.180 align:middle line:90% between the atoms. 00:20:17.180 --> 00:20:19.820 align:middle line:84% So because their process involves, essentially, 00:20:19.820 --> 00:20:22.720 align:middle line:84% the interatomic interactions, that's actually very fast. 00:20:22.720 --> 00:20:24.260 align:middle line:90% It's an instantaneous process. 00:20:24.260 --> 00:20:27.500 align:middle line:84% So, therefore, even if you do this very quickly, 00:20:27.500 --> 00:20:30.750 align:middle line:84% you can still see the thermal expansion there. 00:20:30.750 --> 00:20:33.750 align:middle line:84% Thermal expansion can be considered as instantaneous. 00:20:33.750 --> 00:20:35.650 align:middle line:84% Structural relaxation, on the other hand, 00:20:35.650 --> 00:20:38.510 align:middle line:90% takes time, all right? 00:20:38.510 --> 00:20:43.250 align:middle line:84% So when you change this glass structure instantaneously, 00:20:43.250 --> 00:20:45.330 align:middle line:84% the glass is going to move along this line. 00:20:45.330 --> 00:20:47.830 align:middle line:84% So imagine, if you try to bring it 00:20:47.830 --> 00:20:50.150 align:middle line:84% towards the supercooled liquid, it's 00:20:50.150 --> 00:20:54.550 align:middle line:84% going to go along this dotted line here, 00:20:54.550 --> 00:20:57.010 align:middle line:84% which says that the glass is not relaxing. 00:20:57.010 --> 00:21:00.710 align:middle line:84% It's only changing the volume by thermal expansion. 00:21:00.710 --> 00:21:03.410 align:middle line:84% So now, if you bring it to the supercooled liquid, 00:21:03.410 --> 00:21:05.390 align:middle line:84% then the glass and the supercooled liquid 00:21:05.390 --> 00:21:08.790 align:middle line:84% essentially meets at this point here for this particular glass 00:21:08.790 --> 00:21:11.350 align:middle line:90% line, number 3. 00:21:11.350 --> 00:21:13.210 align:middle line:90% So what does this point mean? 00:21:13.210 --> 00:21:16.510 align:middle line:84% At this point, the glass and the supercooled liquid 00:21:16.510 --> 00:21:19.430 align:middle line:84% has the same volume or the same enthalpy, 00:21:19.430 --> 00:21:22.070 align:middle line:84% depending on what vertical axis is. 00:21:22.070 --> 00:21:25.673 align:middle line:84% So now, what is the driving force or relaxation? 00:21:25.673 --> 00:21:27.590 align:middle line:84% In the previous lecture, we talk about driving 00:21:27.590 --> 00:21:30.388 align:middle line:84% force for volume relaxation is essentially a free volume. 00:21:30.388 --> 00:21:32.430 align:middle line:84% It's a volume difference between the glassy state 00:21:32.430 --> 00:21:34.030 align:middle line:84% and the supercooled liquid state. 00:21:34.030 --> 00:21:37.270 align:middle line:84% Now, with this kind of a system here, 00:21:37.270 --> 00:21:40.030 align:middle line:84% where we have this glass and the supercooled liquid both 00:21:40.030 --> 00:21:41.610 align:middle line:84% coming into this same point here, 00:21:41.610 --> 00:21:43.890 align:middle line:84% the volume difference between them is zero. 00:21:43.890 --> 00:21:46.950 align:middle line:90% 00:21:46.950 --> 00:21:50.310 align:middle line:90% So there's no driving force. 00:21:50.310 --> 00:21:52.710 align:middle line:84% So what that means is that glass is already 00:21:52.710 --> 00:21:56.910 align:middle line:84% in equilibrium with the supercooled liquid state. 00:21:56.910 --> 00:22:00.670 align:middle line:84% So, essentially, fictive temperature 00:22:00.670 --> 00:22:03.330 align:middle line:84% can be used to label the glass structure. 00:22:03.330 --> 00:22:06.510 align:middle line:84% The glass structure, it labels-- essentially, 00:22:06.510 --> 00:22:08.830 align:middle line:84% a fictive temperature allow you to define 00:22:08.830 --> 00:22:10.850 align:middle line:84% a specific supercooled liquid state. 00:22:10.850 --> 00:22:13.790 align:middle line:84% Now, this supercooled liquid state 00:22:13.790 --> 00:22:16.970 align:middle line:84% will be in equilibrium with the corresponding glass structure. 00:22:16.970 --> 00:22:19.950 align:middle line:84% That's what the fictive temperature means. 00:22:19.950 --> 00:22:21.010 align:middle line:90% Again, let me repeat. 00:22:21.010 --> 00:22:22.990 align:middle line:84% Fictive temperature allows you to define 00:22:22.990 --> 00:22:24.993 align:middle line:84% a specific supercooled liquid state. 00:22:24.993 --> 00:22:27.160 align:middle line:84% You can essentially look at the cross-section point, 00:22:27.160 --> 00:22:30.920 align:middle line:84% the supercooled liquid state with a specific temperature, 00:22:30.920 --> 00:22:32.480 align:middle line:90% with the fictive temperature. 00:22:32.480 --> 00:22:35.040 align:middle line:84% Now, the supercooled liquid at this stage, 00:22:35.040 --> 00:22:38.080 align:middle line:84% it will equilibrate with the glass structure that has 00:22:38.080 --> 00:22:40.320 align:middle line:90% the same fictive temperature. 00:22:40.320 --> 00:22:41.340 align:middle line:90% That's what it means. 00:22:41.340 --> 00:22:44.060 align:middle line:90% 00:22:44.060 --> 00:22:46.240 align:middle line:84% Now, while this number seems a bit simple, 00:22:46.240 --> 00:22:50.800 align:middle line:84% the implication of that can be pretty profound 00:22:50.800 --> 00:22:54.040 align:middle line:84% because one thing I'm essentially hinting here 00:22:54.040 --> 00:22:55.720 align:middle line:84% is that I can now use, essentially, 00:22:55.720 --> 00:22:59.080 align:middle line:84% one single parameter, which is fictive temperature, 00:22:59.080 --> 00:23:00.460 align:middle line:90% to describe glass structure. 00:23:00.460 --> 00:23:05.200 align:middle line:90% 00:23:05.200 --> 00:23:08.420 align:middle line:84% If you look at all these different glass structures here, 00:23:08.420 --> 00:23:10.520 align:middle line:84% they all have their unique fictive temperature. 00:23:10.520 --> 00:23:13.440 align:middle line:84% So what we're doing, we are simplifying this very complex 00:23:13.440 --> 00:23:18.120 align:middle line:84% problem of describing glass structure into just giving one 00:23:18.120 --> 00:23:23.100 align:middle line:84% single parameter, fictive temperature. 00:23:23.100 --> 00:23:25.240 align:middle line:84% Obviously, you can probably already suspect 00:23:25.240 --> 00:23:28.420 align:middle line:84% that this is a vastly simplified approximation. 00:23:28.420 --> 00:23:31.020 align:middle line:84% But what I'll show you is that it actually works pretty well, 00:23:31.020 --> 00:23:33.270 align:middle line:84% reasonably well, allows you to predict different glass 00:23:33.270 --> 00:23:35.880 align:middle line:84% properties, all based on one single structural parameter. 00:23:35.880 --> 00:23:40.120 align:middle line:84% So if we take this assumption there, 00:23:40.120 --> 00:23:42.040 align:middle line:84% then one statement I also want to make here 00:23:42.040 --> 00:23:44.248 align:middle line:84% is that the glassy state would now be fully described 00:23:44.248 --> 00:23:47.303 align:middle line:84% by thermodynamic parameters, which is, normally, 00:23:47.303 --> 00:23:48.720 align:middle line:84% if it's something equilibrium, you 00:23:48.720 --> 00:23:50.660 align:middle line:84% can describe it with temperature and pressure. 00:23:50.660 --> 00:23:54.120 align:middle line:84% But now we have additional parameter, Tf. 00:23:54.120 --> 00:23:56.440 align:middle line:84% With the three parameters, you can now uniquely 00:23:56.440 --> 00:24:01.600 align:middle line:84% define the glass structure and, hence, all of its properties. 00:24:01.600 --> 00:24:05.040 align:middle line:84% So if you keep a constant pressure, what that means is 00:24:05.040 --> 00:24:07.480 align:middle line:84% that if you are given the temperature, 00:24:07.480 --> 00:24:11.080 align:middle line:84% the actual temperature, as well as the fictive temperature, 00:24:11.080 --> 00:24:16.360 align:middle line:84% you can now uniquely specify all the glass properties. 00:24:16.360 --> 00:24:19.740 align:middle line:84% It's going to be a single value function of Tf. 00:24:19.740 --> 00:24:23.120 align:middle line:90% 00:24:23.120 --> 00:24:25.570 align:middle line:84% All right, so based on this assumption, 00:24:25.570 --> 00:24:28.450 align:middle line:84% now, let's try to take a look at how we can now 00:24:28.450 --> 00:24:32.210 align:middle line:84% use this fictive temperature theory to now predict 00:24:32.210 --> 00:24:34.170 align:middle line:90% the different glass properties. 00:24:34.170 --> 00:24:35.732 align:middle line:84% So let's take volume as an example. 00:24:35.732 --> 00:24:38.190 align:middle line:84% But, really, you can replace it with any kind of variables. 00:24:38.190 --> 00:24:38.870 align:middle line:90% It can be enthalpy. 00:24:38.870 --> 00:24:39.470 align:middle line:90% It can be entropy. 00:24:39.470 --> 00:24:40.262 align:middle line:90% It can be whatever. 00:24:40.262 --> 00:24:42.210 align:middle line:84% So what we have mentioned there is 00:24:42.210 --> 00:24:45.770 align:middle line:84% that, if you fix the pressure, then the volume 00:24:45.770 --> 00:24:50.370 align:middle line:84% is uniquely defined by only two parameters, temperature, 00:24:50.370 --> 00:24:52.570 align:middle line:84% the actual temperature of the sample, as well 00:24:52.570 --> 00:24:53.945 align:middle line:84% as the fictive temperature, which 00:24:53.945 --> 00:24:55.850 align:middle line:84% gives you the structure of the glass sample. 00:24:55.850 --> 00:25:01.610 align:middle line:90% So now, suppose you want to-- 00:25:01.610 --> 00:25:05.970 align:middle line:84% in a lot of cases, during glass transition, 00:25:05.970 --> 00:25:10.730 align:middle line:84% you may want to predict how this volume changes 00:25:10.730 --> 00:25:14.810 align:middle line:84% as a function of temperature during some kind of a process. 00:25:14.810 --> 00:25:18.243 align:middle line:84% So, ultimately, what you want to do 00:25:18.243 --> 00:25:20.410 align:middle line:84% is you want to try to predict these parameters here, 00:25:20.410 --> 00:25:24.770 align:middle line:84% how the volume changes as a function of temperature. 00:25:24.770 --> 00:25:28.190 align:middle line:84% So now, let's see what we have here. 00:25:28.190 --> 00:25:31.130 align:middle line:90% 00:25:31.130 --> 00:25:33.290 align:middle line:84% So if you are trying to calculate dV, 00:25:33.290 --> 00:25:36.690 align:middle line:84% now, dV has two components because it has two variables, T 00:25:36.690 --> 00:25:37.650 align:middle line:90% and Tf. 00:25:37.650 --> 00:25:40.430 align:middle line:84% The dV change can be caused by temperature change. 00:25:40.430 --> 00:25:41.555 align:middle line:90% It can also be caused by a? 00:25:41.555 --> 00:25:44.013 align:middle line:84% Fictive temperature change, which is essentially structural 00:25:44.013 --> 00:25:44.590 align:middle line:90% change. 00:25:44.590 --> 00:25:47.990 align:middle line:84% So we can then write it in two terms here. 00:25:47.990 --> 00:25:56.210 align:middle line:90% 00:25:56.210 --> 00:26:01.770 align:middle line:84% Let me write it as, essentially, Ve. 00:26:01.770 --> 00:26:04.810 align:middle line:90% 00:26:04.810 --> 00:26:06.310 align:middle line:84% In fact, maybe it's a bit easier. 00:26:06.310 --> 00:26:07.730 align:middle line:90% Let's just look at dV. 00:26:07.730 --> 00:26:09.170 align:middle line:90% Don't worry about the-- 00:26:09.170 --> 00:26:12.770 align:middle line:90% let's just look at Dv T, Tf. 00:26:12.770 --> 00:26:13.850 align:middle line:90% That's a bit easier. 00:26:13.850 --> 00:26:16.110 align:middle line:84% Let's just look at some volume change, 00:26:16.110 --> 00:26:17.430 align:middle line:90% small amount of volume change. 00:26:17.430 --> 00:26:19.330 align:middle line:90% So there are two terms here. 00:26:19.330 --> 00:26:23.580 align:middle line:84% The first term is the volume change 00:26:23.580 --> 00:26:26.227 align:middle line:84% due to a temperature variation, while we keep Tf constant. 00:26:26.227 --> 00:26:28.560 align:middle line:84% It essentially means that the structure is not changing. 00:26:28.560 --> 00:26:31.380 align:middle line:84% You only look at the temperature change. 00:26:31.380 --> 00:26:33.860 align:middle line:84% And, of course, there's a dT here. 00:26:33.860 --> 00:26:36.360 align:middle line:90% The second term is same thing. 00:26:36.360 --> 00:26:38.980 align:middle line:84% You take the partial derivative now with respect to Tf. 00:26:38.980 --> 00:26:42.820 align:middle line:84% You look at volume change due to structure relaxation of the gas, 00:26:42.820 --> 00:26:45.420 align:middle line:84% but with temperature constant, dTf. 00:26:45.420 --> 00:26:50.100 align:middle line:90% 00:26:50.100 --> 00:26:53.860 align:middle line:84% So now, what's the physical significance 00:26:53.860 --> 00:26:57.380 align:middle line:90% of the first term here? 00:26:57.380 --> 00:27:00.380 align:middle line:84% The first term is how the volume changes with respect 00:27:00.380 --> 00:27:01.120 align:middle line:90% to temperature. 00:27:01.120 --> 00:27:04.738 align:middle line:84% Suppose the structure stays constant. 00:27:04.738 --> 00:27:06.280 align:middle line:84% We have already mentioned about that. 00:27:06.280 --> 00:27:08.660 align:middle line:84% If the structure kept constant, then the only thing 00:27:08.660 --> 00:27:10.700 align:middle line:84% that really changes with respect to temperature 00:27:10.700 --> 00:27:12.060 align:middle line:90% is thermal expansion. 00:27:12.060 --> 00:27:15.780 align:middle line:84% So, therefore, you expect the first term, which just basically 00:27:15.780 --> 00:27:19.950 align:middle line:84% gives you the volumetric thermal expansion coefficient. 00:27:19.950 --> 00:27:23.667 align:middle line:84% I add the label subscript here, g. 00:27:23.667 --> 00:27:25.750 align:middle line:84% This is to ensure that this quantity is associated 00:27:25.750 --> 00:27:27.370 align:middle line:90% with the glassy state. 00:27:27.370 --> 00:27:30.270 align:middle line:90% 00:27:30.270 --> 00:27:33.870 align:middle line:84% Later on, I'm going to use another subscript, e, which 00:27:33.870 --> 00:27:37.050 align:middle line:84% is a quantity associated with the supercooled liquid state. 00:27:37.050 --> 00:27:38.443 align:middle line:90% e essentially means equilibrium. 00:27:38.443 --> 00:27:40.110 align:middle line:84% So supercooled liquid state is the state 00:27:40.110 --> 00:27:41.630 align:middle line:90% we are relaxing towards. 00:27:41.630 --> 00:27:44.970 align:middle line:90% So this will be just alpha g dT. 00:27:44.970 --> 00:27:48.150 align:middle line:90% That's the first term. 00:27:48.150 --> 00:27:53.550 align:middle line:84% Now, the second term, for the second term, 00:27:53.550 --> 00:27:55.732 align:middle line:84% we now try to keep temperature constant. 00:27:55.732 --> 00:27:58.190 align:middle line:84% But we're trying to see how the volume changes with respect 00:27:58.190 --> 00:28:00.390 align:middle line:90% to a fictive temperature. 00:28:00.390 --> 00:28:01.510 align:middle line:90% What we can do? 00:28:01.510 --> 00:28:04.790 align:middle line:84% Let's take a look at the plot here. 00:28:04.790 --> 00:28:07.570 align:middle line:84% So this will be, let's say, volume. 00:28:07.570 --> 00:28:09.495 align:middle line:90% This is a temperature. 00:28:09.495 --> 00:28:10.870 align:middle line:84% So what we are trying to see here 00:28:10.870 --> 00:28:13.110 align:middle line:84% is how the volume changes with Tf. 00:28:13.110 --> 00:28:16.450 align:middle line:84% So let's look at two glassy states with different Tf. 00:28:16.450 --> 00:28:18.990 align:middle line:84% So suppose this is a supercooled liquid line, 00:28:18.990 --> 00:28:26.510 align:middle line:84% and then we-- it then turned into two 00:28:26.510 --> 00:28:27.750 align:middle line:90% different glassy states. 00:28:27.750 --> 00:28:29.770 align:middle line:84% Now, these two different glassy states, 00:28:29.770 --> 00:28:31.610 align:middle line:84% they have different fictive temperature. 00:28:31.610 --> 00:28:35.094 align:middle line:90% 00:28:35.094 --> 00:28:38.630 align:middle line:84% So the difference here, this will then give you the dTf. 00:28:38.630 --> 00:28:42.350 align:middle line:90% 00:28:42.350 --> 00:28:44.670 align:middle line:84% What we are trying to get is, what's the volume change 00:28:44.670 --> 00:28:46.770 align:middle line:84% due to the structural transformation 00:28:46.770 --> 00:28:48.610 align:middle line:90% at the same temperature? 00:28:48.610 --> 00:28:50.550 align:middle line:84% So, basically, what we are trying 00:28:50.550 --> 00:28:55.390 align:middle line:84% to get here is, essentially, the length of this segment, 00:28:55.390 --> 00:28:57.470 align:middle line:90% this yellow segment here. 00:28:57.470 --> 00:29:01.550 align:middle line:84% And how do we correlate it with dTf? 00:29:01.550 --> 00:29:04.030 align:middle line:84% So this is actually relatively straightforward 00:29:04.030 --> 00:29:06.830 align:middle line:84% because, what you can do, you can essentially 00:29:06.830 --> 00:29:11.350 align:middle line:84% try to look at this region here, which I fill it up 00:29:11.350 --> 00:29:13.550 align:middle line:90% into a little rectangle. 00:29:13.550 --> 00:29:18.280 align:middle line:84% For this rectangle, this little segment, this little y segment, 00:29:18.280 --> 00:29:19.980 align:middle line:84% the length of this little y segment, 00:29:19.980 --> 00:29:23.160 align:middle line:84% you see that horizontal displacement is dTf? 00:29:23.160 --> 00:29:24.920 align:middle line:84% This vertical segment, essentially, 00:29:24.920 --> 00:29:28.480 align:middle line:84% has to be the slope of the glassy line 00:29:28.480 --> 00:29:32.240 align:middle line:84% times the horizontal displacement. 00:29:32.240 --> 00:29:36.200 align:middle line:84% The slope of the glass line is, basically, alpha g, 00:29:36.200 --> 00:29:40.800 align:middle line:84% the thermal expansion coefficient of glass, times dTf. 00:29:40.800 --> 00:29:43.520 align:middle line:84% So that's what calculate this very short vertical segment 00:29:43.520 --> 00:29:44.600 align:middle line:90% there. 00:29:44.600 --> 00:29:49.140 align:middle line:84% This longer vertical segment, this one, on the other hand, 00:29:49.140 --> 00:29:51.660 align:middle line:84% this is only a supercooled liquid line. 00:29:51.660 --> 00:29:54.240 align:middle line:84% It should be the horizontal displacement 00:29:54.240 --> 00:29:58.840 align:middle line:84% times the slope of your supercooled liquid line. 00:29:58.840 --> 00:30:03.520 align:middle line:84% So as I mentioned, we use the subscript e 00:30:03.520 --> 00:30:05.600 align:middle line:84% to describe any quantity associated 00:30:05.600 --> 00:30:09.340 align:middle line:90% with the supercooled liquid. 00:30:09.340 --> 00:30:11.320 align:middle line:84% e essentially means equilibrium, as I said. 00:30:11.320 --> 00:30:13.960 align:middle line:84% So alpha e, this is basically a thermal expansion coefficient 00:30:13.960 --> 00:30:16.320 align:middle line:90% of the supercooled liquid line. 00:30:16.320 --> 00:30:20.320 align:middle line:84% So then this times dTf, that will give you 00:30:20.320 --> 00:30:23.518 align:middle line:90% this vertical line. 00:30:23.518 --> 00:30:26.060 align:middle line:84% So now this yellow line, that will be pretty straightforward. 00:30:26.060 --> 00:30:28.260 align:middle line:84% It's just going to be the difference of the two. 00:30:28.260 --> 00:30:32.800 align:middle line:90% 00:30:32.800 --> 00:30:41.920 align:middle line:84% So that essentially means second term then 00:30:41.920 --> 00:30:52.720 align:middle line:84% reduces to alpha e minus alpha g times dTf. 00:30:52.720 --> 00:30:54.960 align:middle line:84% That will give you the volume change 00:30:54.960 --> 00:30:57.840 align:middle line:84% due to a modification of structure, 00:30:57.840 --> 00:30:59.720 align:middle line:84% while keeping the same temperature. 00:30:59.720 --> 00:31:02.880 align:middle line:90% 00:31:02.880 --> 00:31:06.480 align:middle line:84% So now, if you want to derive, for example, if you 00:31:06.480 --> 00:31:11.520 align:middle line:84% want to derive, let's say, dV dT, you 00:31:11.520 --> 00:31:14.530 align:middle line:84% want to see how some parameters change with temperature 00:31:14.530 --> 00:31:16.150 align:middle line:84% during some kind of glass process, 00:31:16.150 --> 00:31:20.570 align:middle line:90% then this just becomes alpha g. 00:31:20.570 --> 00:31:22.570 align:middle line:84% The first term essentially just becomes alpha g. 00:31:22.570 --> 00:31:23.850 align:middle line:90% The dT cancels out. 00:31:23.850 --> 00:31:28.310 align:middle line:84% The second term is alpha e minus alpha g. 00:31:28.310 --> 00:31:34.770 align:middle line:90% 00:31:34.770 --> 00:31:39.170 align:middle line:90% We can divide this. 00:31:39.170 --> 00:31:41.730 align:middle line:90% It will be something like this. 00:31:41.730 --> 00:31:44.170 align:middle line:84% What we can do is we can divide both the numerator 00:31:44.170 --> 00:31:48.170 align:middle line:84% and denominator, both with dt, lowercase t, which 00:31:48.170 --> 00:31:49.690 align:middle line:90% represent time. 00:31:49.690 --> 00:31:56.150 align:middle line:84% So if you do that, then this becomes 00:31:56.150 --> 00:32:07.310 align:middle line:90% dTf dTe divided by dT dt. 00:32:07.310 --> 00:32:15.185 align:middle line:90% 00:32:15.185 --> 00:32:19.130 align:middle line:84% So this is basically a dV divided by the temperature 00:32:19.130 --> 00:32:19.810 align:middle line:90% change. 00:32:19.810 --> 00:32:22.090 align:middle line:84% What I'm doing here is I'm dividing both the numerator 00:32:22.090 --> 00:32:24.810 align:middle line:84% and denominator with dt, lowercase t, which 00:32:24.810 --> 00:32:26.050 align:middle line:90% gives you time. 00:32:26.050 --> 00:32:28.750 align:middle line:90% So uppercase T is temperature. 00:32:28.750 --> 00:32:29.950 align:middle line:90% Lowercase t is time. 00:32:29.950 --> 00:32:35.450 align:middle line:90% 00:32:35.450 --> 00:32:41.670 align:middle line:84% So now, if you look at this equation here, alpha g is known. 00:32:41.670 --> 00:32:43.130 align:middle line:90% You can easily measure it. 00:32:43.130 --> 00:32:44.610 align:middle line:84% Alpha e and alpha g, this is also 00:32:44.610 --> 00:32:46.890 align:middle line:84% something you could easily experimentally infer. 00:32:46.890 --> 00:32:50.010 align:middle line:84% dT divided by dt, this is basically 00:32:50.010 --> 00:32:52.730 align:middle line:84% your heating rate or cooling rate, 00:32:52.730 --> 00:32:55.530 align:middle line:84% which also is a known parameter if you know your process. 00:32:55.530 --> 00:33:02.050 align:middle line:84% So the only unknown parameter's this guy, dTf dt, 00:33:02.050 --> 00:33:06.370 align:middle line:84% how the fictive temperature changes over time. 00:33:06.370 --> 00:33:09.610 align:middle line:84% Or in other words, this is how the glass structure 00:33:09.610 --> 00:33:12.620 align:middle line:90% would relaxes over time. 00:33:12.620 --> 00:33:14.220 align:middle line:84% So if you know this parameter, you 00:33:14.220 --> 00:33:16.080 align:middle line:84% can then plug it back into this equation 00:33:16.080 --> 00:33:18.580 align:middle line:84% and then compute, essentially, how the other parameters will 00:33:18.580 --> 00:33:21.180 align:middle line:90% then evolve over time. 00:33:21.180 --> 00:33:23.900 align:middle line:84% I'm using volume as an example, but you 00:33:23.900 --> 00:33:25.820 align:middle line:84% can fairly straightforwardly also 00:33:25.820 --> 00:33:29.100 align:middle line:84% extend this argument into other kinds of parameters here. 00:33:29.100 --> 00:33:31.360 align:middle line:84% On the slides, I show you one additional example. 00:33:31.360 --> 00:33:33.580 align:middle line:84% In this case, this is actually enthalpy. 00:33:33.580 --> 00:33:36.280 align:middle line:84% So for enthalpy, you get this equation here, similarly. 00:33:36.280 --> 00:33:38.900 align:middle line:84% If you know dTf dt, you can also use 00:33:38.900 --> 00:33:41.060 align:middle line:84% it to actually infer how the enthalpy changes 00:33:41.060 --> 00:33:42.860 align:middle line:90% over your process. 00:33:42.860 --> 00:33:46.680 align:middle line:84% So the same thing applies to all the other parameters of classes 00:33:46.680 --> 00:33:50.780 align:middle line:90% if you know this one, dTf dt. 00:33:50.780 --> 00:33:53.460 align:middle line:84% If you know how the relaxation kinetics 00:33:53.460 --> 00:33:55.020 align:middle line:84% of the fictive temperature, then you 00:33:55.020 --> 00:33:58.995 align:middle line:90% know all the glass properties. 00:33:58.995 --> 00:34:01.120 align:middle line:84% All right, so this is what we are going to do next. 00:34:01.120 --> 00:34:04.380 align:middle line:84% We are trying to derive a quantitative expression for how 00:34:04.380 --> 00:34:07.140 align:middle line:84% the Tf is going to change with respect to time. 00:34:07.140 --> 00:34:14.860 align:middle line:90% 00:34:14.860 --> 00:34:19.060 align:middle line:84% So what we want to do, we want to predict 00:34:19.060 --> 00:34:22.260 align:middle line:84% how Tf changes over time, the relaxation 00:34:22.260 --> 00:34:25.260 align:middle line:90% kinetics for the structure. 00:34:25.260 --> 00:34:28.699 align:middle line:84% So I'm plotting here-- this is a pretty generic diagram that 00:34:28.699 --> 00:34:31.739 align:middle line:84% shows how the parameter of a glass 00:34:31.739 --> 00:34:35.060 align:middle line:84% essentially varies as a function of temperature during cooling. 00:34:35.060 --> 00:34:37.587 align:middle line:84% Now, we don't care about the specific cooling process. 00:34:37.587 --> 00:34:39.420 align:middle line:84% What we care about now is just, essentially, 00:34:39.420 --> 00:34:42.000 align:middle line:84% the property of the glass with respect to supercooled liquid. 00:34:42.000 --> 00:34:45.219 align:middle line:84% So we will focus our attention only on these two lines here. 00:34:45.219 --> 00:34:48.260 align:middle line:84% Let's not worry about specific cooling process. 00:34:48.260 --> 00:34:53.540 align:middle line:84% So the cross-point of these two defines Tf, all right? 00:34:53.540 --> 00:34:58.380 align:middle line:84% Then we want to examine now as some other temperature not 00:34:58.380 --> 00:34:59.660 align:middle line:90% at Tf. 00:34:59.660 --> 00:35:01.525 align:middle line:84% What is the structural relaxation kinetics? 00:35:01.525 --> 00:35:03.900 align:middle line:84% Let's suppose we are looking at some temperature, labeled 00:35:03.900 --> 00:35:05.820 align:middle line:90% by this T here. 00:35:05.820 --> 00:35:09.725 align:middle line:84% Now, if the glass is at Tf, then it 00:35:09.725 --> 00:35:11.850 align:middle line:84% will be in equilibrium with the supercooled liquid. 00:35:11.850 --> 00:35:13.475 align:middle line:84% There will be no structural relaxation. 00:35:13.475 --> 00:35:16.550 align:middle line:84% However, if the glass is at some different temperature, 00:35:16.550 --> 00:35:19.790 align:middle line:84% T, then there is actually a driving force 00:35:19.790 --> 00:35:23.530 align:middle line:84% towards equilibrium because, if you look at this one here, 00:35:23.530 --> 00:35:26.990 align:middle line:84% the glass property has some vertical offset with that 00:35:26.990 --> 00:35:28.170 align:middle line:90% of the supercooled liquid. 00:35:28.170 --> 00:35:30.630 align:middle line:84% So this purple vertical line here 00:35:30.630 --> 00:35:35.870 align:middle line:84% essentially give you the driving force of relaxation. 00:35:35.870 --> 00:35:39.282 align:middle line:84% So I'm using P here to represent some generic parameter. 00:35:39.282 --> 00:35:39.990 align:middle line:90% It can be volume. 00:35:39.990 --> 00:35:40.782 align:middle line:90% It can be enthalpy. 00:35:40.782 --> 00:35:43.710 align:middle line:90% It can be whatever. 00:35:43.710 --> 00:35:45.110 align:middle line:84% What I'm going to do on the board 00:35:45.110 --> 00:35:47.935 align:middle line:84% is I will just essentially use volume again as an example. 00:35:47.935 --> 00:35:49.310 align:middle line:84% But think of it as you can always 00:35:49.310 --> 00:35:52.808 align:middle line:84% replace it with any other kind of variable. 00:35:52.808 --> 00:35:54.350 align:middle line:84% So let's start by essentially writing 00:35:54.350 --> 00:35:56.058 align:middle line:84% an expression for the driving force here, 00:35:56.058 --> 00:35:59.890 align:middle line:84% this term here, Pg minus Pe, which, in this case, 00:35:59.890 --> 00:36:01.310 align:middle line:90% we focus on volume. 00:36:01.310 --> 00:36:07.150 align:middle line:84% It's just going to be Vg minus Ve, basically, free volume. 00:36:07.150 --> 00:36:09.047 align:middle line:90% That's what we have. 00:36:09.047 --> 00:36:10.130 align:middle line:90% So what would be this one? 00:36:10.130 --> 00:36:11.130 align:middle line:90% What would be this term? 00:36:11.130 --> 00:36:15.450 align:middle line:84% So to see what this Pg minus Pe is, in fact, 00:36:15.450 --> 00:36:18.510 align:middle line:90% you can look at the plot on top. 00:36:18.510 --> 00:36:21.310 align:middle line:84% The plot, what I have on the board here, the yellow line, 00:36:21.310 --> 00:36:26.050 align:middle line:84% gives you exactly what this Pg minus Pe is. 00:36:26.050 --> 00:36:29.710 align:middle line:90% 00:36:29.710 --> 00:36:35.510 align:middle line:84% If you zoom in and look at, essentially, the diagram here, 00:36:35.510 --> 00:36:37.630 align:middle line:84% what we are trying to do is we are trying to get, 00:36:37.630 --> 00:36:39.390 align:middle line:84% essentially, this vertical offset. 00:36:39.390 --> 00:36:42.310 align:middle line:90% 00:36:42.310 --> 00:36:45.610 align:middle line:84% And we know what's the horizontal offset. 00:36:45.610 --> 00:36:47.110 align:middle line:84% In this case, the horizontal offset, 00:36:47.110 --> 00:36:51.638 align:middle line:84% this is given by Tf minus T. We're 00:36:51.638 --> 00:36:53.930 align:middle line:84% trying to get, essentially, this vertical line segment. 00:36:53.930 --> 00:36:55.990 align:middle line:84% So this vertical line segment, this 00:36:55.990 --> 00:36:59.030 align:middle line:84% is exactly the same as the-- mathematically, 00:36:59.030 --> 00:37:03.710 align:middle line:84% exactly the same as what I have there for that yellow segment, 00:37:03.710 --> 00:37:06.220 align:middle line:84% except instead of the horizontal displacement, 00:37:06.220 --> 00:37:09.240 align:middle line:84% instead of being dTf, it's, basically, Tf minus dt. 00:37:09.240 --> 00:37:12.280 align:middle line:84% So we can use that same expression there. 00:37:12.280 --> 00:37:15.300 align:middle line:84% So the expression will give you-- essentially, in this case, 00:37:15.300 --> 00:37:19.140 align:middle line:84% it will be an alpha e minus alpha g. 00:37:19.140 --> 00:37:26.475 align:middle line:90% 00:37:26.475 --> 00:37:31.680 align:middle line:84% And then we will have dTf, which in this case 00:37:31.680 --> 00:37:45.520 align:middle line:84% will be replaced by Tf minus T. So then what we are trying to do 00:37:45.520 --> 00:37:49.000 align:middle line:84% is we are trying to get a time evolution of Tf. 00:37:49.000 --> 00:37:51.280 align:middle line:84% So what we'll do is let's take the time derivative 00:37:51.280 --> 00:37:53.900 align:middle line:90% on both sides of the equation. 00:37:53.900 --> 00:37:55.960 align:middle line:90% So we essentially do a dT. 00:37:55.960 --> 00:37:58.800 align:middle line:84% So when we do on the left side, we have, essentially, 00:37:58.800 --> 00:38:02.040 align:middle line:90% a term that gives you dVg dt. 00:38:02.040 --> 00:38:05.360 align:middle line:90% 00:38:05.360 --> 00:38:10.440 align:middle line:84% You also have the second term, dVe minus dV dt. 00:38:10.440 --> 00:38:14.138 align:middle line:84% But dV dt is actually not changing 00:38:14.138 --> 00:38:15.680 align:middle line:84% because, what we're trying to do here 00:38:15.680 --> 00:38:18.040 align:middle line:84% is we're trying to calculate, what 00:38:18.040 --> 00:38:19.800 align:middle line:84% is the rate of structural relaxation 00:38:19.800 --> 00:38:22.280 align:middle line:84% at some fixed temperature t here? 00:38:22.280 --> 00:38:27.600 align:middle line:84% So if we fix temperature, the corresponding supercooled liquid 00:38:27.600 --> 00:38:28.817 align:middle line:90% structure is well-defined. 00:38:28.817 --> 00:38:30.400 align:middle line:84% It's basically, this point right here. 00:38:30.400 --> 00:38:34.080 align:middle line:90% 00:38:34.080 --> 00:38:36.920 align:middle line:84% So if the supercooled liquid structure 00:38:36.920 --> 00:38:39.000 align:middle line:84% is fixed at that particular temperature, 00:38:39.000 --> 00:38:44.200 align:middle line:84% then Ve is not going to change as a function of time 00:38:44.200 --> 00:38:46.160 align:middle line:90% in this isothermal process. 00:38:46.160 --> 00:38:47.677 align:middle line:84% So, therefore, the only term that's 00:38:47.677 --> 00:38:49.760 align:middle line:84% left on the left-hand side here will be the only-- 00:38:49.760 --> 00:38:51.440 align:middle line:90% will be the Vg change. 00:38:51.440 --> 00:38:53.460 align:middle line:84% The volume of glass is changing over time. 00:38:53.460 --> 00:38:56.140 align:middle line:90% 00:38:56.140 --> 00:38:59.600 align:middle line:84% On the right-hand side, alpha e minus alpha g, 00:38:59.600 --> 00:39:00.940 align:middle line:90% these are just constant. 00:39:00.940 --> 00:39:02.350 align:middle line:84% They should not change over time. 00:39:02.350 --> 00:39:06.410 align:middle line:84% So that's why you can pull them out as constants. 00:39:06.410 --> 00:39:17.242 align:middle line:84% And then you have dTf dt minus dT dt. 00:39:17.242 --> 00:39:25.850 align:middle line:90% 00:39:25.850 --> 00:39:29.370 align:middle line:84% All right, next thing, let's take a look at this term here. 00:39:29.370 --> 00:39:32.930 align:middle line:84% We have already mentioned about, essentially, the glass structure 00:39:32.930 --> 00:39:36.610 align:middle line:84% relaxation, this, how the volume actually changes over time. 00:39:36.610 --> 00:39:45.450 align:middle line:84% So dVg dt, this is the equivalent to the free volume 00:39:45.450 --> 00:39:46.250 align:middle line:90% change. 00:39:46.250 --> 00:39:49.890 align:middle line:90% 00:39:49.890 --> 00:39:55.610 align:middle line:84% And in the previous class, we have already made the assumption 00:39:55.610 --> 00:39:58.810 align:middle line:84% that, if you assume a simple single exponential, 00:39:58.810 --> 00:40:05.370 align:middle line:84% relaxation kinetics, then the kinetics would then 00:40:05.370 --> 00:40:08.370 align:middle line:84% be proportional to the driving force with the proportionality 00:40:08.370 --> 00:40:11.730 align:middle line:84% constant being a 1 over tau relaxation. 00:40:11.730 --> 00:40:26.610 align:middle line:84% So in this case, the relaxation rate is going to be the driving 00:40:26.610 --> 00:40:30.570 align:middle line:84% force-- this is the free volume-- 00:40:30.570 --> 00:40:43.210 align:middle line:90% normalized with tau relaxation 00:40:43.210 --> 00:40:45.330 align:middle line:84% Now, we also have this expression here, 00:40:45.330 --> 00:40:48.210 align:middle line:90% Vg minus Ve equal to this guy. 00:40:48.210 --> 00:40:51.530 align:middle line:84% We can plug it into this equation here. 00:40:51.530 --> 00:40:58.050 align:middle line:84% So then it becomes negative 1 over tau relaxation, then 00:40:58.050 --> 00:40:58.880 align:middle line:90% times-- 00:40:58.880 --> 00:41:02.540 align:middle line:84% so Vg minus Ve, we substitute it with everything on top. 00:41:02.540 --> 00:41:29.420 align:middle line:84% It becomes alpha e minus alpha g, and then Tf minus T. 00:41:29.420 --> 00:41:32.753 align:middle line:84% One thing I should also change a little bit is actually here. 00:41:32.753 --> 00:41:34.920 align:middle line:84% For this equation, when we take the time derivative, 00:41:34.920 --> 00:41:36.660 align:middle line:84% we already mentioned that what we are trying to do here 00:41:36.660 --> 00:41:39.460 align:middle line:84% is, essentially, we are looking at the structural relaxation 00:41:39.460 --> 00:41:42.160 align:middle line:84% at a constant temperature and see how it changes over time. 00:41:42.160 --> 00:41:45.360 align:middle line:84% So because we are actually doing it at constant temperature, 00:41:45.360 --> 00:41:48.300 align:middle line:84% this term, dT dt, should really be 0. 00:41:48.300 --> 00:41:51.517 align:middle line:84% So I should remove this term here 00:41:51.517 --> 00:41:53.600 align:middle line:84% because the temperature is not changing over time. 00:41:53.600 --> 00:41:56.058 align:middle line:84% We are only looking at the structural relaxation over time. 00:41:56.058 --> 00:42:00.823 align:middle line:90% 00:42:00.823 --> 00:42:02.740 align:middle line:84% So now you can see where I'm trying to combat. 00:42:02.740 --> 00:42:07.900 align:middle line:84% These last two equations there, they're both equal to dVg dt. 00:42:07.900 --> 00:42:10.140 align:middle line:84% So they should be equal to each other. 00:42:10.140 --> 00:42:12.660 align:middle line:90% 00:42:12.660 --> 00:42:14.300 align:middle line:84% If you set them equal to each other, 00:42:14.300 --> 00:42:18.460 align:middle line:84% this alpha e minus alpha g term actually cancels out. 00:42:18.460 --> 00:42:28.780 align:middle line:84% So what you are left behind, now you have dTf dt. 00:42:28.780 --> 00:42:30.040 align:middle line:90% Let's see what's left behind. 00:42:30.040 --> 00:42:35.780 align:middle line:84% You have a negative tau relaxation. 00:42:35.780 --> 00:42:38.200 align:middle line:90% You also have a Tf minus t. 00:42:38.200 --> 00:42:56.030 align:middle line:90% 00:42:56.030 --> 00:42:59.270 align:middle line:90% This is Tool's equation. 00:42:59.270 --> 00:43:00.830 align:middle line:84% It's actually not really surprising 00:43:00.830 --> 00:43:02.370 align:middle line:84% that we come up with this result. 00:43:02.370 --> 00:43:05.410 align:middle line:84% You see that it follows the same mathematical formalism. 00:43:05.410 --> 00:43:09.390 align:middle line:84% It also follows the simple first order relaxation. 00:43:09.390 --> 00:43:15.170 align:middle line:84% You have the kinetics scales with the driving force. 00:43:15.170 --> 00:43:19.070 align:middle line:84% So what it says is the driving force for structural relaxation 00:43:19.070 --> 00:43:22.150 align:middle line:84% is the difference between Tf and the temperature 00:43:22.150 --> 00:43:26.350 align:middle line:90% T, actual temperature T. 00:43:26.350 --> 00:43:28.350 align:middle line:84% So what's the physical significance there? 00:43:28.350 --> 00:43:31.830 align:middle line:84% I've already mentioned that if, let's suppose, 00:43:31.830 --> 00:43:36.070 align:middle line:84% you conduct your experiment at a temperature equaling to Tf. 00:43:36.070 --> 00:43:38.630 align:middle line:84% At Tf, we mentioned that the glass 00:43:38.630 --> 00:43:41.230 align:middle line:84% should equilibrate with the supercooled liquid state 00:43:41.230 --> 00:43:42.870 align:middle line:90% at that same temperature. 00:43:42.870 --> 00:43:46.283 align:middle line:84% So if T equals to Tf, glass and supercooled liquid, 00:43:46.283 --> 00:43:47.950 align:middle line:84% they are in equilibrium with each other. 00:43:47.950 --> 00:43:50.510 align:middle line:84% There should be no driving force. 00:43:50.510 --> 00:43:53.910 align:middle line:84% So anytime your actual glass temperature 00:43:53.910 --> 00:43:57.070 align:middle line:84% deviates from its fictive temperature, 00:43:57.070 --> 00:43:59.230 align:middle line:84% then you move away from equilibrium 00:43:59.230 --> 00:44:02.910 align:middle line:84% and then you have your relaxation. 00:44:02.910 --> 00:44:05.550 align:middle line:84% So that's why, from the perspective 00:44:05.550 --> 00:44:09.350 align:middle line:84% of fictive temperature, this difference here, Tf and T, 00:44:09.350 --> 00:44:11.790 align:middle line:84% the difference between the two, is the driving force 00:44:11.790 --> 00:44:13.050 align:middle line:90% for structural relaxation. 00:44:13.050 --> 00:44:16.800 align:middle line:90% 00:44:16.800 --> 00:44:18.550 align:middle line:84% So now if you have this equation, 00:44:18.550 --> 00:44:22.250 align:middle line:84% while it looks-- seems very simple, you have this equation. 00:44:22.250 --> 00:44:24.753 align:middle line:84% You can now put it back into the generic equation 00:44:24.753 --> 00:44:26.170 align:middle line:84% that I'm [INAUDIBLE] listing here. 00:44:26.170 --> 00:44:28.070 align:middle line:84% You can put it into this term here. 00:44:28.070 --> 00:44:30.490 align:middle line:84% As long as you know how Tf changes with time, 00:44:30.490 --> 00:44:32.270 align:middle line:84% you can use it to actually predict 00:44:32.270 --> 00:44:34.790 align:middle line:84% the evolution of all the other kind of parameters 00:44:34.790 --> 00:44:36.630 align:middle line:90% with respect to the glass. 00:44:36.630 --> 00:44:40.270 align:middle line:84% In fact, what you can do, if you actually 00:44:40.270 --> 00:44:44.650 align:middle line:84% substitute Tool's equation back into this particular equation 00:44:44.650 --> 00:44:46.030 align:middle line:84% I put on the board here, you will 00:44:46.030 --> 00:44:47.790 align:middle line:84% get the free isovolumetric relaxation 00:44:47.790 --> 00:44:53.382 align:middle line:84% theory, which is what we talked about in the previous lecture. 00:44:53.382 --> 00:44:54.840 align:middle line:84% Again, that's not really surprising 00:44:54.840 --> 00:44:57.000 align:middle line:84% because this is all like a circular argument. 00:44:57.000 --> 00:44:59.720 align:middle line:84% We derive the two equation by assuming, essentially, 00:44:59.720 --> 00:45:02.737 align:middle line:84% the free isovolumetric relaxation theory in this case. 00:45:02.737 --> 00:45:04.320 align:middle line:84% But the idea is that it applies to all 00:45:04.320 --> 00:45:06.800 align:middle line:84% the other thermodynamic parameters as well. 00:45:06.800 --> 00:45:12.240 align:middle line:84% Now, to show you one example about how to actually originally 00:45:12.240 --> 00:45:14.160 align:middle line:84% come up with the theory and how we actually 00:45:14.160 --> 00:45:16.495 align:middle line:84% use it to predict glass structural evolution, 00:45:16.495 --> 00:45:19.120 align:middle line:84% these are actually some figures quoted from the original paper, 00:45:19.120 --> 00:45:20.175 align:middle line:90% published by Tool. 00:45:20.175 --> 00:45:21.800 align:middle line:84% At the time, he was a scientist working 00:45:21.800 --> 00:45:24.560 align:middle line:84% at the National Bureau of Standards, which is basically 00:45:24.560 --> 00:45:25.660 align:middle line:90% NIST today. 00:45:25.660 --> 00:45:28.640 align:middle line:84% So he did some precision measurements, 00:45:28.640 --> 00:45:31.060 align:middle line:84% looking into the thermal expansion of glass materials. 00:45:31.060 --> 00:45:32.840 align:middle line:84% So that's actually what's plotted here. 00:45:32.840 --> 00:45:35.600 align:middle line:84% So vertical axis is, basically, expansion. 00:45:35.600 --> 00:45:37.540 align:middle line:84% You can treat it as volume, effectively. 00:45:37.540 --> 00:45:38.920 align:middle line:90% Horizontal axis is temperature. 00:45:38.920 --> 00:45:41.040 align:middle line:84% He prepared two glass samples that he 00:45:41.040 --> 00:45:43.600 align:middle line:90% labeled as 1 and 1 prime. 00:45:43.600 --> 00:45:45.640 align:middle line:84% And these thoughts, essentially, give you 00:45:45.640 --> 00:45:49.378 align:middle line:84% how the volume of these two samples actually changes over-- 00:45:49.378 --> 00:45:50.920 align:middle line:84% when you actually change temperature. 00:45:50.920 --> 00:45:53.560 align:middle line:84% So you can clearly see how it essentially 00:45:53.560 --> 00:45:56.060 align:middle line:84% expands, and relaxes, and then kind of comes back up, 00:45:56.060 --> 00:45:57.860 align:middle line:84% so fairly typical glass relaxation. 00:45:57.860 --> 00:46:03.880 align:middle line:84% So from this plot, you can then go back to this equation here. 00:46:03.880 --> 00:46:09.080 align:middle line:84% So in this equation here, now, all these parameters here, 00:46:09.080 --> 00:46:13.160 align:middle line:84% you can measure dT dt, how the volume changes with temperature. 00:46:13.160 --> 00:46:14.960 align:middle line:84% This is basically represented by this plot 00:46:14.960 --> 00:46:16.260 align:middle line:90% here, from experimental data. 00:46:16.260 --> 00:46:17.877 align:middle line:90% So know this parameter. 00:46:17.877 --> 00:46:19.960 align:middle line:84% You know all these thermal expansion coefficients. 00:46:19.960 --> 00:46:22.600 align:middle line:84% You know the heating rate, dT dt. 00:46:22.600 --> 00:46:28.040 align:middle line:84% So you can then use that to calculate, what is the dTf dt? 00:46:28.040 --> 00:46:30.400 align:middle line:84% From this experimental data, you can actually 00:46:30.400 --> 00:46:34.320 align:middle line:84% predict how the Tf changes over time. 00:46:34.320 --> 00:46:37.380 align:middle line:84% So, basically, by looking at experimental data, 00:46:37.380 --> 00:46:40.600 align:middle line:84% you can infer how the fictive temperature of these two glass 00:46:40.600 --> 00:46:44.145 align:middle line:84% samples changes in your experiment. 00:46:44.145 --> 00:46:46.020 align:middle line:84% So that's actually what this plot represents. 00:46:46.020 --> 00:46:48.520 align:middle line:84% He basically inferred fictive temperature of the two glass 00:46:48.520 --> 00:46:53.090 align:middle line:84% samples, and then as a function of temperature in this case. 00:46:53.090 --> 00:46:54.850 align:middle line:84% So the next thing you do is that, now you 00:46:54.850 --> 00:46:56.348 align:middle line:84% know the fictive temperature, you 00:46:56.348 --> 00:46:57.890 align:middle line:84% can then use that to actually predict 00:46:57.890 --> 00:46:59.610 align:middle line:90% other thermodynamic properties. 00:46:59.610 --> 00:47:02.290 align:middle line:84% So in this case, he tried to use that to actually predict 00:47:02.290 --> 00:47:03.690 align:middle line:90% enthalpy. 00:47:03.690 --> 00:47:05.450 align:middle line:90% So now, you know this term here. 00:47:05.450 --> 00:47:06.930 align:middle line:84% You can plug it into this equation 00:47:06.930 --> 00:47:08.770 align:middle line:84% here and calculate how the enthalpy changes 00:47:08.770 --> 00:47:11.890 align:middle line:90% over the experimental process. 00:47:11.890 --> 00:47:14.690 align:middle line:84% Now, experimentally, what is the physical significance 00:47:14.690 --> 00:47:16.450 align:middle line:90% of enthalpy change? 00:47:16.450 --> 00:47:18.650 align:middle line:84% Enthalpy change gives you the amount of heat 00:47:18.650 --> 00:47:21.090 align:middle line:84% that's absorbed or released by a sample, which 00:47:21.090 --> 00:47:25.730 align:middle line:84% you can experimentally measure using calorimetry. 00:47:25.730 --> 00:47:28.850 align:middle line:84% So that's what he did in the end, on the right-hand side 00:47:28.850 --> 00:47:29.810 align:middle line:90% here. 00:47:29.810 --> 00:47:31.310 align:middle line:84% On the right-hand side, he basically 00:47:31.310 --> 00:47:36.213 align:middle line:84% did some differential thermal analysis, DTA. 00:47:36.213 --> 00:47:38.130 align:middle line:84% We're going to talk later all about these kind 00:47:38.130 --> 00:47:39.830 align:middle line:84% of thermal analysis techniques there. 00:47:39.830 --> 00:47:41.810 align:middle line:84% But long story short, it basically 00:47:41.810 --> 00:47:44.250 align:middle line:84% is a measure of how the amount of heat that's 00:47:44.250 --> 00:47:46.450 align:middle line:84% released and absorbed as a function of temperature 00:47:46.450 --> 00:47:47.630 align:middle line:90% will heat up the sample. 00:47:47.630 --> 00:47:54.050 align:middle line:84% So these two curves, this is the corresponding amount 00:47:54.050 --> 00:48:00.850 align:middle line:84% of, basically, a DTA trace, DTA measure curve, during-- 00:48:00.850 --> 00:48:03.090 align:middle line:84% on the right-hand panel, this is actually 00:48:03.090 --> 00:48:06.450 align:middle line:84% the theoretical curve that's predicted based 00:48:06.450 --> 00:48:09.850 align:middle line:90% on the fictive temperature. 00:48:09.850 --> 00:48:11.310 align:middle line:84% What's on the left-hand side here, 00:48:11.310 --> 00:48:13.770 align:middle line:84% this is actually experimentally-measured data. 00:48:13.770 --> 00:48:17.170 align:middle line:84% So now you see this striking similarity. 00:48:17.170 --> 00:48:20.090 align:middle line:84% The Tools equation, the fictive temperature, 00:48:20.090 --> 00:48:22.610 align:middle line:84% now allows you to actually predict, essentially, 00:48:22.610 --> 00:48:25.650 align:middle line:84% the enthalpy change based on volume expansion coefficient 00:48:25.650 --> 00:48:26.410 align:middle line:90% data. 00:48:26.410 --> 00:48:30.290 align:middle line:84% Or, sorry, volume expansion data. 00:48:30.290 --> 00:48:32.370 align:middle line:84% So what this result really shows you, 00:48:32.370 --> 00:48:35.010 align:middle line:84% that this remarkable agreement is that you can now-- 00:48:35.010 --> 00:48:39.270 align:middle line:84% essentially, all these different parameters, temperature-- sorry, 00:48:39.270 --> 00:48:42.310 align:middle line:84% all the different glass properties, volume or enthalpy, 00:48:42.310 --> 00:48:45.250 align:middle line:84% they can all be correlated into this one single parameter, which 00:48:45.250 --> 00:48:46.550 align:middle line:90% is fictive temperature. 00:48:46.550 --> 00:48:49.032 align:middle line:84% That's what this experiment verifies. 00:48:49.032 --> 00:48:50.740 align:middle line:84% Essentially, they use fictive temperature 00:48:50.740 --> 00:48:52.580 align:middle line:84% as a bridge that allows you to correlate 00:48:52.580 --> 00:48:54.860 align:middle line:84% two sets of data that simply would be quite different. 00:48:54.860 --> 00:48:57.475 align:middle line:90% One is volume, one is enthalpy. 00:48:57.475 --> 00:48:59.100 align:middle line:84% This comparison shows that both of them 00:48:59.100 --> 00:49:01.540 align:middle line:84% can be predicted by using one single parameter, which 00:49:01.540 --> 00:49:05.260 align:middle line:90% is fictive temperature. 00:49:05.260 --> 00:49:07.600 align:middle line:84% So that goes back to the assumption there. 00:49:07.600 --> 00:49:09.220 align:middle line:84% You can label glass structure, which 00:49:09.220 --> 00:49:13.460 align:middle line:90% is one single parameter, Tf. 00:49:13.460 --> 00:49:15.560 align:middle line:84% All right, so while that looks pretty good, 00:49:15.560 --> 00:49:19.100 align:middle line:84% I also want to mention that this theory, obviously, 00:49:19.100 --> 00:49:20.880 align:middle line:90% is oversimplified. 00:49:20.880 --> 00:49:23.100 align:middle line:84% Glass structure is very, very complex. 00:49:23.100 --> 00:49:25.060 align:middle line:84% You don't imagine that you can simply just 00:49:25.060 --> 00:49:27.820 align:middle line:84% summarize it with one single parameter there. 00:49:27.820 --> 00:49:32.100 align:middle line:84% So a few years later, there was another experiment carried out 00:49:32.100 --> 00:49:35.482 align:middle line:84% by one of his postdocs, one of Tool's postdocs. 00:49:35.482 --> 00:49:37.940 align:middle line:84% So this is one of the rare cases where the postdoc actually 00:49:37.940 --> 00:49:42.620 align:middle line:84% disproved the theory proposed by his advisor, but whatever. 00:49:42.620 --> 00:49:43.825 align:middle line:90% So this is Ritland. 00:49:43.825 --> 00:49:45.200 align:middle line:84% It's called Ritland's experiment. 00:49:45.200 --> 00:49:47.860 align:middle line:84% So, essentially, what Ritland did 00:49:47.860 --> 00:49:50.940 align:middle line:84% is he has two groups of glass samples. 00:49:50.940 --> 00:49:52.660 align:middle line:90% They have the same composition. 00:49:52.660 --> 00:49:55.620 align:middle line:84% They go through two different heat treatment process. 00:49:55.620 --> 00:50:00.860 align:middle line:84% And in the end, they get the same refractive index. 00:50:00.860 --> 00:50:02.180 align:middle line:90% So what does that mean? 00:50:02.180 --> 00:50:04.658 align:middle line:84% According to Tool's theory, all the parameters 00:50:04.658 --> 00:50:07.200 align:middle line:84% correlates with, essentially, just one single parameter here, 00:50:07.200 --> 00:50:08.300 align:middle line:90% which is Tf. 00:50:08.300 --> 00:50:11.180 align:middle line:84% So if the two glass samples have the same refractive index, 00:50:11.180 --> 00:50:13.680 align:middle line:84% refractive index actually correlates with probability, 00:50:13.680 --> 00:50:15.220 align:middle line:90% which then relates to density. 00:50:15.220 --> 00:50:17.608 align:middle line:84% It's an amount of polarizable electrons 00:50:17.608 --> 00:50:19.900 align:middle line:84% within a certain volume that can respond to oscillating 00:50:19.900 --> 00:50:21.240 align:middle line:90% electromagnetic field. 00:50:21.240 --> 00:50:24.140 align:middle line:84% But, anyway, this is a parameter of the glass. 00:50:24.140 --> 00:50:27.260 align:middle line:84% So if the two glass samples have the same fictive temperature 00:50:27.260 --> 00:50:29.220 align:middle line:84% at the same temperature, it means 00:50:29.220 --> 00:50:32.980 align:middle line:84% that they must have the same fictive temperature. 00:50:32.980 --> 00:50:34.880 align:middle line:84% The two glass must have the same structure. 00:50:34.880 --> 00:50:38.220 align:middle line:90% 00:50:38.220 --> 00:50:41.220 align:middle line:84% So now, if you look at how you obtain these two groups, one 00:50:41.220 --> 00:50:43.260 align:middle line:84% of them, you essentially just keep it 00:50:43.260 --> 00:50:45.870 align:middle line:84% at some temperature for a long period of time. 00:50:45.870 --> 00:50:47.653 align:middle line:84% So after the long period of time, 00:50:47.653 --> 00:50:49.070 align:middle line:84% you assume this sample has already 00:50:49.070 --> 00:50:50.410 align:middle line:90% reached equilibrium state. 00:50:50.410 --> 00:50:52.110 align:middle line:84% The sample should already equilibrate 00:50:52.110 --> 00:50:53.830 align:middle line:84% with the supercooled liquid, also 00:50:53.830 --> 00:50:56.550 align:middle line:90% at the same 530 degrees Celsius. 00:50:56.550 --> 00:51:01.450 align:middle line:84% So, therefore, by definition, for group A, this glass, 00:51:01.450 --> 00:51:02.950 align:middle line:84% it should have a fictive temperature 00:51:02.950 --> 00:51:05.950 align:middle line:90% of 530 degrees Celsius. 00:51:05.950 --> 00:51:09.710 align:middle line:84% That's just the definition of fictive temperature. 00:51:09.710 --> 00:51:12.870 align:middle line:84% If you leave it at such a long, long time when 00:51:12.870 --> 00:51:16.230 align:middle line:84% you reach equilibrium, the T should then equal to Tf. 00:51:16.230 --> 00:51:19.150 align:middle line:90% 00:51:19.150 --> 00:51:20.758 align:middle line:84% So then what about the second group? 00:51:20.758 --> 00:51:22.550 align:middle line:84% The second group goes through, essentially, 00:51:22.550 --> 00:51:25.110 align:middle line:84% some process, cool at some certain rate, whatever. 00:51:25.110 --> 00:51:26.150 align:middle line:90% It doesn't matter. 00:51:26.150 --> 00:51:29.110 align:middle line:84% But based on Tool's theory, the second group 00:51:29.110 --> 00:51:30.830 align:middle line:84% should also have the fictive temperature 00:51:30.830 --> 00:51:33.990 align:middle line:90% of 530 degrees Celsius. 00:51:33.990 --> 00:51:36.230 align:middle line:84% So now, what happens if you put both of them 00:51:36.230 --> 00:51:39.290 align:middle line:84% in a furnace that's held at 530 degrees Celsius? 00:51:39.290 --> 00:51:41.470 align:middle line:84% What would Tool's theory predict? 00:51:41.470 --> 00:51:44.190 align:middle line:84% Tool's theory would predict that, because both samples has 00:51:44.190 --> 00:51:48.750 align:middle line:84% a Tf that exactly equals to T, 530 degrees Celsius, 00:51:48.750 --> 00:51:51.190 align:middle line:84% so no structural relaxation should take place. 00:51:51.190 --> 00:51:59.950 align:middle line:84% Both samples should exhibit no change whatsoever 00:51:59.950 --> 00:52:02.258 align:middle line:84% because the Tf both equal to actual temperature, T. 00:52:02.258 --> 00:52:04.550 align:middle line:84% So according to this equation, no structural relaxation 00:52:04.550 --> 00:52:08.270 align:middle line:84% should take place for both samples. 00:52:08.270 --> 00:52:11.330 align:middle line:90% Does that make sense? 00:52:11.330 --> 00:52:15.590 align:middle line:84% AUDIENCE: Because group B has been cooled down to reach 530? 00:52:15.590 --> 00:52:19.193 align:middle line:84% JUEJUN HU: Cooled down for 530 degrees Celsius, yeah. 00:52:19.193 --> 00:52:20.110 align:middle line:90% AUDIENCE: Stays there. 00:52:20.110 --> 00:52:20.950 align:middle line:84% JUEJUN HU: Yeah, and, essentially, 00:52:20.950 --> 00:52:23.242 align:middle line:84% because they have the same refractive index, what we're 00:52:23.242 --> 00:52:25.190 align:middle line:84% trying to infer here, according to our theory, 00:52:25.190 --> 00:52:27.520 align:middle line:84% is that they should have the same structure. 00:52:27.520 --> 00:52:30.830 align:middle line:84% AUDIENCE: Wait, so for group B, they basically 00:52:30.830 --> 00:52:32.110 align:middle line:90% cooled it all the way down? 00:52:32.110 --> 00:52:33.630 align:middle line:84% JUEJUN HU: You just cool it all the way down. 00:52:33.630 --> 00:52:35.338 align:middle line:84% AUDIENCE: Based on the slope of the line, 00:52:35.338 --> 00:52:37.190 align:middle line:84% it should have fictive temperature of 530? 00:52:37.190 --> 00:52:38.110 align:middle line:84% JUEJUN HU: I think they just cool it down 00:52:38.110 --> 00:52:39.630 align:middle line:90% to 530 degrees Celsius. 00:52:39.630 --> 00:52:42.220 align:middle line:84% For a melt, that's a much higher temperature. 00:52:42.220 --> 00:52:45.160 align:middle line:84% They cool down with a specific rate to 530. 00:52:45.160 --> 00:52:48.390 align:middle line:84% So then the refractive index at 530 is the same as the group A. 00:52:48.390 --> 00:52:51.390 align:middle line:84% AUDIENCE: How is that different from group A? 00:52:51.390 --> 00:52:54.160 align:middle line:84% JUEJUN HU: Oh, it's just thermal history. 00:52:54.160 --> 00:52:57.080 align:middle line:84% It's just cool down immediately to 530, and then that's it. 00:52:57.080 --> 00:53:00.868 align:middle line:84% You don't sit there for a long period of time. 00:53:00.868 --> 00:53:03.160 align:middle line:84% AUDIENCE: So group A is at equilibrium and group B is-- 00:53:03.160 --> 00:53:03.910 align:middle line:90% JUEJUN HU: Is not. 00:53:03.910 --> 00:53:06.168 align:middle line:84% It's just like as cooled, essentially. 00:53:06.168 --> 00:53:07.460 align:middle line:90% AUDIENCE: It's passing through. 00:53:07.460 --> 00:53:09.668 align:middle line:84% JUEJUN HU: It's passing through 530, yeah, basically. 00:53:09.668 --> 00:53:11.760 align:middle line:84% It's not staying there for a long time. 00:53:11.760 --> 00:53:14.560 align:middle line:84% At 530, with that cooling rate, it also 00:53:14.560 --> 00:53:16.760 align:middle line:90% gives you the refractive index. 00:53:16.760 --> 00:53:19.080 align:middle line:84% Essentially, there are two sets of samples 00:53:19.080 --> 00:53:21.700 align:middle line:84% with different thermal history with the same refractive index. 00:53:21.700 --> 00:53:24.340 align:middle line:90% 00:53:24.340 --> 00:53:25.480 align:middle line:90% AUDIENCE: Yes. 00:53:25.480 --> 00:53:27.940 align:middle line:84% JUEJUN HU: Yeah, one is that you confirm it's equilibrated. 00:53:27.940 --> 00:53:31.740 align:middle line:84% The second one is that you tune the thermal history. 00:53:31.740 --> 00:53:34.567 align:middle line:84% So it also happens to have the same refractive index, 00:53:34.567 --> 00:53:36.900 align:middle line:84% but it does not go through the same equilibrium process. 00:53:36.900 --> 00:53:40.810 align:middle line:90% 00:53:40.810 --> 00:53:42.450 align:middle line:84% But according to our theory, they 00:53:42.450 --> 00:53:44.170 align:middle line:84% should be the same because they have the same fictive 00:53:44.170 --> 00:53:44.790 align:middle line:90% temperature. 00:53:44.790 --> 00:53:45.890 align:middle line:90% They have the same index. 00:53:45.890 --> 00:53:48.140 align:middle line:84% So there should be no structural relaxation whatsoever 00:53:48.140 --> 00:53:50.210 align:middle line:90% because they are the same. 00:53:50.210 --> 00:53:53.390 align:middle line:84% Now, that's obviously not the case. 00:53:53.390 --> 00:53:54.890 align:middle line:84% This is actually a refractive index, 00:53:54.890 --> 00:53:57.182 align:middle line:84% how the refractive index changes for the second sample, 00:53:57.182 --> 00:53:58.090 align:middle line:90% this trace there. 00:53:58.090 --> 00:54:00.010 align:middle line:84% You can clearly see, it actually continues 00:54:00.010 --> 00:54:03.030 align:middle line:84% to drop a little bit after keeping at 530 C. 00:54:03.030 --> 00:54:04.450 align:middle line:90% And then it goes back up. 00:54:04.450 --> 00:54:05.930 align:middle line:84% Eventually it will, of course, goes 00:54:05.930 --> 00:54:07.763 align:middle line:84% back to the same temperature because it then 00:54:07.763 --> 00:54:10.530 align:middle line:84% converges to the same state as in glass A. 00:54:10.530 --> 00:54:13.370 align:middle line:84% But the fact that you are seeing the time evolution 00:54:13.370 --> 00:54:16.130 align:middle line:84% of the refractive index says that the structure of these two 00:54:16.130 --> 00:54:18.350 align:middle line:84% glasses, despite they have the same refractive index, 00:54:18.350 --> 00:54:21.370 align:middle line:84% is different, which should not be something that comes 00:54:21.370 --> 00:54:23.193 align:middle line:84% as a surprise because you imagine, 00:54:23.193 --> 00:54:24.610 align:middle line:84% glass structure is so complicated, 00:54:24.610 --> 00:54:27.947 align:middle line:84% you just cannot describe it with one single parameter. 00:54:27.947 --> 00:54:29.530 align:middle line:84% Or another way to understand it is you 00:54:29.530 --> 00:54:31.630 align:middle line:84% can think of, that for different parameters, 00:54:31.630 --> 00:54:33.330 align:middle line:90% they have different Tf. 00:54:33.330 --> 00:54:35.010 align:middle line:84% In this case, for refractive index, 00:54:35.010 --> 00:54:36.593 align:middle line:90% they may have the same Tf. 00:54:36.593 --> 00:54:38.010 align:middle line:84% But then the glass structure still 00:54:38.010 --> 00:54:42.370 align:middle line:84% can be different when it comes to other parameters. 00:54:42.370 --> 00:54:45.530 align:middle line:84% So bottom line is that you can calculate 00:54:45.530 --> 00:54:48.410 align:middle line:84% glass structure fully using just one single parameter, Tf. 00:54:48.410 --> 00:54:51.530 align:middle line:84% In fact, sort of a quick way to explain the Ritland experiment 00:54:51.530 --> 00:54:54.370 align:middle line:84% is that there are actually multiple relaxation mechanisms 00:54:54.370 --> 00:54:56.090 align:middle line:90% that can currently take place. 00:54:56.090 --> 00:54:57.990 align:middle line:84% They have different kinetic constants. 00:54:57.990 --> 00:55:00.730 align:middle line:84% So that if you superimpose these two curves together, 00:55:00.730 --> 00:55:02.230 align:middle line:90% you get a shape like this. 00:55:02.230 --> 00:55:04.230 align:middle line:84% You have a faster relaxation that goes this way. 00:55:04.230 --> 00:55:06.050 align:middle line:84% You have another slow relaxation that 00:55:06.050 --> 00:55:07.450 align:middle line:90% goes in the opposite direction. 00:55:07.450 --> 00:55:12.149 align:middle line:90% 00:55:12.149 --> 00:55:15.010 align:middle line:84% So takeaway message from relaxation experiment 00:55:15.010 --> 00:55:19.490 align:middle line:84% is that structural glass is very complex. 00:55:19.490 --> 00:55:22.330 align:middle line:84% You cannot simply characterize it using one single parameter, 00:55:22.330 --> 00:55:24.750 align:middle line:84% Tf, even though it's a pretty good approximation, 00:55:24.750 --> 00:55:27.610 align:middle line:84% as in the case that's proved by 2. 00:55:27.610 --> 00:55:30.130 align:middle line:84% The more refined theory is called 00:55:30.130 --> 00:55:34.970 align:middle line:84% a Tool-Narayanaswami-Moynihan, or TNM model. 00:55:34.970 --> 00:55:37.420 align:middle line:84% I've included a link there that talks a little bit more 00:55:37.420 --> 00:55:39.020 align:middle line:90% about the theory. 00:55:39.020 --> 00:55:40.800 align:middle line:84% It's outside the scope of this lecture. 00:55:40.800 --> 00:55:43.540 align:middle line:84% But if you really want to study the relaxation, 00:55:43.540 --> 00:55:46.580 align:middle line:84% this is the theory people use in, essentially, 00:55:46.580 --> 00:55:51.660 align:middle line:84% scientific investigations or industrial applications. 00:55:51.660 --> 00:55:54.980 align:middle line:84% In fact, the authors of this theory, 00:55:54.980 --> 00:55:58.380 align:middle line:84% originally, they were all affiliated with General Motors. 00:55:58.380 --> 00:56:01.860 align:middle line:84% So General Motors has a very large glass research center. 00:56:01.860 --> 00:56:04.938 align:middle line:84% They actually look into relaxation because, of course, 00:56:04.938 --> 00:56:06.980 align:middle line:84% they worry about relaxation of windshield glasses 00:56:06.980 --> 00:56:08.200 align:middle line:90% over a long period of time. 00:56:08.200 --> 00:56:09.760 align:middle line:90% So that's why. 00:56:09.760 --> 00:56:11.260 align:middle line:84% They actually come up with this kind 00:56:11.260 --> 00:56:14.460 align:middle line:84% of a very fundamental scientific investigation in this model 00:56:14.460 --> 00:56:17.180 align:middle line:90% there. 00:56:17.180 --> 00:56:20.540 align:middle line:84% So that's actually all I want to cover here 00:56:20.540 --> 00:56:23.046 align:middle line:84% in this lecture about relaxation. 00:56:23.046 --> 00:56:24.340 align:middle line:90% AUDIENCE: Love it. 00:56:24.340 --> 00:56:25.720 align:middle line:90% JUEJUN HU: That's from ChatGPT. 00:56:25.720 --> 00:56:26.926 align:middle line:90% [LAUGHTER] 00:56:26.926 --> 00:56:28.240 align:middle line:90% 00:56:28.240 --> 00:56:30.340 align:middle line:90% So now, question for you? 00:56:30.340 --> 00:56:31.218 align:middle line:90% What is relaxation? 00:56:31.218 --> 00:56:32.760 align:middle line:84% Obviously, that's not our definition. 00:56:32.760 --> 00:56:35.720 align:middle line:84% So what do you think is the relaxation? 00:56:35.720 --> 00:56:39.020 align:middle line:90% 00:56:39.020 --> 00:56:41.205 align:middle line:84% So the formal definition of relaxation, 00:56:41.205 --> 00:56:42.580 align:middle line:84% which, interestingly, we have not 00:56:42.580 --> 00:56:44.780 align:middle line:84% talked about throughout the entire lecture here. 00:56:44.780 --> 00:56:46.740 align:middle line:84% We just give you descriptive terms. 00:56:46.740 --> 00:56:49.620 align:middle line:84% Relaxation is the return of a perturbed system 00:56:49.620 --> 00:56:50.800 align:middle line:90% into equilibrium. 00:56:50.800 --> 00:56:57.030 align:middle line:90% 00:56:57.030 --> 00:56:59.400 align:middle line:84% If you look at this definition here, 00:56:59.400 --> 00:57:01.020 align:middle line:84% there are actually three key things 00:57:01.020 --> 00:57:03.940 align:middle line:84% you have to know about the relaxation process. 00:57:03.940 --> 00:57:07.660 align:middle line:84% You need to know, what is the equilibrium state the system is 00:57:07.660 --> 00:57:10.360 align:middle line:90% returning to? 00:57:10.360 --> 00:57:13.713 align:middle line:84% You need to know, what is the driving force for this process? 00:57:13.713 --> 00:57:15.380 align:middle line:84% And, finally, you need to know, what are 00:57:15.380 --> 00:57:17.100 align:middle line:90% the kinetics of the process? 00:57:17.100 --> 00:57:18.660 align:middle line:84% These are three key parameters you 00:57:18.660 --> 00:57:22.020 align:middle line:84% need to know anytime you deal with a relaxation, equilibrium 00:57:22.020 --> 00:57:25.200 align:middle line:84% state, driving force, as well as the kinetics. 00:57:25.200 --> 00:57:31.820 align:middle line:90% 00:57:31.820 --> 00:57:36.590 align:middle line:84% We mentioned that this kind of a relaxation process, this 00:57:36.590 --> 00:57:37.775 align:middle line:90% is obviously time dependent. 00:57:37.775 --> 00:57:39.150 align:middle line:84% It occurs in the glass transition 00:57:39.150 --> 00:57:42.070 align:middle line:84% regime, where the Deborah number is of the order of unity. 00:57:42.070 --> 00:57:44.230 align:middle line:84% We talk about three different examples here, 00:57:44.230 --> 00:57:46.653 align:middle line:84% stress-strain relaxation in viscoelastic solids, 00:57:46.653 --> 00:57:48.070 align:middle line:84% free volume relaxation, as well as 00:57:48.070 --> 00:57:49.770 align:middle line:84% the more general structural relaxation, 00:57:49.770 --> 00:57:50.790 align:middle line:90% which is Tool's theory. 00:57:50.790 --> 00:57:53.230 align:middle line:84% In all three examples, you can look 00:57:53.230 --> 00:57:55.370 align:middle line:90% at what the three elements are. 00:57:55.370 --> 00:57:58.870 align:middle line:84% So for the equilibrium state, strain-stress relaxation 00:57:58.870 --> 00:58:01.790 align:middle line:84% is when the residual strain or stress completely goes to 0. 00:58:01.790 --> 00:58:03.890 align:middle line:90% That's equilibrium state. 00:58:03.890 --> 00:58:05.990 align:middle line:84% And the driving force for the relaxation process, 00:58:05.990 --> 00:58:09.710 align:middle line:84% obviously, would be the residual strain or stress. 00:58:09.710 --> 00:58:11.730 align:middle line:84% For free isovolumetric relaxation, 00:58:11.730 --> 00:58:15.190 align:middle line:84% the equilibrium state is the supercooled liquid state. 00:58:15.190 --> 00:58:18.830 align:middle line:84% And the driving force would be the free volume, the difference 00:58:18.830 --> 00:58:20.910 align:middle line:84% of the glass volume with respect to the volume 00:58:20.910 --> 00:58:23.270 align:middle line:90% of the supercooled liquid. 00:58:23.270 --> 00:58:25.853 align:middle line:84% For the general structural relaxation, the equilibrium 00:58:25.853 --> 00:58:27.770 align:middle line:84% state, again, is the supercooled liquid state. 00:58:27.770 --> 00:58:29.187 align:middle line:84% And the driving force is basically 00:58:29.187 --> 00:58:33.030 align:middle line:84% this, the difference of Tf, the glass structure, 00:58:33.030 --> 00:58:36.950 align:middle line:84% and the temperature at which the sample actually sits. 00:58:36.950 --> 00:58:39.830 align:middle line:84% That's the driving force for structural relaxation. 00:58:39.830 --> 00:58:46.922 align:middle line:84% Now, in terms of kinetics, these are the main equations 00:58:46.922 --> 00:58:48.630 align:middle line:84% you can actually use to essentially model 00:58:48.630 --> 00:58:51.150 align:middle line:84% all kinds of relaxation kinetics, the most important one 00:58:51.150 --> 00:58:55.070 align:middle line:84% being this single first order theory, where you assume 00:58:55.070 --> 00:58:58.417 align:middle line:84% the relaxation rate of some parameter p scales 00:58:58.417 --> 00:59:00.750 align:middle line:84% with the driving force with the proportionality constant 00:59:00.750 --> 00:59:03.650 align:middle line:84% of a negative 1 over tau relaxation. 00:59:03.650 --> 00:59:04.810 align:middle line:90% That's the most simple one. 00:59:04.810 --> 00:59:06.310 align:middle line:84% And that will give you, essentially, 00:59:06.310 --> 00:59:11.710 align:middle line:84% a single exponential decay when a response to a perturbation. 00:59:11.710 --> 00:59:13.610 align:middle line:84% If you want to get a more refined model, 00:59:13.610 --> 00:59:15.190 align:middle line:84% you can, of course, take into account 00:59:15.190 --> 00:59:17.190 align:middle line:84% that there may be multiple relaxation mechanisms 00:59:17.190 --> 00:59:18.490 align:middle line:90% concurrently taking place. 00:59:18.490 --> 00:59:20.323 align:middle line:84% And that's where you're actually introducing 00:59:20.323 --> 00:59:23.190 align:middle line:84% a stretched exponential there to describe the dispersion 00:59:23.190 --> 00:59:24.990 align:middle line:90% of relaxation time. 00:59:24.990 --> 00:59:28.150 align:middle line:84% Once you know this relaxation kinetics, in the end, 00:59:28.150 --> 00:59:30.590 align:middle line:84% you can then plug this function, psi, 00:59:30.590 --> 00:59:33.880 align:middle line:84% that gives you relaxation kinetics into the Boltzmann 00:59:33.880 --> 00:59:35.840 align:middle line:84% Superposition Principle, which then gives you 00:59:35.840 --> 00:59:39.720 align:middle line:84% the full description about how the relaxation is 00:59:39.720 --> 00:59:43.480 align:middle line:84% going to respond as any kind of time-dependent perturbation. 00:59:43.480 --> 00:59:45.805 align:middle line:84% So, basically, what we have-- what 00:59:45.805 --> 00:59:47.180 align:middle line:84% you have learned in this lecture, 00:59:47.180 --> 00:59:50.520 align:middle line:84% in essence, is that you will be able to cope 00:59:50.520 --> 00:59:53.100 align:middle line:84% with any kind of realization problem, the kinetics, 00:59:53.100 --> 00:59:55.960 align:middle line:84% as long as you know the relaxation kinetics 00:59:55.960 --> 00:59:59.360 align:middle line:84% model, as well as the time-dependent perturbation. 00:59:59.360 --> 01:00:01.840 align:middle line:84% By combining them into the Boltzmann Superposition 01:00:01.840 --> 01:00:04.300 align:middle line:84% Principle, you can predict all the parameters you need. 01:00:04.300 --> 01:00:08.143 align:middle line:90% 01:00:08.143 --> 01:00:09.560 align:middle line:84% Yeah, this is just a quick summary 01:00:09.560 --> 01:00:12.520 align:middle line:84% of all the other theory, which I'm not going to go through yet. 01:00:12.520 --> 01:00:15.400 align:middle line:84% Any more questions about relaxation 01:00:15.400 --> 01:00:17.700 align:middle line:84% before I move on to talk about mechanical properties? 01:00:17.700 --> 01:00:24.040 align:middle line:90% 01:00:24.040 --> 01:00:28.640 align:middle line:84% All right, so now let's switch gears a little bit 01:00:28.640 --> 01:00:33.680 align:middle line:84% and let's talk about mechanical properties. 01:00:33.680 --> 01:00:37.440 align:middle line:84% So talking about speaking of mechanical properties, 01:00:37.440 --> 01:00:40.700 align:middle line:84% so if you are talking about inorganic network glasses-- 01:00:40.700 --> 01:00:42.408 align:middle line:84% so let's exclude metallic glasses for now 01:00:42.408 --> 01:00:44.117 align:middle line:84% because they are quite different in terms 01:00:44.117 --> 01:00:45.280 align:middle line:90% of mechanical properties. 01:00:45.280 --> 01:00:47.670 align:middle line:84% If you think about inorganic network glass, 01:00:47.670 --> 01:00:49.920 align:middle line:84% what would be the first word that comes into your mind 01:00:49.920 --> 01:00:51.750 align:middle line:84% when talking about mechanical properties? 01:00:51.750 --> 01:00:52.680 align:middle line:90% AUDIENCE: Fragile. 01:00:52.680 --> 01:00:56.440 align:middle line:84% JUEJUN HU: Fragile, brittle, stiff, whatever. 01:00:56.440 --> 01:00:59.760 align:middle line:84% Now, is that perspective supported by actual numbers? 01:00:59.760 --> 01:01:00.540 align:middle line:90% We are scientists. 01:01:00.540 --> 01:01:01.460 align:middle line:90% We go by numbers. 01:01:01.460 --> 01:01:04.200 align:middle line:90% So this is what we have. 01:01:04.200 --> 01:01:06.760 align:middle line:84% If you look at the ultimate tensile stress, 01:01:06.760 --> 01:01:10.640 align:middle line:84% for iron, 35 megapascal, structural steel, 01:01:10.640 --> 01:01:12.620 align:middle line:90% 550 megapascal. 01:01:12.620 --> 01:01:16.440 align:middle line:84% But for glass, glass fiber, they are like an order of magnitude 01:01:16.440 --> 01:01:18.880 align:middle line:84% higher in terms of the tensile strength compared 01:01:18.880 --> 01:01:21.020 align:middle line:90% to steel, high-quality steel. 01:01:21.020 --> 01:01:24.120 align:middle line:90% 01:01:24.120 --> 01:01:25.560 align:middle line:90% So what does that mean? 01:01:25.560 --> 01:01:29.710 align:middle line:84% This contradicts our conventional wisdom. 01:01:29.710 --> 01:01:33.170 align:middle line:84% It means glass is actually very, very strong. 01:01:33.170 --> 01:01:38.610 align:middle line:84% So, in fact, exactly, that's what we have. 01:01:38.610 --> 01:01:41.830 align:middle line:84% Why all the superhero movies, why do they have Iron Man? 01:01:41.830 --> 01:01:44.010 align:middle line:90% You should have Glass Man. 01:01:44.010 --> 01:01:45.830 align:middle line:84% Glass is a lot stronger than steel. 01:01:45.830 --> 01:01:47.270 align:middle line:90% Why do you get Iron Man there? 01:01:47.270 --> 01:01:50.250 align:middle line:90% 01:01:50.250 --> 01:01:50.880 align:middle line:90% What's wrong? 01:01:50.880 --> 01:01:52.130 align:middle line:90% AUDIENCE: Sell that to Marvel. 01:01:52.130 --> 01:01:53.953 align:middle line:84% I said, you can sell that to Marvel. 01:01:53.953 --> 01:01:54.870 align:middle line:90% JUEJUN HU: We can try. 01:01:54.870 --> 01:01:57.295 align:middle line:90% [LAUGHS] Yeah? 01:01:57.295 --> 01:01:58.170 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:01:58.170 --> 01:01:59.970 align:middle line:84% JUEJUN HU: Why we have this impression 01:01:59.970 --> 01:02:02.210 align:middle line:84% that glass is fragile and brittle, 01:02:02.210 --> 01:02:05.770 align:middle line:90% they are much weaker than steel? 01:02:05.770 --> 01:02:08.070 align:middle line:84% Well, we have to step back a little bit. 01:02:08.070 --> 01:02:09.810 align:middle line:84% And let's start by essentially talking about, essentially, 01:02:09.810 --> 01:02:11.370 align:middle line:84% the different measures, how we actually measure 01:02:11.370 --> 01:02:12.590 align:middle line:90% the strength of a material. 01:02:12.590 --> 01:02:14.173 align:middle line:84% So when we talk about strength, that's 01:02:14.173 --> 01:02:15.550 align:middle line:90% a pretty loosely-defined term. 01:02:15.550 --> 01:02:16.730 align:middle line:84% In fact, there are two-- when we talk 01:02:16.730 --> 01:02:17.730 align:middle line:84% about mechanical properties, there 01:02:17.730 --> 01:02:18.870 align:middle line:84% are two parameters we talk about. 01:02:18.870 --> 01:02:19.670 align:middle line:90% There's the strength. 01:02:19.670 --> 01:02:20.628 align:middle line:90% There's also toughness. 01:02:20.628 --> 01:02:22.590 align:middle line:84% We should probably learn in mechanics class. 01:02:22.590 --> 01:02:24.890 align:middle line:84% Strength is basically the amount of stress 01:02:24.890 --> 01:02:28.530 align:middle line:84% you apply to a material before it caused some kind of change, 01:02:28.530 --> 01:02:30.210 align:middle line:90% like failure. 01:02:30.210 --> 01:02:32.210 align:middle line:84% Toughness, on the other hand, is actually 01:02:32.210 --> 01:02:34.530 align:middle line:84% measuring the energy the material will absorb 01:02:34.530 --> 01:02:37.050 align:middle line:90% before it comes to failure. 01:02:37.050 --> 01:02:39.290 align:middle line:84% And that's actually different because we look at, 01:02:39.290 --> 01:02:40.790 align:middle line:84% for example, when I'm plotting here, 01:02:40.790 --> 01:02:42.850 align:middle line:84% this is basically a strain-stress curve 01:02:42.850 --> 01:02:46.370 align:middle line:84% for two different classes of material. 01:02:46.370 --> 01:02:49.170 align:middle line:84% Essentially, our left-hand side, this is a ductile material. 01:02:49.170 --> 01:02:51.130 align:middle line:84% Right-hand side, this is a fragile material. 01:02:51.130 --> 01:02:53.450 align:middle line:84% Or, actually, brittle material, sorry. 01:02:53.450 --> 01:02:55.450 align:middle line:84% So on the left-hand side, this would 01:02:55.450 --> 01:02:57.075 align:middle line:84% be for things like polymers and metals, 01:02:57.075 --> 01:02:59.242 align:middle line:84% where they can actually produce a significant amount 01:02:59.242 --> 01:03:00.510 align:middle line:90% of plastic deformation. 01:03:00.510 --> 01:03:03.710 align:middle line:84% So then if you start to apply stress, then what happens there, 01:03:03.710 --> 01:03:06.150 align:middle line:84% initially, you get this linear deformation there, 01:03:06.150 --> 01:03:07.390 align:middle line:90% linear elastic deformation. 01:03:07.390 --> 01:03:08.723 align:middle line:90% This is the [? hooking ?] range. 01:03:08.723 --> 01:03:11.130 align:middle line:84% At some point, you start introducing significant amount 01:03:11.130 --> 01:03:12.550 align:middle line:90% of plastic deformation. 01:03:12.550 --> 01:03:14.570 align:middle line:84% This is how you define your stress. 01:03:14.570 --> 01:03:17.450 align:middle line:84% This is when the plastic deformation reaches 0.2%. 01:03:17.450 --> 01:03:20.710 align:middle line:84% If you continue to deform it, at some point, 01:03:20.710 --> 01:03:22.810 align:middle line:84% you're going to reach the ultimate strength. 01:03:22.810 --> 01:03:26.300 align:middle line:84% Beyond that, you see the stress actually decreases. 01:03:26.300 --> 01:03:28.060 align:middle line:90% This is because of necking. 01:03:28.060 --> 01:03:30.760 align:middle line:84% So some regions of the sample essentially becomes narrower. 01:03:30.760 --> 01:03:32.220 align:middle line:90% The diameter becomes smaller. 01:03:32.220 --> 01:03:35.240 align:middle line:84% So as a result, this leads to a concentration of the stress. 01:03:35.240 --> 01:03:38.580 align:middle line:84% And that's why you see, essentially, the stress actually 01:03:38.580 --> 01:03:42.986 align:middle line:84% sort of decreases and then eventually fractures. 01:03:42.986 --> 01:03:46.820 align:middle line:84% Now, the major toughness, you are looking 01:03:46.820 --> 01:03:49.400 align:middle line:90% at the area under the curve. 01:03:49.400 --> 01:03:51.020 align:middle line:84% The area under the curve gives you 01:03:51.020 --> 01:03:55.720 align:middle line:84% essentially the energy absorbed by the material per unit volume. 01:03:55.720 --> 01:03:57.340 align:middle line:84% So, essentially, it's W divided by V, 01:03:57.340 --> 01:03:59.540 align:middle line:84% where W is the absorbed energy and V 01:03:59.540 --> 01:04:00.880 align:middle line:90% is the volume of the material. 01:04:00.880 --> 01:04:03.450 align:middle line:90% 01:04:03.450 --> 01:04:07.300 align:middle line:84% So for a ductile material, because there's 01:04:07.300 --> 01:04:09.940 align:middle line:84% a very large plastic deformation, 01:04:09.940 --> 01:04:14.020 align:middle line:84% the area under the curve is also very large. 01:04:14.020 --> 01:04:15.820 align:middle line:84% It means that this plastic deformation 01:04:15.820 --> 01:04:19.660 align:middle line:84% allows you to absorb a lot of the energy before it breaks. 01:04:19.660 --> 01:04:24.900 align:middle line:84% And this is in contrast to a brittle material, like glass. 01:04:24.900 --> 01:04:30.528 align:middle line:84% For glass, essentially, you have this elastic deformation. 01:04:30.528 --> 01:04:32.820 align:middle line:84% And then that very little amount of plastic deformation 01:04:32.820 --> 01:04:33.695 align:middle line:90% actually takes place. 01:04:33.695 --> 01:04:35.220 align:middle line:90% It just cracks. 01:04:35.220 --> 01:04:36.300 align:middle line:90% It just breaks. 01:04:36.300 --> 01:04:40.160 align:middle line:84% So because of this small amount of plastic deformation, 01:04:40.160 --> 01:04:42.220 align:middle line:84% it means that the area under the curve 01:04:42.220 --> 01:04:46.460 align:middle line:84% can be much smaller, even though the stress, strength, can 01:04:46.460 --> 01:04:47.500 align:middle line:90% be very high. 01:04:47.500 --> 01:04:50.140 align:middle line:84% Strength is what you measure the stress 01:04:50.140 --> 01:04:52.260 align:middle line:90% you need to break the material. 01:04:52.260 --> 01:04:55.860 align:middle line:84% So high strength does not always equate to high toughness. 01:04:55.860 --> 01:04:58.500 align:middle line:84% Ductile material tend to be tougher, 01:04:58.500 --> 01:05:03.500 align:middle line:84% even for the same amount of ultimate strength. 01:05:03.500 --> 01:05:07.980 align:middle line:84% So this explains partly why, material like steel, people 01:05:07.980 --> 01:05:09.940 align:middle line:90% consider it to be stronger. 01:05:09.940 --> 01:05:12.740 align:middle line:84% It's because we are mostly talking about toughness there. 01:05:12.740 --> 01:05:15.660 align:middle line:84% Steel can absorb a lot more deformation, a lot more energy, 01:05:15.660 --> 01:05:19.640 align:middle line:84% compared to glass, even if they have the same strength. 01:05:19.640 --> 01:05:21.060 align:middle line:90% That's one part of the story. 01:05:21.060 --> 01:05:26.950 align:middle line:84% Second part of the story is that the strength of the glass 01:05:26.950 --> 01:05:31.430 align:middle line:84% material tend to be also very sensitive to defects. 01:05:31.430 --> 01:05:34.990 align:middle line:84% So you can get very high-quality glass fibers 01:05:34.990 --> 01:05:36.510 align:middle line:90% with a high strength there. 01:05:36.510 --> 01:05:40.030 align:middle line:84% But if you're talking about the glass, well, we use every day, 01:05:40.030 --> 01:05:42.190 align:middle line:84% the actual strength is actually much lower 01:05:42.190 --> 01:05:44.850 align:middle line:84% compared to glass fiber because of the presence of defects. 01:05:44.850 --> 01:05:47.410 align:middle line:90% 01:05:47.410 --> 01:05:51.670 align:middle line:84% So to see, what will be the strength, theoretical strength, 01:05:51.670 --> 01:05:53.590 align:middle line:84% of a brittle material, let's take a look 01:05:53.590 --> 01:05:55.590 align:middle line:90% at this very simple model here. 01:05:55.590 --> 01:05:59.150 align:middle line:84% So what we are trying to predict is we are trying to predict, 01:05:59.150 --> 01:06:02.390 align:middle line:84% what is the ultimate strength of glass 01:06:02.390 --> 01:06:04.910 align:middle line:90% from a theoretical perspective? 01:06:04.910 --> 01:06:07.150 align:middle line:84% So if you look at the process here, 01:06:07.150 --> 01:06:10.230 align:middle line:84% when you actually apply a tensile stress on a brittle 01:06:10.230 --> 01:06:14.110 align:middle line:84% glass, what you are really doing is increasing the spacing 01:06:14.110 --> 01:06:16.432 align:middle line:90% between these atoms. 01:06:16.432 --> 01:06:18.390 align:middle line:84% You are stretching the material and, therefore, 01:06:18.390 --> 01:06:22.070 align:middle line:84% increasing the average spacing between the atoms. 01:06:22.070 --> 01:06:25.110 align:middle line:84% So in this process, when the average spacing between atoms 01:06:25.110 --> 01:06:27.750 align:middle line:84% increases, the attractive forces becomes larger 01:06:27.750 --> 01:06:29.270 align:middle line:90% than repulsive forces. 01:06:29.270 --> 01:06:31.130 align:middle line:84% So the atoms want to stick together. 01:06:31.130 --> 01:06:33.990 align:middle line:84% And that's why you see there's actually a stress accumulating 01:06:33.990 --> 01:06:35.190 align:middle line:90% inside the material. 01:06:35.190 --> 01:06:39.350 align:middle line:84% And it increases initially with increasing amount of strain. 01:06:39.350 --> 01:06:41.390 align:middle line:84% Now, at some point, you imagine that, when 01:06:41.390 --> 01:06:44.310 align:middle line:84% atoms become so far apart, you start to pull them so far apart, 01:06:44.310 --> 01:06:45.790 align:middle line:84% the interaction between them also 01:06:45.790 --> 01:06:48.150 align:middle line:84% starts to get weaker because the longer distance. 01:06:48.150 --> 01:06:53.310 align:middle line:84% So the stress will have to decrease after some point. 01:06:53.310 --> 01:06:56.950 align:middle line:84% So now if you plot the stress-strain curve, 01:06:56.950 --> 01:06:58.570 align:middle line:84% it should look something like this. 01:06:58.570 --> 01:07:01.310 align:middle line:84% It will increase initially and then decreases. 01:07:01.310 --> 01:07:04.950 align:middle line:84% And then the top point here will now 01:07:04.950 --> 01:07:08.510 align:middle line:84% defines, essentially, the ultimate strength 01:07:08.510 --> 01:07:09.270 align:middle line:90% of the material. 01:07:09.270 --> 01:07:12.710 align:middle line:90% 01:07:12.710 --> 01:07:17.030 align:middle line:84% The slope of this curve when the strain is small, 01:07:17.030 --> 01:07:20.600 align:middle line:84% this then defines the elastic constant, Young's modulus. 01:07:20.600 --> 01:07:23.280 align:middle line:84% E will be Young's modulus, is the proportionality constant, 01:07:23.280 --> 01:07:25.760 align:middle line:84% [? pure ?] in the hooking can range. 01:07:25.760 --> 01:07:32.040 align:middle line:84% So if you look from the energy perspective, what happens there? 01:07:32.040 --> 01:07:33.380 align:middle line:90% You perform some work. 01:07:33.380 --> 01:07:34.460 align:middle line:90% You stretch the glass. 01:07:34.460 --> 01:07:36.140 align:middle line:90% Eventually, you break it. 01:07:36.140 --> 01:07:39.000 align:middle line:84% The energy has to turn into something. 01:07:39.000 --> 01:07:41.640 align:middle line:84% So in this particular case here, the energy 01:07:41.640 --> 01:07:45.600 align:middle line:84% contributes to breaking of the chemical bonds 01:07:45.600 --> 01:07:47.040 align:middle line:84% When you break the chemical bonds 01:07:47.040 --> 01:07:52.600 align:middle line:84% between the glass or atoms, you essentially create new surfaces. 01:07:52.600 --> 01:07:55.280 align:middle line:84% So the work you perform now gets stored in the system 01:07:55.280 --> 01:07:57.180 align:middle line:90% in the form of surface energy. 01:07:57.180 --> 01:07:59.920 align:middle line:90% 01:07:59.920 --> 01:08:06.440 align:middle line:84% So what we argue is that the area under the curve 01:08:06.440 --> 01:08:09.240 align:middle line:84% should equate to the surface energy 01:08:09.240 --> 01:08:12.220 align:middle line:84% you generated due to the breakage of the glass. 01:08:12.220 --> 01:08:29.160 align:middle line:90% 01:08:29.160 --> 01:08:31.880 align:middle line:84% So now, based on this insight, we can then actually 01:08:31.880 --> 01:08:34.380 align:middle line:84% quickly try to derive, what is the theoretical strength 01:08:34.380 --> 01:08:35.380 align:middle line:90% of the brittle material? 01:08:35.380 --> 01:08:38.800 align:middle line:84% We're trying to calculate, what's sigma m? 01:08:38.800 --> 01:08:43.500 align:middle line:84% So we can make a first approximation here. 01:08:43.500 --> 01:08:45.640 align:middle line:84% If you look at the shape of this curve here, 01:08:45.640 --> 01:08:47.359 align:middle line:90% it increases and then decreases. 01:08:47.359 --> 01:08:50.040 align:middle line:84% We can approximate it as like a sine curve, 01:08:50.040 --> 01:08:51.720 align:middle line:90% just to make it simpler. 01:08:51.720 --> 01:08:54.300 align:middle line:84% So if we approximate it as a sine curve, 01:08:54.300 --> 01:08:56.960 align:middle line:84% then the stress should then equals 01:08:56.960 --> 01:09:03.840 align:middle line:90% to a sign, some kind of strain. 01:09:03.840 --> 01:09:10.200 align:middle line:84% Now, the maximum point is defined by sigma m. 01:09:10.200 --> 01:09:15.439 align:middle line:84% Now, the coefficient before strain, this 01:09:15.439 --> 01:09:19.210 align:middle line:84% would be essentially related to the period of the sine curve 01:09:19.210 --> 01:09:20.050 align:middle line:90% there. 01:09:20.050 --> 01:09:23.290 align:middle line:84% So if you define this point here as lambda, 01:09:23.290 --> 01:09:26.170 align:middle line:84% then the [? period ?] will basically 2 times lambda. 01:09:26.170 --> 01:09:32.810 align:middle line:84% So we can then write this as, this would be lambda, 01:09:32.810 --> 01:09:36.609 align:middle line:90% this would be a pi. 01:09:36.609 --> 01:09:39.810 align:middle line:84% So when epsilon equals to 2 lambda, when this train equal 01:09:39.810 --> 01:09:41.310 align:middle line:90% to lambda, this becomes 2 pi. 01:09:41.310 --> 01:09:43.529 align:middle line:84% You go through a full period of the sine curve. 01:09:43.529 --> 01:09:50.689 align:middle line:90% 01:09:50.689 --> 01:09:55.210 align:middle line:84% So next thing is that we know that the slope of this curve 01:09:55.210 --> 01:09:59.250 align:middle line:84% must equal to the Young's modulus at this point here. 01:09:59.250 --> 01:10:01.970 align:middle line:84% So, mathematically, that basically 01:10:01.970 --> 01:10:04.730 align:middle line:84% says, when you take the partial-- 01:10:04.730 --> 01:10:14.970 align:middle line:84% when you do the d sigma, d epsilon, when your epsilon is 0, 01:10:14.970 --> 01:10:16.330 align:middle line:90% this should equal to a? 01:10:16.330 --> 01:10:20.393 align:middle line:90% E. 01:10:20.393 --> 01:10:21.310 align:middle line:90% So what would that be? 01:10:21.310 --> 01:10:24.170 align:middle line:84% If you take the derivative of the sine at 0, 01:10:24.170 --> 01:10:30.530 align:middle line:84% that would just be a sigma m pi lambda. 01:10:30.530 --> 01:10:38.370 align:middle line:90% 01:10:38.370 --> 01:10:42.170 align:middle line:84% So from that equation, you can now infer what lambda is. 01:10:42.170 --> 01:10:46.330 align:middle line:84% Lambda should just be-- let's see. 01:10:46.330 --> 01:11:08.290 align:middle line:84% Lambda is pi sigma m divided by E. So now, the next thing, 01:11:08.290 --> 01:11:10.510 align:middle line:84% we want to look at the area under the curve. 01:11:10.510 --> 01:11:14.140 align:middle line:84% This is absorbed energy, so w divided by v. This 01:11:14.140 --> 01:11:16.560 align:middle line:84% would be the area under the curve. 01:11:16.560 --> 01:11:18.060 align:middle line:84% And what you would do, of course you 01:11:18.060 --> 01:11:21.500 align:middle line:84% do integration of sigma over d epsilon. 01:11:21.500 --> 01:11:24.040 align:middle line:90% You integrate from 0 to lambda. 01:11:24.040 --> 01:11:28.420 align:middle line:90% 01:11:28.420 --> 01:11:31.460 align:middle line:90% This is a sine curve. 01:11:31.460 --> 01:11:33.460 align:middle line:84% So, in fact, the area under the sine curve, 01:11:33.460 --> 01:11:34.980 align:middle line:84% there's a fairly simple conclusion, 01:11:34.980 --> 01:11:43.180 align:middle line:84% saying that it will be a factor of pi over 2, 2 over pi times 01:11:43.180 --> 01:11:45.700 align:middle line:84% the area under the rectangle that covers the entire sine 01:11:45.700 --> 01:11:46.200 align:middle line:90% curve. 01:11:46.200 --> 01:11:48.180 align:middle line:84% So the area under the rectangle will 01:11:48.180 --> 01:11:51.907 align:middle line:90% be just sigma m times lambda. 01:11:51.907 --> 01:11:53.240 align:middle line:90% You can also do the integration. 01:11:53.240 --> 01:11:55.480 align:middle line:84% You will come up with the same conclusion, basically. 01:11:55.480 --> 01:12:00.060 align:middle line:90% 01:12:00.060 --> 01:12:05.843 align:middle line:84% So now you can plug-in the expression there for lambda. 01:12:05.843 --> 01:12:06.760 align:middle line:90% Let's see what we get. 01:12:06.760 --> 01:12:16.600 align:middle line:84% So it will be sigma m, E. Pi cancels out. 01:12:16.600 --> 01:12:18.300 align:middle line:84% And there's another factor, sigma m. 01:12:18.300 --> 01:12:19.160 align:middle line:90% Pi cancels out. 01:12:19.160 --> 01:12:20.980 align:middle line:84% That's also a factor of 2, so, yeah. 01:12:20.980 --> 01:12:29.340 align:middle line:90% 01:12:29.340 --> 01:12:32.060 align:middle line:84% So now the final question goes back to, 01:12:32.060 --> 01:12:37.380 align:middle line:84% what is the volume here that we should define? 01:12:37.380 --> 01:12:45.020 align:middle line:84% So if you look at the vector w divided by v, 01:12:45.020 --> 01:12:47.200 align:middle line:90% w is the surface energy. 01:12:47.200 --> 01:12:49.580 align:middle line:84% It's the newly generated surface due to this breaking 01:12:49.580 --> 01:12:50.280 align:middle line:90% of the glass. 01:12:50.280 --> 01:12:54.500 align:middle line:84% So it can be described as gamma, which 01:12:54.500 --> 01:12:58.340 align:middle line:90% is the surface energy, times s. 01:12:58.340 --> 01:13:00.285 align:middle line:90% This would be the surface area. 01:13:00.285 --> 01:13:02.660 align:middle line:84% This would be the area of, essentially, the cross-section 01:13:02.660 --> 01:13:04.460 align:middle line:90% of glass that breaks. 01:13:04.460 --> 01:13:06.920 align:middle line:84% We want to time a factor of 2 there. 01:13:06.920 --> 01:13:07.420 align:middle line:90% Why? 01:13:07.420 --> 01:13:08.795 align:middle line:84% Because anytime glass breaks, you 01:13:08.795 --> 01:13:11.430 align:middle line:84% generate two surfaces instead of one. 01:13:11.430 --> 01:13:15.190 align:middle line:84% So divide it by v. So what you see 01:13:15.190 --> 01:13:16.950 align:middle line:84% is that this vector now goes back 01:13:16.950 --> 01:13:21.790 align:middle line:84% to essentially the surface-to-volume ratio, 01:13:21.790 --> 01:13:24.350 align:middle line:90% surface-to-volume ratio. 01:13:24.350 --> 01:13:29.245 align:middle line:84% So volume, in this case, here, volume divided by surface area 01:13:29.245 --> 01:13:30.870 align:middle line:84% is just going to give you, essentially, 01:13:30.870 --> 01:13:33.710 align:middle line:84% the thickness of the glass sample we're talking about here. 01:13:33.710 --> 01:13:38.990 align:middle line:84% So this would just be 2 gamma, thickness of the glass. 01:13:38.990 --> 01:13:41.290 align:middle line:84% So what would be the thickness of glass, a0 here? 01:13:41.290 --> 01:13:44.830 align:middle line:90% 01:13:44.830 --> 01:13:47.510 align:middle line:84% So this would be the thickness of glass that actually generates 01:13:47.510 --> 01:13:51.423 align:middle line:90% two fresh, new surfaces. 01:13:51.423 --> 01:13:53.090 align:middle line:84% So now, if you think about this process, 01:13:53.090 --> 01:13:55.507 align:middle line:84% the way I'm drawing it here can be a little bit misleading 01:13:55.507 --> 01:13:57.270 align:middle line:84% because the way I'm drawing it here 01:13:57.270 --> 01:13:59.850 align:middle line:84% seems like you have a large piece of glass. 01:13:59.850 --> 01:14:03.550 align:middle line:84% When you start pulling it, only one-- 01:14:03.550 --> 01:14:08.310 align:middle line:84% you only get elongation within one of the atomic planes here, 01:14:08.310 --> 01:14:10.510 align:middle line:84% which is not the case in the ideal scenario, 01:14:10.510 --> 01:14:12.670 align:middle line:84% if it is completely [? defect-free. ?] When you 01:14:12.670 --> 01:14:15.230 align:middle line:84% stretch this class, due to symmetry, 01:14:15.230 --> 01:14:19.550 align:middle line:84% all the chemical bonds should be stretched proportionally. 01:14:19.550 --> 01:14:22.390 align:middle line:84% So when you break, you don't break at just one 01:14:22.390 --> 01:14:23.650 align:middle line:90% of these atomic planes there. 01:14:23.650 --> 01:14:24.557 align:middle line:90% In fact, all of them. 01:14:24.557 --> 01:14:26.390 align:middle line:84% You essentially break the entire glass piece 01:14:26.390 --> 01:14:28.290 align:middle line:84% into a set of single atomic plates. 01:14:28.290 --> 01:14:31.350 align:middle line:84% They should break all at the same time for the chemical bond, 01:14:31.350 --> 01:14:33.310 align:middle line:90% if it's uniform. 01:14:33.310 --> 01:14:35.430 align:middle line:84% So, therefore, what it means is that, 01:14:35.430 --> 01:14:37.570 align:middle line:84% to generate two fresh new surfaces, 01:14:37.570 --> 01:14:40.290 align:middle line:84% you are really looking at only one single atomic plane. 01:14:40.290 --> 01:14:45.110 align:middle line:84% Each atomic plane generate two surfaces. 01:14:45.110 --> 01:14:48.670 align:middle line:84% The other way to think about it, just think about simple two-- 01:14:48.670 --> 01:14:50.510 align:middle line:90% just a bilayer glass. 01:14:50.510 --> 01:14:52.157 align:middle line:84% Start to stretch the bilayer glass, 01:14:52.157 --> 01:14:53.490 align:middle line:90% eventually, it's going to break. 01:14:53.490 --> 01:14:56.250 align:middle line:90% You generate two flat surfaces. 01:14:56.250 --> 01:14:59.110 align:middle line:84% That's exactly the process, what we're describing here, 01:14:59.110 --> 01:15:01.550 align:middle line:84% if you just stretch a bilayer glass. 01:15:01.550 --> 01:15:03.990 align:middle line:84% So in this case, the thickness of glass, 01:15:03.990 --> 01:15:06.770 align:middle line:90% a0 would just be atomic spacing. 01:15:06.770 --> 01:15:11.680 align:middle line:90% 01:15:11.680 --> 01:15:14.620 align:middle line:84% If you have a thicker glass, you actually generate more surfaces. 01:15:14.620 --> 01:15:18.840 align:middle line:90% 01:15:18.840 --> 01:15:21.800 align:middle line:84% So you increase v, but you increase the s proportionally 01:15:21.800 --> 01:15:22.580 align:middle line:90% at the same time. 01:15:22.580 --> 01:15:25.880 align:middle line:90% 01:15:25.880 --> 01:15:30.020 align:middle line:84% So now you can then finally put it back into the equation here. 01:15:30.020 --> 01:15:32.440 align:middle line:84% Now, you know whether w equals to vs. 01:15:32.440 --> 01:15:39.440 align:middle line:84% This is 2 gamma divided by a0, which is atomic spacing. 01:15:39.440 --> 01:15:42.200 align:middle line:90% You cancel out 2, cancel out 2. 01:15:42.200 --> 01:15:57.760 align:middle line:84% So now, finally, your sigma m becomes gamma E a0 square root. 01:15:57.760 --> 01:16:01.640 align:middle line:90% 01:16:01.640 --> 01:16:05.437 align:middle line:84% So this would be the theoretical strength of a glass. 01:16:05.437 --> 01:16:07.520 align:middle line:84% This is the stress you will need to actually break 01:16:07.520 --> 01:16:10.340 align:middle line:84% the glass sample to essentially create these fresh new surfaces. 01:16:10.340 --> 01:16:16.960 align:middle line:90% 01:16:16.960 --> 01:16:19.460 align:middle line:84% All right, I'm going to quickly just end here. 01:16:19.460 --> 01:16:21.960 align:middle line:84% I just want to mention that now you have all these numbers. 01:16:21.960 --> 01:16:23.710 align:middle line:84% With this equation, you can go and predict 01:16:23.710 --> 01:16:26.200 align:middle line:84% the theoretical strength of glass. 01:16:26.200 --> 01:16:31.200 align:middle line:84% You come up with a remarkable number of 35 gigapascal. 01:16:31.200 --> 01:16:35.640 align:middle line:84% This is way higher than steel, way higher 01:16:35.640 --> 01:16:37.500 align:middle line:84% than any glass you actually measure. 01:16:37.500 --> 01:16:40.178 align:middle line:84% Typical glass, you actually get 30 megapascals. 01:16:40.178 --> 01:16:41.720 align:middle line:84% So it's like three order of magnitude 01:16:41.720 --> 01:16:43.640 align:middle line:84% lower compared to physical strength. 01:16:43.640 --> 01:16:46.360 align:middle line:84% However, if you look at silicon nanowire, 01:16:46.360 --> 01:16:48.200 align:middle line:84% they do get to the strength at least 01:16:48.200 --> 01:16:50.560 align:middle line:90% at the same order of magnitude. 01:16:50.560 --> 01:16:53.040 align:middle line:84% So what it says, basically, is that there's 01:16:53.040 --> 01:16:56.240 align:middle line:84% some kind of defects in glass, microscopic defects. 01:16:56.240 --> 01:16:59.280 align:middle line:84% The defects compromise the strength of the glass material. 01:16:59.280 --> 01:17:02.280 align:middle line:84% However, if you go to a nanowire, because it's so small, 01:17:02.280 --> 01:17:04.850 align:middle line:84% the probability of having a defect-free nanowire 01:17:04.850 --> 01:17:06.910 align:middle line:84% is much higher compared to bulk glass. 01:17:06.910 --> 01:17:09.330 align:middle line:84% So that's why you can approach this theoretical limit 01:17:09.330 --> 01:17:12.730 align:middle line:84% in silicon nanowire, but not in a macroscopic glass piece. 01:17:12.730 --> 01:17:15.850 align:middle line:84% So we're going to go into a bit more details about the theory 01:17:15.850 --> 01:17:17.250 align:middle line:90% later in the next lecture. 01:17:17.250 --> 01:17:18.345 align:middle line:90% Let's actually stop here. 01:17:18.345 --> 01:17:19.470 align:middle line:90% We are running out of time. 01:17:19.470 --> 01:17:20.410 align:middle line:90% Any more questions? 01:17:20.410 --> 01:17:21.715 align:middle line:90% Yes? 01:17:21.715 --> 01:17:24.090 align:middle line:84% AUDIENCE: And what will we consider 01:17:24.090 --> 01:17:28.570 align:middle line:84% as a defect, defect within [INAUDIBLE] silicon nanowires 01:17:28.570 --> 01:17:32.123 align:middle line:84% since there's not any structural bias? 01:17:32.123 --> 01:17:34.290 align:middle line:84% JUEJUN HU: You can still have defects, if you have-- 01:17:34.290 --> 01:17:36.890 align:middle line:84% I don't know-- maybe a tiny, little surface 01:17:36.890 --> 01:17:39.090 align:middle line:90% dent missing a few atoms. 01:17:39.090 --> 01:17:40.450 align:middle line:90% Or you just have non-uniformity. 01:17:40.450 --> 01:17:41.950 align:middle line:90% Diameter is non-uniform. 01:17:41.950 --> 01:17:44.530 align:middle line:84% There's some atom rearrangement or whatever 01:17:44.530 --> 01:17:46.210 align:middle line:90% having some-- one location. 01:17:46.210 --> 01:17:47.850 align:middle line:90% Or you have stacking difference. 01:17:47.850 --> 01:17:49.870 align:middle line:84% So you can have some defects there. 01:17:49.870 --> 01:17:51.590 align:middle line:90% But, again, it's so small. 01:17:51.590 --> 01:17:54.510 align:middle line:84% And the defect we are talking about, as you'll see later, 01:17:54.510 --> 01:17:57.262 align:middle line:84% they are essentially microcracks. 01:17:57.262 --> 01:17:58.970 align:middle line:84% Those microcracks typically have a length 01:17:58.970 --> 01:18:03.570 align:middle line:84% of a few microns, which is much bigger than the nanowire. 01:18:03.570 --> 01:18:04.410 align:middle line:90% So that's why. 01:18:04.410 --> 01:18:05.955 align:middle line:84% In nanowire, in lots of cases, it 01:18:05.955 --> 01:18:07.330 align:middle line:84% can be considered as defect free. 01:18:07.330 --> 01:18:08.970 align:middle line:84% And that's why you can really approach 01:18:08.970 --> 01:18:10.830 align:middle line:84% the theoretical strength so closely. 01:18:10.830 --> 01:18:14.070 align:middle line:90% 01:18:14.070 --> 01:18:14.910 align:middle line:90% Any more questions? 01:18:14.910 --> 01:18:17.530 align:middle line:90% 01:18:17.530 --> 01:18:20.040 align:middle line:90% All right, thank you. 01:18:20.040 --> 01:19:03.000 align:middle line:90%