WEBVTT 00:00:00.000 --> 00:00:02.460 align:middle line:90% [SQUEAKING] 00:00:02.460 --> 00:00:04.420 align:middle line:90% [RUSTLING] 00:00:04.420 --> 00:00:06.380 align:middle line:90% [CLICKING] 00:00:06.380 --> 00:00:17.980 align:middle line:90% 00:00:17.980 --> 00:00:19.880 align:middle line:90% JUEJUN HU: OK. 00:00:19.880 --> 00:00:22.280 align:middle line:84% So in this lecture, we are going to start our discussion 00:00:22.280 --> 00:00:26.112 align:middle line:84% about viscoelasticity and relaxation. 00:00:26.112 --> 00:00:28.320 align:middle line:84% Now, I want to mention that this type of organization 00:00:28.320 --> 00:00:33.040 align:middle line:84% is uncommon in most textbooks about glass science. 00:00:33.040 --> 00:00:36.920 align:middle line:84% They typically treat viscoelasticity and relaxation 00:00:36.920 --> 00:00:39.280 align:middle line:84% in two completely different sections there. 00:00:39.280 --> 00:00:41.880 align:middle line:84% I decided to put them together for a few reasons. 00:00:41.880 --> 00:00:45.440 align:middle line:84% One is that they both occur during the glass transition 00:00:45.440 --> 00:00:48.892 align:middle line:84% regime, and they have a very similar structural origin. 00:00:48.892 --> 00:00:50.600 align:middle line:84% They essentially all relate to relaxation 00:00:50.600 --> 00:00:51.820 align:middle line:90% of the glass structure. 00:00:51.820 --> 00:00:54.680 align:middle line:84% And as you also see later, they can 00:00:54.680 --> 00:00:58.240 align:middle line:84% be described using identical mathematical formalisms. 00:00:58.240 --> 00:01:01.200 align:middle line:84% So it helps a lot if we develop mathematical models 00:01:01.200 --> 00:01:05.620 align:middle line:84% and then use it as a way to lead to the discussion 00:01:05.620 --> 00:01:07.180 align:middle line:90% about relaxation kinetics. 00:01:07.180 --> 00:01:08.740 align:middle line:90% OK. 00:01:08.740 --> 00:01:09.420 align:middle line:90% OK. 00:01:09.420 --> 00:01:13.340 align:middle line:84% So as I mentioned, both of these phenomena 00:01:13.340 --> 00:01:17.340 align:middle line:84% are characters of the glass transition regime. 00:01:17.340 --> 00:01:20.140 align:middle line:84% So the glass transition regime scientifically 00:01:20.140 --> 00:01:24.660 align:middle line:84% is probably the most intriguing part about the whole glassy 00:01:24.660 --> 00:01:25.580 align:middle line:90% state. 00:01:25.580 --> 00:01:29.000 align:middle line:84% So as one example here, what I'm showing you here, 00:01:29.000 --> 00:01:33.660 align:middle line:84% this is the Journal of Science in their 125th anniversary. 00:01:33.660 --> 00:01:37.220 align:middle line:84% They posted the list of 125 questions that 00:01:37.220 --> 00:01:40.900 align:middle line:84% summarizes some of the most significant outstanding 00:01:40.900 --> 00:01:43.340 align:middle line:84% challenges in all fields of science. 00:01:43.340 --> 00:01:46.660 align:middle line:84% So one of them is about glass, and it's 00:01:46.660 --> 00:01:48.360 align:middle line:90% right about glass transition. 00:01:48.360 --> 00:01:51.020 align:middle line:84% So what it says is, "what is the nature of the glassy state? 00:01:51.020 --> 00:01:53.020 align:middle line:84% Molecules in a glass are arranged much 00:01:53.020 --> 00:01:55.540 align:middle line:84% like those in liquids, but are more tightly packed. 00:01:55.540 --> 00:02:00.680 align:middle line:84% Where and why does liquid end and glass begin?" 00:02:00.680 --> 00:02:02.640 align:middle line:90% That's glass transition. 00:02:02.640 --> 00:02:04.440 align:middle line:90% OK. 00:02:04.440 --> 00:02:09.000 align:middle line:84% So there has been lots of advances 00:02:09.000 --> 00:02:11.320 align:middle line:84% in this field, obviously, over the past 20 years 00:02:11.320 --> 00:02:13.120 align:middle line:90% after the question was posed. 00:02:13.120 --> 00:02:16.040 align:middle line:84% But suffice it to say that this is still 00:02:16.040 --> 00:02:18.780 align:middle line:84% an area of very active scientific investigation. 00:02:18.780 --> 00:02:25.240 align:middle line:84% In fact, I also want to quote the physicist David Weitz from 00:02:25.240 --> 00:02:26.815 align:middle line:84% Harvard University, where he says, 00:02:26.815 --> 00:02:29.440 align:middle line:84% "there are more theories of the glass transition than there are 00:02:29.440 --> 00:02:31.720 align:middle line:90% theorists who propose them." 00:02:31.720 --> 00:02:36.600 align:middle line:84% So what we are going to do in this lecture here-- 00:02:36.600 --> 00:02:38.360 align:middle line:84% I'm actually not going to try to take 00:02:38.360 --> 00:02:41.880 align:middle line:84% a deep dive into the microstructure nature of glass 00:02:41.880 --> 00:02:42.640 align:middle line:90% transition. 00:02:42.640 --> 00:02:44.440 align:middle line:84% Instead, what I want to do is I want 00:02:44.440 --> 00:02:47.320 align:middle line:84% to focus on what I consider as a defining 00:02:47.320 --> 00:02:52.080 align:middle line:84% feature of glass transition regime, which is the relaxation. 00:02:52.080 --> 00:02:56.123 align:middle line:84% Now, to look into relaxation, let's step back a little bit. 00:02:56.123 --> 00:02:58.540 align:middle line:84% Let's take a look at this slide, which I quoted, actually, 00:02:58.540 --> 00:03:01.220 align:middle line:84% from a previous lecture, that essentially looks 00:03:01.220 --> 00:03:02.700 align:middle line:84% at the mechanical characteristics, 00:03:02.700 --> 00:03:04.740 align:middle line:84% or rheological characteristics, of glasses 00:03:04.740 --> 00:03:06.180 align:middle line:90% across different temperatures. 00:03:06.180 --> 00:03:07.900 align:middle line:84% We mentioned that at low temperature, 00:03:07.900 --> 00:03:10.860 align:middle line:84% glasses is basically an elastic solid, exhibit 00:03:10.860 --> 00:03:12.460 align:middle line:84% all the characteristics of a solid 00:03:12.460 --> 00:03:16.463 align:middle line:84% that has instantaneous response, follows Hookean's law, and such. 00:03:16.463 --> 00:03:18.880 align:middle line:84% At high temperature, where it becomes a super cool liquid, 00:03:18.880 --> 00:03:21.720 align:middle line:84% it is obviously liquid, so it will flow viscously, 00:03:21.720 --> 00:03:24.900 align:middle line:84% and it can be described more or less as a Newtonian fluid. 00:03:24.900 --> 00:03:27.300 align:middle line:84% Now, if you look at things in between, 00:03:27.300 --> 00:03:29.660 align:middle line:84% this is where the glass transition comes in, OK? 00:03:29.660 --> 00:03:32.260 align:middle line:84% And in between regime here, there 00:03:32.260 --> 00:03:35.600 align:middle line:84% will exhibit properties that combine both solid and liquid, 00:03:35.600 --> 00:03:37.180 align:middle line:90% which is viscoelasticity. 00:03:37.180 --> 00:03:39.620 align:middle line:84% And this is also the regime where you actually 00:03:39.620 --> 00:03:42.940 align:middle line:84% see significant structural relaxation. 00:03:42.940 --> 00:03:45.080 align:middle line:84% So before I move on to the next slide, 00:03:45.080 --> 00:03:46.880 align:middle line:84% I just want to quickly mention one thing. 00:03:46.880 --> 00:03:48.320 align:middle line:84% Throughout this entire slide deck, 00:03:48.320 --> 00:03:51.080 align:middle line:84% I'm using this kind of a color coding scheme here. 00:03:51.080 --> 00:03:54.060 align:middle line:84% Anything that's red is associated 00:03:54.060 --> 00:03:56.640 align:middle line:90% with a solid or elasticity. 00:03:56.640 --> 00:04:00.760 align:middle line:84% Anything that's blue is basically the liquid or viscous 00:04:00.760 --> 00:04:01.840 align:middle line:90% contribution. 00:04:01.840 --> 00:04:02.960 align:middle line:90% OK? 00:04:02.960 --> 00:04:03.760 align:middle line:90% All right. 00:04:03.760 --> 00:04:06.320 align:middle line:84% So I made the statement that relaxation 00:04:06.320 --> 00:04:11.320 align:middle line:84% is only significant in the glass transition regime. 00:04:11.320 --> 00:04:12.840 align:middle line:90% Why? 00:04:12.840 --> 00:04:16.519 align:middle line:84% Let's see what happens if you go through the liquid phase. 00:04:16.519 --> 00:04:19.560 align:middle line:84% So previously, when we were talking about the laboratory 00:04:19.560 --> 00:04:21.839 align:middle line:84% definition of glass transition, we 00:04:21.839 --> 00:04:25.680 align:middle line:84% said that the liquid regime is essentially ergodic. 00:04:25.680 --> 00:04:29.520 align:middle line:84% All the molecules or atoms, they have sufficiently high energy, 00:04:29.520 --> 00:04:31.600 align:middle line:84% they can easily move around, and therefore, they 00:04:31.600 --> 00:04:35.120 align:middle line:84% can access all the possible microscopic states. 00:04:35.120 --> 00:04:38.320 align:middle line:84% What that also means is the system tends to stay 00:04:38.320 --> 00:04:41.515 align:middle line:90% in the "equilibrium" state, OK? 00:04:41.515 --> 00:04:43.140 align:middle line:84% "Equilibrium" with the quotation marks. 00:04:43.140 --> 00:04:45.280 align:middle line:84% I'm going to talk about that later. 00:04:45.280 --> 00:04:48.440 align:middle line:84% But the idea is that because molecular motion is 00:04:48.440 --> 00:04:52.600 align:middle line:84% so efficient and so swift, the relaxation is actually 00:04:52.600 --> 00:04:53.460 align:middle line:90% very, very fast. 00:04:53.460 --> 00:04:55.400 align:middle line:84% Any structure change is almost instantaneous. 00:04:55.400 --> 00:04:58.050 align:middle line:84% That's why it tends to stay at the state that corresponds 00:04:58.050 --> 00:05:00.890 align:middle line:84% to the lowest Gibbs free energy, which is the "equilibrium" 00:05:00.890 --> 00:05:02.810 align:middle line:84% state, as we are talking about here. 00:05:02.810 --> 00:05:07.970 align:middle line:84% So liquid, by definition, is always in the equilibrium state. 00:05:07.970 --> 00:05:09.490 align:middle line:90% Now, what about solid? 00:05:09.490 --> 00:05:12.370 align:middle line:84% If you cool it down, then all the atomic motion 00:05:12.370 --> 00:05:14.010 align:middle line:90% becomes very sluggish. 00:05:14.010 --> 00:05:16.370 align:middle line:84% Then kinetically, any relaxation is 00:05:16.370 --> 00:05:18.790 align:middle line:84% getting inhibited despite that, in this case, 00:05:18.790 --> 00:05:21.450 align:middle line:84% you have a very large thermodynamic driving force 00:05:21.450 --> 00:05:23.530 align:middle line:84% towards relaxing to the equilibrium state. 00:05:23.530 --> 00:05:25.190 align:middle line:84% But kinetically, it won't happen. 00:05:25.190 --> 00:05:27.550 align:middle line:84% So that's why, in the glassy state, 00:05:27.550 --> 00:05:30.530 align:middle line:84% you also don't see a lot of structural relaxation. 00:05:30.530 --> 00:05:32.788 align:middle line:84% Right in between, the glass transition, 00:05:32.788 --> 00:05:34.330 align:middle line:84% is where you have the right condition 00:05:34.330 --> 00:05:36.490 align:middle line:84% to induce significant relaxation. 00:05:36.490 --> 00:05:39.370 align:middle line:84% Kinetically, it is sufficiently fast. 00:05:39.370 --> 00:05:44.350 align:middle line:84% It will take place during the duration of our experiment. 00:05:44.350 --> 00:05:47.610 align:middle line:84% And thermodynamically, that's also adequate driving force. 00:05:47.610 --> 00:05:50.450 align:middle line:84% The glass line has already deviated 00:05:50.450 --> 00:05:51.710 align:middle line:90% from supercooled liquid line. 00:05:51.710 --> 00:05:53.710 align:middle line:84% So there is actually thermodynamic driving force 00:05:53.710 --> 00:05:55.190 align:middle line:90% for such relaxation. 00:05:55.190 --> 00:05:57.870 align:middle line:84% So we can characterize the significance 00:05:57.870 --> 00:05:59.550 align:middle line:84% of this kind of relaxation phenomena 00:05:59.550 --> 00:06:01.990 align:middle line:84% by again comparing two quantities here. 00:06:01.990 --> 00:06:06.030 align:middle line:84% So tau with the subscript ie, this 00:06:06.030 --> 00:06:10.190 align:middle line:84% is the time scale for structural relaxation. 00:06:10.190 --> 00:06:14.550 align:middle line:84% And t obs, this would be the time of observation 00:06:14.550 --> 00:06:16.570 align:middle line:84% or basically the duration of the experiment. 00:06:16.570 --> 00:06:20.110 align:middle line:84% So in the glass transition regime, 00:06:20.110 --> 00:06:23.410 align:middle line:84% we have the two time scales more or less are comparable. 00:06:23.410 --> 00:06:26.950 align:middle line:84% This is where you will see significant relaxation. 00:06:26.950 --> 00:06:28.950 align:middle line:90% OK. 00:06:28.950 --> 00:06:31.670 align:middle line:84% Quantitatively, people define this quantity here 00:06:31.670 --> 00:06:33.310 align:middle line:90% called Deborah Number-- 00:06:33.310 --> 00:06:37.550 align:middle line:84% it's basically a ratio of tau over the t observation. 00:06:37.550 --> 00:06:41.670 align:middle line:84% So it's equivalent to saying that the Deborah number has 00:06:41.670 --> 00:06:44.070 align:middle line:84% to be of the order of unity in order 00:06:44.070 --> 00:06:46.710 align:middle line:84% to see significant structural relaxation, which correspond 00:06:46.710 --> 00:06:47.945 align:middle line:90% to glass transition regime. 00:06:47.945 --> 00:06:50.070 align:middle line:84% Now, you wonder where the Deborah number, this name 00:06:50.070 --> 00:06:50.750 align:middle line:90% comes from? 00:06:50.750 --> 00:06:52.410 align:middle line:90% It's actually from the Bible. 00:06:52.410 --> 00:06:58.290 align:middle line:84% It basically says that the prophetess Deborah, 00:06:58.290 --> 00:07:00.730 align:middle line:84% over a time period of billions of years, 00:07:00.730 --> 00:07:03.970 align:middle line:84% they observed that the mountains flowed before the Lord. 00:07:03.970 --> 00:07:07.465 align:middle line:90% So that's viscoelasticity. 00:07:07.465 --> 00:07:10.090 align:middle line:84% It essentially means that if you give it long enough relaxation 00:07:10.090 --> 00:07:12.350 align:middle line:84% time, anything that appears like a solid, 00:07:12.350 --> 00:07:16.250 align:middle line:84% they can exhibit some viscoelastic response. 00:07:16.250 --> 00:07:21.290 align:middle line:84% So that's why people call it the Deborah number. 00:07:21.290 --> 00:07:23.010 align:middle line:84% So as I mentioned, we're going to talk 00:07:23.010 --> 00:07:26.290 align:middle line:84% about structural relaxation and viscoelasticity here. 00:07:26.290 --> 00:07:29.210 align:middle line:84% Viscoelasticity, in itself, is the manifestation 00:07:29.210 --> 00:07:30.830 align:middle line:84% of structural relaxation in glass. 00:07:30.830 --> 00:07:32.338 align:middle line:90% That's the microscopic origin. 00:07:32.338 --> 00:07:34.130 align:middle line:84% Now, before we go to the microscopic origin 00:07:34.130 --> 00:07:36.210 align:middle line:84% here, what we will focus on is focusing 00:07:36.210 --> 00:07:39.250 align:middle line:84% on the mathematical description of viscoelasticity. 00:07:39.250 --> 00:07:42.010 align:middle line:84% And as I said, this will serve as the basis 00:07:42.010 --> 00:07:45.090 align:middle line:84% to derive the mathematical formalism to describe 00:07:45.090 --> 00:07:46.450 align:middle line:90% structural relaxation. 00:07:46.450 --> 00:07:48.610 align:middle line:84% I have promised you before that we 00:07:48.610 --> 00:07:50.690 align:middle line:84% are going to derive a mathematical theory that 00:07:50.690 --> 00:07:52.590 align:middle line:84% allows you to quantitatively predict 00:07:52.590 --> 00:07:56.510 align:middle line:84% what is the shape of this curve as the glasses cool down 00:07:56.510 --> 00:07:59.425 align:middle line:84% and reheat up, going through the glass transition region. 00:07:59.425 --> 00:08:01.550 align:middle line:84% That's what we're going to do in this lecture here. 00:08:01.550 --> 00:08:02.790 align:middle line:90% OK. 00:08:02.790 --> 00:08:03.350 align:middle line:90% OK. 00:08:03.350 --> 00:08:08.030 align:middle line:84% So speaking of viscoelasticity, what I'm introducing here 00:08:08.030 --> 00:08:10.670 align:middle line:84% basically is just a basic mathematical formalism 00:08:10.670 --> 00:08:12.950 align:middle line:84% to describe the mechanical properties of solids 00:08:12.950 --> 00:08:14.910 align:middle line:90% in a viscoelastic regime. 00:08:14.910 --> 00:08:17.330 align:middle line:84% You may have seen this a little bit in 3013. 00:08:17.330 --> 00:08:20.870 align:middle line:84% But we will be considering a complex shear 00:08:20.870 --> 00:08:23.150 align:middle line:84% modulus rather than a pure real shear modulus, 00:08:23.150 --> 00:08:25.190 align:middle line:90% like elastic solids. 00:08:25.190 --> 00:08:29.710 align:middle line:84% And to define a complex elastic modulus, what you would do 00:08:29.710 --> 00:08:35.970 align:middle line:84% is considering a stimulus that's applied in a sinusoidal manner. 00:08:35.970 --> 00:08:37.549 align:middle line:90% It can be stress or strain. 00:08:37.549 --> 00:08:40.830 align:middle line:84% In this case, we actually choose to apply a sinusoidally varying 00:08:40.830 --> 00:08:41.789 align:middle line:90% shear strain. 00:08:41.789 --> 00:08:42.429 align:middle line:90% OK? 00:08:42.429 --> 00:08:44.030 align:middle line:84% So mathematically, you can describe 00:08:44.030 --> 00:08:46.870 align:middle line:90% it using a complex number. 00:08:46.870 --> 00:08:48.870 align:middle line:84% And if you think about elastic response, 00:08:48.870 --> 00:08:50.050 align:middle line:90% it's pretty straightforward. 00:08:50.050 --> 00:08:52.550 align:middle line:84% You just time it with the elastic constant. 00:08:52.550 --> 00:08:57.050 align:middle line:84% So what we see here is that the stress 00:08:57.050 --> 00:09:00.070 align:middle line:84% from elastic response and the strain, 00:09:00.070 --> 00:09:02.330 align:middle line:84% they are always in phase with each other because 00:09:02.330 --> 00:09:04.410 align:middle line:90% of instantaneous response. 00:09:04.410 --> 00:09:08.450 align:middle line:84% Now, if it's a viscous response, the constitutive relation, 00:09:08.450 --> 00:09:11.210 align:middle line:84% especially given by this one here, 00:09:11.210 --> 00:09:17.130 align:middle line:84% this Newtonian flow, now, if you take a time derivative 00:09:17.130 --> 00:09:20.370 align:middle line:84% with respect to the strain here, because 00:09:20.370 --> 00:09:22.250 align:middle line:84% of the complex differential, when 00:09:22.250 --> 00:09:24.210 align:middle line:84% you take the time derivative, you 00:09:24.210 --> 00:09:28.410 align:middle line:90% pull out a factor of i omega. 00:09:28.410 --> 00:09:34.410 align:middle line:84% So that means, effectively, now the stress 00:09:34.410 --> 00:09:40.290 align:middle line:84% differs from the strain by a factor of i. 00:09:40.290 --> 00:09:42.650 align:middle line:84% There's an imaginary number coming into play here. 00:09:42.650 --> 00:09:45.290 align:middle line:84% The imaginary number in the complex plane means that these 00:09:45.290 --> 00:09:51.310 align:middle line:84% two quantities, their phase, differ by 1/2 pi. 00:09:51.310 --> 00:09:52.990 align:middle line:84% So therefore, there is a phase lag 00:09:52.990 --> 00:09:56.050 align:middle line:84% between the stress and the corresponding strain. 00:09:56.050 --> 00:10:00.550 align:middle line:84% They are out of phase if you are looking at the viscous response. 00:10:00.550 --> 00:10:03.350 align:middle line:84% So now, if you combine both together, then 00:10:03.350 --> 00:10:07.590 align:middle line:84% phenomenologically, the effective shear modulus 00:10:07.590 --> 00:10:10.950 align:middle line:84% would then contain both real part and imaginary part, 00:10:10.950 --> 00:10:15.590 align:middle line:84% where the real part comes from the elastic contribution, 00:10:15.590 --> 00:10:20.330 align:middle line:84% whereas the imaginary part comes from the viscous contribution. 00:10:20.330 --> 00:10:23.465 align:middle line:90% 00:10:23.465 --> 00:10:26.430 align:middle line:84% So that's why the real part is what people call, 00:10:26.430 --> 00:10:27.610 align:middle line:90% basically, a shear modulus. 00:10:27.610 --> 00:10:29.850 align:middle line:84% Sometimes people also call it a storage modulus, 00:10:29.850 --> 00:10:32.110 align:middle line:84% because any elastic response, you 00:10:32.110 --> 00:10:35.310 align:middle line:84% are converting the external force, 00:10:35.310 --> 00:10:38.750 align:middle line:84% the energy of external force or work done by external force, 00:10:38.750 --> 00:10:40.370 align:middle line:90% into elastic energy. 00:10:40.370 --> 00:10:41.530 align:middle line:90% So it's stored there. 00:10:41.530 --> 00:10:42.970 align:middle line:90% You have elastic energy stored. 00:10:42.970 --> 00:10:47.170 align:middle line:84% But if it's a viscous flow, because it's internal friction, 00:10:47.170 --> 00:10:50.330 align:middle line:84% you are moving molecules past each other, 00:10:50.330 --> 00:10:52.410 align:middle line:84% any work performed by external shear force 00:10:52.410 --> 00:10:54.450 align:middle line:84% would be dissipated in the form of heat. 00:10:54.450 --> 00:10:55.850 align:middle line:84% So that's why this is also called 00:10:55.850 --> 00:10:58.770 align:middle line:84% the loss modulus, the imaginary part of the shear modulus. 00:10:58.770 --> 00:11:02.890 align:middle line:84% It's also considered as a way to dissipate mechanical work. 00:11:02.890 --> 00:11:04.690 align:middle line:90% OK. 00:11:04.690 --> 00:11:05.190 align:middle line:90% OK. 00:11:05.190 --> 00:11:07.250 align:middle line:84% So that's the high-level framework. 00:11:07.250 --> 00:11:09.770 align:middle line:84% Now, to derive the mathematical formalism 00:11:09.770 --> 00:11:12.490 align:middle line:84% of how to actually look at relaxation 00:11:12.490 --> 00:11:16.850 align:middle line:84% phenomena in a viscoelastic solid, 00:11:16.850 --> 00:11:21.170 align:middle line:84% we have to introduce a couple of phenomenological models. 00:11:21.170 --> 00:11:24.050 align:middle line:84% Of course, Hookean spring and Newtonian dashpot, 00:11:24.050 --> 00:11:26.210 align:middle line:84% which I suppose you have seen in 3013? 00:11:26.210 --> 00:11:26.770 align:middle line:90% Yeah. 00:11:26.770 --> 00:11:29.190 align:middle line:90% So Hookean spring-- simple. 00:11:29.190 --> 00:11:31.210 align:middle line:90% It's strain equals stress. 00:11:31.210 --> 00:11:34.770 align:middle line:84% Dashpot is something characterized ideal properties 00:11:34.770 --> 00:11:37.490 align:middle line:90% of a Newtonian fluid. 00:11:37.490 --> 00:11:39.290 align:middle line:84% The assumption there is that because we 00:11:39.290 --> 00:11:41.530 align:middle line:84% are assuming linear relations-- so practically, 00:11:41.530 --> 00:11:44.570 align:middle line:84% we are assuming we are operating in the linear regime. 00:11:44.570 --> 00:11:47.450 align:middle line:84% So linear regime, rigorously speaking, 00:11:47.450 --> 00:11:49.950 align:middle line:84% applies only when the strain is very small. 00:11:49.950 --> 00:11:52.410 align:middle line:84% And so the strain rate is also very small. 00:11:52.410 --> 00:11:53.730 align:middle line:90% That's why things stay linear. 00:11:53.730 --> 00:11:58.110 align:middle line:84% And as we will see later, what's really important there 00:11:58.110 --> 00:11:59.750 align:middle line:84% is that every time you're introducing 00:11:59.750 --> 00:12:04.590 align:middle line:84% one phenomenological model containing a Newtonian dashpot, 00:12:04.590 --> 00:12:09.070 align:middle line:84% each dashpot corresponds to one microscopic structural 00:12:09.070 --> 00:12:11.310 align:middle line:90% relaxation mechanism. 00:12:11.310 --> 00:12:14.390 align:middle line:84% And so as you'll see later, each Newtonian dashpot 00:12:14.390 --> 00:12:17.670 align:middle line:84% is associated with one characteristic relaxation time 00:12:17.670 --> 00:12:18.670 align:middle line:90% scale. 00:12:18.670 --> 00:12:20.990 align:middle line:84% So different microscopic mechanisms 00:12:20.990 --> 00:12:22.550 align:middle line:84% can have very different time scales. 00:12:22.550 --> 00:12:25.430 align:middle line:84% So for example, if you think of polymers, when you're actually 00:12:25.430 --> 00:12:28.830 align:middle line:84% introducing plastic deformation or a viscoelastic deformation 00:12:28.830 --> 00:12:31.630 align:middle line:84% in polymers, the shape change can 00:12:31.630 --> 00:12:34.910 align:middle line:84% be introduced by changing the conformation of the polymer 00:12:34.910 --> 00:12:35.470 align:middle line:90% chain. 00:12:35.470 --> 00:12:39.410 align:middle line:84% You can have a linear chain getting folded or twisted. 00:12:39.410 --> 00:12:43.270 align:middle line:84% And then you can also have, for example, bone 00:12:43.270 --> 00:12:46.190 align:middle line:84% breaking or having some kind of chemical reaction. 00:12:46.190 --> 00:12:48.360 align:middle line:84% So these two have very different time scales. 00:12:48.360 --> 00:12:50.920 align:middle line:84% So one's reflected on the phenomenological model 00:12:50.920 --> 00:12:53.100 align:middle line:84% of the material in terms of mechanical property, 00:12:53.100 --> 00:12:55.860 align:middle line:84% is that you then need two Newtonian dashpots 00:12:55.860 --> 00:12:58.360 align:middle line:84% to correspond to each of these kind of microscopic structure 00:12:58.360 --> 00:13:00.080 align:middle line:90% relaxation mechanisms. 00:13:00.080 --> 00:13:02.500 align:middle line:90% OK? 00:13:02.500 --> 00:13:06.520 align:middle line:84% So now, in order to introduce in the mathematical formulation 00:13:06.520 --> 00:13:08.120 align:middle line:84% here, we look at a few different ways 00:13:08.120 --> 00:13:10.560 align:middle line:84% to assemble these basic elements together 00:13:10.560 --> 00:13:13.100 align:middle line:84% to describe the viscoelastic response. 00:13:13.100 --> 00:13:14.240 align:middle line:90% The most simple one-- 00:13:14.240 --> 00:13:15.380 align:middle line:90% Maxwell element. 00:13:15.380 --> 00:13:16.920 align:middle line:90% Just put the two in series. 00:13:16.920 --> 00:13:18.760 align:middle line:84% If you put them in series, obviously, 00:13:18.760 --> 00:13:23.000 align:middle line:84% the stress is constant across the two elements. 00:13:23.000 --> 00:13:26.200 align:middle line:84% But the strain is going to be the sum of the strain that's 00:13:26.200 --> 00:13:29.520 align:middle line:84% sustained by the spring and the dashpot. 00:13:29.520 --> 00:13:35.520 align:middle line:84% So now we can consider what is its response 00:13:35.520 --> 00:13:39.660 align:middle line:84% of the system to two kinds of external stimuli, OK? 00:13:39.660 --> 00:13:42.320 align:middle line:84% Constant stress versus constant strain. 00:13:42.320 --> 00:13:45.680 align:middle line:84% Constant stress is relatively straightforward. 00:13:45.680 --> 00:13:48.900 align:middle line:84% If you apply a constant stress to a system, 00:13:48.900 --> 00:13:52.180 align:middle line:84% because the stress is uniform across both elements-- 00:13:52.180 --> 00:13:54.300 align:middle line:84% so same stress applied to both elements-- 00:13:54.300 --> 00:13:58.780 align:middle line:84% the stress is going to elongate the spring to a fixed amount 00:13:58.780 --> 00:14:00.820 align:middle line:90% dictated by Hookean's law. 00:14:00.820 --> 00:14:04.300 align:middle line:84% For the dashpot, a constant stress essentially 00:14:04.300 --> 00:14:06.820 align:middle line:90% means constant strain rate. 00:14:06.820 --> 00:14:08.640 align:middle line:84% So if you have constant strain rate, 00:14:08.640 --> 00:14:11.340 align:middle line:84% it means that the strain is going 00:14:11.340 --> 00:14:15.020 align:middle line:90% to grow linearly over time. 00:14:15.020 --> 00:14:19.700 align:middle line:84% So what we're seeing here, basically, is a creep. 00:14:19.700 --> 00:14:22.020 align:middle line:84% So this happens in lots of cases in high temperature. 00:14:22.020 --> 00:14:23.423 align:middle line:90% For example, lots of materials. 00:14:23.423 --> 00:14:25.340 align:middle line:84% If you apply constant stress, then eventually, 00:14:25.340 --> 00:14:28.700 align:middle line:84% the material continuously deforms slowly, OK? 00:14:28.700 --> 00:14:31.100 align:middle line:90% So that's what happens there. 00:14:31.100 --> 00:14:34.900 align:middle line:84% Now, what's more interesting is actually constant strain. 00:14:34.900 --> 00:14:37.520 align:middle line:84% This is more relevant to relaxation there, OK? 00:14:37.520 --> 00:14:39.725 align:middle line:84% Because constant strain leads to stress relaxation. 00:14:39.725 --> 00:14:41.100 align:middle line:84% So if it's constant strain, let's 00:14:41.100 --> 00:14:43.660 align:middle line:84% first take a look at what's going to happen intuitively. 00:14:43.660 --> 00:14:46.340 align:middle line:84% If you [? apply ?] constant strain, initially, 00:14:46.340 --> 00:14:50.880 align:middle line:84% keep in mind that a dashpot cannot respond to strain 00:14:50.880 --> 00:14:52.640 align:middle line:90% immediately. 00:14:52.640 --> 00:14:56.080 align:middle line:84% It takes time to elongate the dashpot, 00:14:56.080 --> 00:14:58.040 align:middle line:84% because you apply any force, you're 00:14:58.040 --> 00:15:00.320 align:middle line:84% introducing a finite strain rate. 00:15:00.320 --> 00:15:01.920 align:middle line:90% It's not a finite strain. 00:15:01.920 --> 00:15:05.240 align:middle line:84% So it takes time to build up a strain in a dashpot. 00:15:05.240 --> 00:15:08.720 align:middle line:84% So therefore, if you apply a constant strain, 00:15:08.720 --> 00:15:13.080 align:middle line:84% then, initially, all the strain is going to get concentrated 00:15:13.080 --> 00:15:16.440 align:middle line:90% on this Hookean spring. 00:15:16.440 --> 00:15:18.920 align:middle line:84% But then this Hookean spring, when it's stretched, 00:15:18.920 --> 00:15:23.440 align:middle line:84% it's then going to apply a stretching force 00:15:23.440 --> 00:15:28.040 align:middle line:84% on the dashpot, which then slowly elongates the dashpot. 00:15:28.040 --> 00:15:31.960 align:middle line:84% And as the dashpot elongates, because the total strain is 00:15:31.960 --> 00:15:35.720 align:middle line:84% constant, the strain distributed on the Hookean spring 00:15:35.720 --> 00:15:38.140 align:middle line:90% will now decrease. 00:15:38.140 --> 00:15:40.520 align:middle line:84% The equilibrium state will be then 00:15:40.520 --> 00:15:42.640 align:middle line:84% this dashpot that's fully stretched, 00:15:42.640 --> 00:15:46.820 align:middle line:84% but the spring now experiences no stress. 00:15:46.820 --> 00:15:50.180 align:middle line:84% So what we're seeing here is a typical phenomena of relaxation. 00:15:50.180 --> 00:15:53.140 align:middle line:84% You start out from a state that's not in equilibrium. 00:15:53.140 --> 00:15:54.780 align:middle line:84% The initial state is when you have 00:15:54.780 --> 00:15:56.700 align:middle line:90% the spring getting stretched. 00:15:56.700 --> 00:16:00.580 align:middle line:84% The end state, this stress completely goes away. 00:16:00.580 --> 00:16:02.560 align:middle line:84% This will be the equilibrium state. 00:16:02.560 --> 00:16:05.140 align:middle line:90% 00:16:05.140 --> 00:16:08.460 align:middle line:84% So now what will be the kinetics of this process? 00:16:08.460 --> 00:16:10.840 align:middle line:84% So this will also be pretty straightforward. 00:16:10.840 --> 00:16:13.680 align:middle line:84% We can just write the total strain. 00:16:13.680 --> 00:16:16.260 align:middle line:84% This is the sum of the elastic component 00:16:16.260 --> 00:16:19.340 align:middle line:90% plus the viscous component. 00:16:19.340 --> 00:16:20.580 align:middle line:90% OK? 00:16:20.580 --> 00:16:27.060 align:middle line:84% So for elastic component, we know that the stress would just 00:16:27.060 --> 00:16:32.220 align:middle line:84% simply be G times the epsilon E. This 00:16:32.220 --> 00:16:38.020 align:middle line:84% would be the stress on the whole system-- 00:16:38.020 --> 00:16:39.940 align:middle line:84% on the spring but also the whole system. 00:16:39.940 --> 00:16:43.880 align:middle line:84% So now this stress is also what's driving the viscous flow 00:16:43.880 --> 00:16:45.080 align:middle line:90% on the dashpot, yeah? 00:16:45.080 --> 00:16:50.500 align:middle line:84% So this should also equal to eta d epsilon v, dt. 00:16:50.500 --> 00:16:55.400 align:middle line:90% 00:16:55.400 --> 00:17:00.160 align:middle line:84% Because epsilon v is basically epsilon minus epsilon E, 00:17:00.160 --> 00:17:02.200 align:middle line:84% so we can plug into the equation here. 00:17:02.200 --> 00:17:06.659 align:middle line:84% This becomes negative d epsilon E, dt. 00:17:06.659 --> 00:17:10.319 align:middle line:90% 00:17:10.319 --> 00:17:15.280 align:middle line:84% So then if you look at these two terms here, what this says 00:17:15.280 --> 00:17:19.800 align:middle line:84% is that the epsilon E will have to follow an exponentially 00:17:19.800 --> 00:17:21.900 align:middle line:90% decaying formalism. 00:17:21.900 --> 00:17:28.319 align:middle line:84% So epsilon E, this will have to be proportional to exponential. 00:17:28.319 --> 00:17:28.860 align:middle line:90% Let's see. 00:17:28.860 --> 00:17:30.360 align:middle line:90% So this will be a-- 00:17:30.360 --> 00:17:31.160 align:middle line:90% divide over here. 00:17:31.160 --> 00:17:35.160 align:middle line:90% Negative g divided by eta. 00:17:35.160 --> 00:17:40.600 align:middle line:84% So this would be a negative G divided by eta, t. 00:17:40.600 --> 00:17:45.820 align:middle line:84% So the stress would have to decay exponentially. 00:17:45.820 --> 00:17:49.900 align:middle line:84% And we need to determine the specific coefficients of this 00:17:49.900 --> 00:17:52.700 align:middle line:84% by looking at the boundary condition where, in this case, 00:17:52.700 --> 00:17:56.820 align:middle line:84% is the added stress, initially, when t equals 0, 00:17:56.820 --> 00:18:00.180 align:middle line:90% it has to equals to the strain. 00:18:00.180 --> 00:18:01.360 align:middle line:90% All the strain is sustained. 00:18:01.360 --> 00:18:02.110 align:middle line:90% Sorry, the strain. 00:18:02.110 --> 00:18:05.160 align:middle line:84% All the strain is sustained by the Hookean spring. 00:18:05.160 --> 00:18:07.580 align:middle line:84% So that's why, initially, it has to equal epsilon. 00:18:07.580 --> 00:18:12.820 align:middle line:84% Over time, when t goes to infinity, it decays to 0, OK? 00:18:12.820 --> 00:18:17.420 align:middle line:84% We can now also introduce in this quantity here tau, which 00:18:17.420 --> 00:18:24.740 align:middle line:84% would be our relaxation time for this particular system 00:18:24.740 --> 00:18:27.340 align:middle line:84% as eta divided by G. Then this one 00:18:27.340 --> 00:18:35.180 align:middle line:84% becomes epsilon exponential negative t divided by tau. 00:18:35.180 --> 00:18:36.620 align:middle line:90% OK? 00:18:36.620 --> 00:18:39.528 align:middle line:84% So we use a single exponential to describe 00:18:39.528 --> 00:18:41.320 align:middle line:84% this kind of relaxation kinetics if there's 00:18:41.320 --> 00:18:46.120 align:middle line:84% only one relaxation mechanism involved in the material. 00:18:46.120 --> 00:18:48.920 align:middle line:84% OK, every relaxation mechanism corresponds 00:18:48.920 --> 00:18:51.920 align:middle line:90% to one relaxation time scale. 00:18:51.920 --> 00:18:53.400 align:middle line:84% Now, we can also further manipulate 00:18:53.400 --> 00:18:55.440 align:middle line:90% this equation a little bit, OK? 00:18:55.440 --> 00:18:59.120 align:middle line:84% So what's the driving force for this relaxation? 00:18:59.120 --> 00:19:01.120 align:middle line:84% The driving force for this relaxation 00:19:01.120 --> 00:19:02.880 align:middle line:90% is actually internal stress. 00:19:02.880 --> 00:19:06.060 align:middle line:84% It's a stress that's sustained by the spring. 00:19:06.060 --> 00:19:07.960 align:middle line:84% This is what's stretching the dashpot, 00:19:07.960 --> 00:19:10.680 align:middle line:84% eventually leading to relaxation there, OK, to a deformation 00:19:10.680 --> 00:19:11.560 align:middle line:90% relaxation. 00:19:11.560 --> 00:19:17.400 align:middle line:84% So we can write the kinetics here in the form of-- 00:19:17.400 --> 00:19:19.840 align:middle line:90% I mean, we can also write sigma. 00:19:19.840 --> 00:19:23.640 align:middle line:84% This will also equal to epsilon divided 00:19:23.640 --> 00:19:27.600 align:middle line:84% by g, this starting state, and then it exponentially decays. 00:19:27.600 --> 00:19:30.280 align:middle line:84% We can write the relaxation kinetics also 00:19:30.280 --> 00:19:34.220 align:middle line:84% in the form of the driving force there. 00:19:34.220 --> 00:19:41.540 align:middle line:90% 00:19:41.540 --> 00:19:45.340 align:middle line:90% So this is our epsilon E-- 00:19:45.340 --> 00:19:46.800 align:middle line:90% this is our stress here. 00:19:46.800 --> 00:19:50.540 align:middle line:90% 00:19:50.540 --> 00:19:52.780 align:middle line:84% So if you look at how the stress-- let's 00:19:52.780 --> 00:19:54.520 align:middle line:90% look at the relaxation rate. 00:19:54.520 --> 00:19:56.780 align:middle line:90% So d sigma, dt-- 00:19:56.780 --> 00:20:01.260 align:middle line:90% how the stress decays over time. 00:20:01.260 --> 00:20:02.680 align:middle line:90% It's complex exponential. 00:20:02.680 --> 00:20:04.347 align:middle line:84% So it's actually pretty straightforward. 00:20:04.347 --> 00:20:09.600 align:middle line:84% It's actually negative tau, 1 over, sigma. 00:20:09.600 --> 00:20:12.957 align:middle line:90% 00:20:12.957 --> 00:20:14.540 align:middle line:84% Now, this is actually a very important 00:20:14.540 --> 00:20:16.020 align:middle line:90% mathematical formalism. 00:20:16.020 --> 00:20:18.980 align:middle line:84% When we are assuming that we have a single exponential 00:20:18.980 --> 00:20:23.700 align:middle line:84% of decay there, what we are basically assuming implicitly 00:20:23.700 --> 00:20:28.060 align:middle line:84% is that the kinetics of relaxation scale 00:20:28.060 --> 00:20:35.340 align:middle line:84% linearly with the driving force, which makes intuitive sense. 00:20:35.340 --> 00:20:37.140 align:middle line:84% The larger the driving force, the faster 00:20:37.140 --> 00:20:40.380 align:middle line:84% the relaxation kinetics, the faster the rate of change. 00:20:40.380 --> 00:20:43.080 align:middle line:90% 00:20:43.080 --> 00:20:45.380 align:middle line:84% Now, this is the same formalism we 00:20:45.380 --> 00:20:46.880 align:middle line:84% are going to use later on when we're 00:20:46.880 --> 00:20:49.200 align:middle line:84% deriving the structural relaxation in glass materials. 00:20:49.200 --> 00:20:51.520 align:middle line:84% If assuming a single relaxation mechanism, 00:20:51.520 --> 00:20:54.960 align:middle line:84% rate proportional to the driving force. 00:20:54.960 --> 00:20:55.460 align:middle line:90% OK. 00:20:55.460 --> 00:20:58.400 align:middle line:90% 00:20:58.400 --> 00:20:58.900 align:middle line:90% All right. 00:20:58.900 --> 00:21:00.040 align:middle line:84% That's sort of the key conclusion 00:21:00.040 --> 00:21:02.220 align:middle line:84% I wanted to come to there, that particular equation. 00:21:02.220 --> 00:21:04.240 align:middle line:84% So I'm going to leave it on the board. 00:21:04.240 --> 00:21:05.660 align:middle line:90% We will be using it later. 00:21:05.660 --> 00:21:06.160 align:middle line:90% OK? 00:21:06.160 --> 00:21:10.600 align:middle line:90% 00:21:10.600 --> 00:21:11.220 align:middle line:90% All right. 00:21:11.220 --> 00:21:17.440 align:middle line:84% So then the rest I can just very quickly go over it. 00:21:17.440 --> 00:21:19.460 align:middle line:84% In addition to this kind of a constant strain, 00:21:19.460 --> 00:21:23.160 align:middle line:84% constant stress, you can also apply an oscillatory strain, 00:21:23.160 --> 00:21:25.520 align:middle line:84% which, these are the sinusoidally varying strain 00:21:25.520 --> 00:21:26.040 align:middle line:90% and stress. 00:21:26.040 --> 00:21:30.000 align:middle line:84% For that, the main quantity I want to mention here 00:21:30.000 --> 00:21:32.160 align:middle line:90% is the so-called loss tangent. 00:21:32.160 --> 00:21:34.080 align:middle line:90% This tangent theta here. 00:21:34.080 --> 00:21:38.210 align:middle line:84% So tangent theta, this is basically 00:21:38.210 --> 00:21:45.930 align:middle line:84% defined as the ratio between the imaginary part of the shear 00:21:45.930 --> 00:21:49.370 align:middle line:84% modulus and the real part of the modulus. 00:21:49.370 --> 00:21:53.410 align:middle line:84% So physical significance of this loss tangent 00:21:53.410 --> 00:22:01.610 align:middle line:84% is that it is the amount of energy loss per cycle 00:22:01.610 --> 00:22:07.610 align:middle line:84% in this sinusoidal strain normalized by the elastic energy 00:22:07.610 --> 00:22:09.810 align:middle line:90% that's stored in the system. 00:22:09.810 --> 00:22:13.730 align:middle line:84% So the higher the loss tangent, the more energy 00:22:13.730 --> 00:22:17.430 align:middle line:84% dissipation you get per cycle during the oscillatory strain. 00:22:17.430 --> 00:22:20.370 align:middle line:90% 00:22:20.370 --> 00:22:23.090 align:middle line:84% What we are seeing here is that, in this particular case 00:22:23.090 --> 00:22:29.290 align:middle line:84% for Maxwell element, this loss tangent monotonically decreases 00:22:29.290 --> 00:22:32.570 align:middle line:90% as you increase omega. 00:22:32.570 --> 00:22:33.070 align:middle line:90% OK? 00:22:33.070 --> 00:22:35.190 align:middle line:84% The horizontal axis here is basically omega times tau, 00:22:35.190 --> 00:22:37.470 align:middle line:84% where tau is the relaxation time, which makes sense. 00:22:37.470 --> 00:22:39.470 align:middle line:84% Because if you have higher frequency, obviously, 00:22:39.470 --> 00:22:40.160 align:middle line:90% a shorter time. 00:22:40.160 --> 00:22:41.910 align:middle line:84% In fact, the elastic constant contribution 00:22:41.910 --> 00:22:45.050 align:middle line:84% becomes more significant because any kind of elastic-- or sorry, 00:22:45.050 --> 00:22:47.030 align:middle line:84% any kind of viscous component, it 00:22:47.030 --> 00:22:50.750 align:middle line:84% takes time to respond to external changes. 00:22:50.750 --> 00:22:55.190 align:middle line:84% Also, having a shorter cycle, a shorter period 00:22:55.190 --> 00:22:58.210 align:middle line:84% also decreases the amount of energy loss per cycle. 00:22:58.210 --> 00:23:01.350 align:middle line:84% So that's why you see this decreasing trend here, OK? 00:23:01.350 --> 00:23:03.590 align:middle line:84% But in general, this would be a very general quantity 00:23:03.590 --> 00:23:06.510 align:middle line:84% that you can use to gauge whether the system is conserving 00:23:06.510 --> 00:23:08.750 align:middle line:90% energy or dissipating energy. 00:23:08.750 --> 00:23:10.950 align:middle line:90% OK? 00:23:10.950 --> 00:23:11.670 align:middle line:90% All right. 00:23:11.670 --> 00:23:13.388 align:middle line:84% So I'm going to go pretty quickly here. 00:23:13.388 --> 00:23:14.930 align:middle line:84% These are some of the other elements. 00:23:14.930 --> 00:23:18.030 align:middle line:84% You can also assemble the Hookean spring and the dashpot 00:23:18.030 --> 00:23:18.935 align:middle line:90% in different ways. 00:23:18.935 --> 00:23:20.810 align:middle line:84% You can put them in parallel with each other. 00:23:20.810 --> 00:23:23.830 align:middle line:84% This is called the Voigt-Kelvin element. 00:23:23.830 --> 00:23:30.390 align:middle line:84% For this kind of element here, if you apply a constant strain, 00:23:30.390 --> 00:23:36.770 align:middle line:84% then all the stress is going to be sustained by the spring 00:23:36.770 --> 00:23:38.170 align:middle line:84% here, because dashpot, it doesn't 00:23:38.170 --> 00:23:40.008 align:middle line:90% respond to constant strain. 00:23:40.008 --> 00:23:42.050 align:middle line:84% For constant strain, the dashpot doesn't give you 00:23:42.050 --> 00:23:46.310 align:middle line:84% any friction force, because strain rate is zero. 00:23:46.310 --> 00:23:48.130 align:middle line:84% So that's why all the stress will now 00:23:48.130 --> 00:23:49.670 align:middle line:90% be sustained by the spring. 00:23:49.670 --> 00:23:52.570 align:middle line:84% That's why we have this equation here. 00:23:52.570 --> 00:23:55.010 align:middle line:84% It also exhibits stress relaxation, 00:23:55.010 --> 00:23:56.470 align:middle line:90% unlike the Maxwell element. 00:23:56.470 --> 00:23:59.850 align:middle line:84% Maxwell element-- sorry, strain relaxation. 00:23:59.850 --> 00:24:02.650 align:middle line:84% Maxwell element exhibits stress relaxation 00:24:02.650 --> 00:24:05.490 align:middle line:84% whereas for this Voigt-Kelvin element, 00:24:05.490 --> 00:24:07.910 align:middle line:90% it exhibits a strain relaxation. 00:24:07.910 --> 00:24:10.330 align:middle line:84% If you apply a constant stress, then what 00:24:10.330 --> 00:24:13.170 align:middle line:84% happens there is that initially, again, all the stress is going 00:24:13.170 --> 00:24:16.130 align:middle line:84% to get sustained by this spring element, 00:24:16.130 --> 00:24:18.650 align:middle line:84% but then over time, the stress is then 00:24:18.650 --> 00:24:21.130 align:middle line:84% going to lead to elongation of the dashpot. 00:24:21.130 --> 00:24:29.130 align:middle line:84% So in the equilibrium state, if you apply a constant stress, 00:24:29.130 --> 00:24:30.850 align:middle line:90% you want to deform the system. 00:24:30.850 --> 00:24:34.070 align:middle line:84% But you cannot deform this dashboard immediately, 00:24:34.070 --> 00:24:37.350 align:middle line:84% because as I said, it takes time for the viscous element 00:24:37.350 --> 00:24:39.510 align:middle line:84% to actually respond to any changes. 00:24:39.510 --> 00:24:41.950 align:middle line:84% So initially, the strain is zero. 00:24:41.950 --> 00:24:44.610 align:middle line:84% And that's why the spring is not elongating. 00:24:44.610 --> 00:24:46.910 align:middle line:84% There's no elastic stress component. 00:24:46.910 --> 00:24:50.030 align:middle line:84% But then over time, this dashpot starts 00:24:50.030 --> 00:24:52.230 align:middle line:84% to slowly increase in length, which 00:24:52.230 --> 00:24:54.670 align:middle line:84% means that more and more of the stress 00:24:54.670 --> 00:24:57.750 align:middle line:84% is actually now getting sustained by the spring instead 00:24:57.750 --> 00:24:58.750 align:middle line:90% of dashpot. 00:24:58.750 --> 00:25:02.570 align:middle line:84% So in the equilibrium state, the dashboard will stop elongating, 00:25:02.570 --> 00:25:05.030 align:middle line:84% but then all the stress will now be 00:25:05.030 --> 00:25:07.610 align:middle line:90% sustained by the spring layer. 00:25:07.610 --> 00:25:08.130 align:middle line:90% OK? 00:25:08.130 --> 00:25:11.810 align:middle line:84% So we can still describe this kind of relaxation phenomena 00:25:11.810 --> 00:25:15.590 align:middle line:84% similarly using an exponential decay here, looking 00:25:15.590 --> 00:25:17.110 align:middle line:90% at the kinetics. 00:25:17.110 --> 00:25:19.310 align:middle line:90% OK? 00:25:19.310 --> 00:25:20.810 align:middle line:90% OK. 00:25:20.810 --> 00:25:24.190 align:middle line:84% So now, in the end, if you want to describe a general material 00:25:24.190 --> 00:25:27.090 align:middle line:84% that potentially can have many relaxation mechanisms, 00:25:27.090 --> 00:25:29.250 align:middle line:84% you can use this kind of generalized Maxwell model. 00:25:29.250 --> 00:25:31.670 align:middle line:84% Basically, you put a bunch of these Maxwell elements 00:25:31.670 --> 00:25:35.330 align:middle line:84% in parallel with respect to each other. 00:25:35.330 --> 00:25:39.170 align:middle line:84% And then the response of the entire material system 00:25:39.170 --> 00:25:42.170 align:middle line:84% would then be the sum of the individual components. 00:25:42.170 --> 00:25:45.050 align:middle line:84% Now, the key takeaway message here 00:25:45.050 --> 00:25:49.730 align:middle line:84% is that for this kind of system, every dashpot 00:25:49.730 --> 00:25:53.450 align:middle line:84% corresponds to one structural relaxation mechanism. 00:25:53.450 --> 00:25:56.690 align:middle line:84% So in real material, there can be many complex relaxation 00:25:56.690 --> 00:25:59.090 align:middle line:84% mechanisms that occur concurrently. 00:25:59.090 --> 00:26:01.090 align:middle line:84% And that's why you need to introduce 00:26:01.090 --> 00:26:02.690 align:middle line:84% a large number of these dashpots here 00:26:02.690 --> 00:26:06.050 align:middle line:84% to characterize the dispersion of relaxation time. 00:26:06.050 --> 00:26:09.830 align:middle line:84% So now if you look at the actual time domain response, 00:26:09.830 --> 00:26:12.470 align:middle line:84% what happens if you have a stress relaxation? 00:26:12.470 --> 00:26:12.987 align:middle line:90% Same thing. 00:26:12.987 --> 00:26:14.570 align:middle line:84% You apply constant strain, look at how 00:26:14.570 --> 00:26:18.090 align:middle line:84% the internal stress of the material relaxes over time. 00:26:18.090 --> 00:26:20.050 align:middle line:84% You will get, phenomenologically, 00:26:20.050 --> 00:26:22.030 align:middle line:84% an expression like what I'm showing here. 00:26:22.030 --> 00:26:23.290 align:middle line:90% OK? 00:26:23.290 --> 00:26:25.610 align:middle line:84% Instead of having one single exponential, 00:26:25.610 --> 00:26:31.050 align:middle line:84% you now have to look at the sum of many exponential functions. 00:26:31.050 --> 00:26:37.270 align:middle line:84% Each of the tau i here gives you the characteristic relaxation 00:26:37.270 --> 00:26:41.190 align:middle line:84% time for one particular microscopic mechanism. 00:26:41.190 --> 00:26:43.630 align:middle line:90% OK? 00:26:43.630 --> 00:26:49.470 align:middle line:84% This sum, you can also write it in terms of what's 00:26:49.470 --> 00:26:51.870 align:middle line:90% called a Prony series. 00:26:51.870 --> 00:26:54.630 align:middle line:84% Sum of single exponentials can be 00:26:54.630 --> 00:26:58.130 align:middle line:84% expressed as this so-called stretched exponential function 00:26:58.130 --> 00:26:59.170 align:middle line:90% there. 00:26:59.170 --> 00:27:00.670 align:middle line:84% In a stretched exponential function, 00:27:00.670 --> 00:27:02.670 align:middle line:84% this looks very similar to a single exponential, 00:27:02.670 --> 00:27:06.590 align:middle line:84% except that we add the exponents here, beta. 00:27:06.590 --> 00:27:07.350 align:middle line:90% OK? 00:27:07.350 --> 00:27:09.230 align:middle line:84% So if you look at, functionally, what 00:27:09.230 --> 00:27:11.190 align:middle line:84% this stretched exponential function 00:27:11.190 --> 00:27:17.590 align:middle line:84% looks like, let's say this is stress, 00:27:17.590 --> 00:27:19.990 align:middle line:90% this is how it varies over time. 00:27:19.990 --> 00:27:23.390 align:middle line:84% If it's single exponential, this is what the function 00:27:23.390 --> 00:27:26.470 align:middle line:90% would be looking like. 00:27:26.470 --> 00:27:31.810 align:middle line:84% And then-- it's not very smooth, but that's OK. 00:27:31.810 --> 00:27:35.810 align:middle line:84% When time equals to tau, the stress 00:27:35.810 --> 00:27:40.210 align:middle line:84% relaxes to 1 over e of the starting value. 00:27:40.210 --> 00:27:42.100 align:middle line:90% OK? 00:27:42.100 --> 00:27:45.370 align:middle line:84% Now, if it's a stretched exponential, when you have beta 00:27:45.370 --> 00:27:48.610 align:middle line:84% there, if beta is less than 1, this 00:27:48.610 --> 00:27:49.830 align:middle line:90% is what you're going to get. 00:27:49.830 --> 00:28:03.390 align:middle line:84% You will have a function that initially relaxes faster. 00:28:03.390 --> 00:28:23.290 align:middle line:90% 00:28:23.290 --> 00:28:25.970 align:middle line:84% You will have a function that initially relaxes faster 00:28:25.970 --> 00:28:27.690 align:middle line:84% compared to a complex exponential. 00:28:27.690 --> 00:28:29.410 align:middle line:84% And then eventually, the relaxation 00:28:29.410 --> 00:28:32.150 align:middle line:90% is going to get slower, OK? 00:28:32.150 --> 00:28:32.650 align:middle line:90% Let's see. 00:28:32.650 --> 00:28:42.510 align:middle line:90% 00:28:42.510 --> 00:28:44.750 align:middle line:84% So the reason you get the initial fast relaxation 00:28:44.750 --> 00:28:47.750 align:middle line:84% is because now you have many kinds of relaxation mechanisms. 00:28:47.750 --> 00:28:51.670 align:middle line:84% They have a difference in terms of relaxation time scale, OK? 00:28:51.670 --> 00:28:53.310 align:middle line:84% So there are always some mechanisms 00:28:53.310 --> 00:28:55.990 align:middle line:84% that exhibit the faster relaxation than others. 00:28:55.990 --> 00:28:59.350 align:middle line:84% So those are what lead to initial fast decay there. 00:28:59.350 --> 00:29:02.390 align:middle line:84% And then eventually, the slower decay 00:29:02.390 --> 00:29:05.070 align:middle line:84% is when the first components has decayed away. 00:29:05.070 --> 00:29:07.510 align:middle line:84% You now are left behind with the slow components. 00:29:07.510 --> 00:29:09.270 align:middle line:84% So that's why this with a longer decay 00:29:09.270 --> 00:29:11.410 align:middle line:84% tail compared to a conventional exponential decay. 00:29:11.410 --> 00:29:11.910 align:middle line:90% OK? 00:29:11.910 --> 00:29:14.750 align:middle line:90% 00:29:14.750 --> 00:29:15.750 align:middle line:90% OK. 00:29:15.750 --> 00:29:19.630 align:middle line:84% So basically, what we have discussed here up 00:29:19.630 --> 00:29:25.190 align:middle line:84% to this point is that we have shown you 00:29:25.190 --> 00:29:27.870 align:middle line:84% that there are three types of response we can look at here, 00:29:27.870 --> 00:29:28.370 align:middle line:90% OK? 00:29:28.370 --> 00:29:30.730 align:middle line:84% We have classic elastic response, 00:29:30.730 --> 00:29:33.160 align:middle line:84% we have viscous response, and the combination of them 00:29:33.160 --> 00:29:35.320 align:middle line:84% gives you the viscoelastic response. 00:29:35.320 --> 00:29:37.640 align:middle line:84% If it's an elastic response, that everything 00:29:37.640 --> 00:29:41.560 align:middle line:84% is instantaneous, if you apply a stress on a profile like what 00:29:41.560 --> 00:29:43.000 align:middle line:84% I'm showing you on the top there, 00:29:43.000 --> 00:29:45.280 align:middle line:84% you see the strain profile essentially 00:29:45.280 --> 00:29:48.120 align:middle line:84% replicates exactly what you have for the stress. 00:29:48.120 --> 00:29:48.640 align:middle line:90% OK? 00:29:48.640 --> 00:29:51.800 align:middle line:84% With a proportionality constant given by your elastic constants, 00:29:51.800 --> 00:29:52.560 align:middle line:90% OK? 00:29:52.560 --> 00:30:01.560 align:middle line:84% For stress, the strain rate will be proportional to the stress. 00:30:01.560 --> 00:30:04.360 align:middle line:84% So then a constant stress gives rise 00:30:04.360 --> 00:30:07.320 align:middle line:84% to a linearly varying strain here. 00:30:07.320 --> 00:30:09.600 align:middle line:84% When you remove the stress, any strain 00:30:09.600 --> 00:30:11.660 align:middle line:84% that has already been induced will be permanent. 00:30:11.660 --> 00:30:13.640 align:middle line:90% It will stay there. 00:30:13.640 --> 00:30:15.853 align:middle line:84% Whereas for viscoelastic strain, it's 00:30:15.853 --> 00:30:18.120 align:middle line:84% sort of the combination of the two. 00:30:18.120 --> 00:30:21.000 align:middle line:84% If you apply a stress like what I'm showing here, 00:30:21.000 --> 00:30:23.160 align:middle line:84% as soon as you apply the stress, you're 00:30:23.160 --> 00:30:25.880 align:middle line:84% introducing a little bit of an elastic response that's 00:30:25.880 --> 00:30:27.800 align:middle line:90% instantaneous. 00:30:27.800 --> 00:30:31.980 align:middle line:84% But then over time, you have relaxation. 00:30:31.980 --> 00:30:32.820 align:middle line:90% OK? 00:30:32.820 --> 00:30:38.780 align:middle line:84% You get a time-varying strain response. 00:30:38.780 --> 00:30:40.660 align:middle line:84% When you remove the stress, you are 00:30:40.660 --> 00:30:43.220 align:middle line:84% going to get, again, some instantaneous response. 00:30:43.220 --> 00:30:48.380 align:middle line:84% But then as time proceeds, you get some additional variation 00:30:48.380 --> 00:30:50.330 align:middle line:90% there. 00:30:50.330 --> 00:30:50.830 align:middle line:90% OK? 00:30:50.830 --> 00:30:53.740 align:middle line:84% So the key feature for the viscoelastic response 00:30:53.740 --> 00:30:58.060 align:middle line:90% is that this is time-dependent. 00:30:58.060 --> 00:31:01.100 align:middle line:84% The state of a material at one particular instant of time 00:31:01.100 --> 00:31:04.460 align:middle line:84% not only depends on the instantaneous stress that's 00:31:04.460 --> 00:31:06.020 align:middle line:84% applied right at that time point, 00:31:06.020 --> 00:31:10.420 align:middle line:84% but it also depends on the history of the stress that's 00:31:10.420 --> 00:31:12.100 align:middle line:90% applied on the material. 00:31:12.100 --> 00:31:14.540 align:middle line:84% So this slide shows you some examples 00:31:14.540 --> 00:31:16.500 align:middle line:84% where you have the material exhibiting 00:31:16.500 --> 00:31:18.007 align:middle line:90% typical viscoelastic response. 00:31:18.007 --> 00:31:19.632 align:middle line:84% A lot of them are essentially some kind 00:31:19.632 --> 00:31:23.700 align:middle line:84% of amorphous material, where the operation temperature resides 00:31:23.700 --> 00:31:25.520 align:middle line:84% right along the glass transition regime. 00:31:25.520 --> 00:31:26.895 align:middle line:84% This is where the viscoelasticity 00:31:26.895 --> 00:31:28.480 align:middle line:90% is most pronounced. 00:31:28.480 --> 00:31:30.480 align:middle line:90% Of course, mozzarella cheese. 00:31:30.480 --> 00:31:33.180 align:middle line:84% You all have experienced when they're hot, 00:31:33.180 --> 00:31:35.560 align:middle line:84% when you try to remove them from the oven. 00:31:35.560 --> 00:31:37.560 align:middle line:84% They will deform continuously even if you're not 00:31:37.560 --> 00:31:39.440 align:middle line:90% trying to stretch them. 00:31:39.440 --> 00:31:41.440 align:middle line:90% Human skin. 00:31:41.440 --> 00:31:43.040 align:middle line:84% It's quite obvious that human skin 00:31:43.040 --> 00:31:47.280 align:middle line:84% has a sort of time-dependent, delayed response, OK? 00:31:47.280 --> 00:31:51.280 align:middle line:84% If it's a baby, then the response will be more elastic. 00:31:51.280 --> 00:31:54.380 align:middle line:84% If someone like me, you see a significant viscous component. 00:31:54.380 --> 00:31:57.400 align:middle line:84% That's a good sign that I'm actually getting old, OK? 00:31:57.400 --> 00:31:58.540 align:middle line:90% Turbine blades. 00:31:58.540 --> 00:32:00.360 align:middle line:84% These are things that were traditionally-- 00:32:00.360 --> 00:32:02.780 align:middle line:84% if it's a low temperature, this is basically an elastic solid. 00:32:02.780 --> 00:32:04.947 align:middle line:84% But when they're operating at very high temperature, 00:32:04.947 --> 00:32:07.240 align:middle line:84% you start to see responses like creep kicking in, which 00:32:07.240 --> 00:32:09.220 align:middle line:90% is due to the viscoelasticity. 00:32:09.220 --> 00:32:12.160 align:middle line:90% Volcanic lava. 00:32:12.160 --> 00:32:13.640 align:middle line:90% This is a good example. 00:32:13.640 --> 00:32:16.080 align:middle line:84% Basically, glass-forming fluids, OK? 00:32:16.080 --> 00:32:18.183 align:middle line:84% That's why they exhibit the viscoelastic response. 00:32:18.183 --> 00:32:19.600 align:middle line:84% In terms of practical applications 00:32:19.600 --> 00:32:21.560 align:middle line:84% of these viscoelastic materials, people 00:32:21.560 --> 00:32:24.800 align:middle line:84% use them for things like propellers for naval ships 00:32:24.800 --> 00:32:27.380 align:middle line:84% because having the viscous component 00:32:27.380 --> 00:32:28.940 align:middle line:84% is helpful in that they can actually 00:32:28.940 --> 00:32:33.340 align:middle line:84% absorb a lot of the vibrations or dampen those vibrations 00:32:33.340 --> 00:32:34.900 align:middle line:90% through dissipating the energy. 00:32:34.900 --> 00:32:36.960 align:middle line:84% And that's important for things like naval ships, 00:32:36.960 --> 00:32:39.960 align:middle line:84% because they want to make sure that they remain quiet, 00:32:39.960 --> 00:32:42.500 align:middle line:84% they produce less amount of noise, so they don't get easily 00:32:42.500 --> 00:32:50.580 align:middle line:84% spotted by sonar by ships from the adversaries. 00:32:50.580 --> 00:32:52.277 align:middle line:84% In terms of everyday applications, 00:32:52.277 --> 00:32:53.860 align:middle line:84% I have to scratch my head a little bit 00:32:53.860 --> 00:32:56.180 align:middle line:84% to think about what is the actual practical use 00:32:56.180 --> 00:32:57.680 align:middle line:90% of a viscoelastic material? 00:32:57.680 --> 00:32:59.362 align:middle line:90% But that's one-- memory foam. 00:32:59.362 --> 00:33:01.820 align:middle line:84% That is also a classical example of a viscoelastic material 00:33:01.820 --> 00:33:02.800 align:middle line:90% in your everyday life. 00:33:02.800 --> 00:33:03.300 align:middle line:90% OK? 00:33:03.300 --> 00:33:07.140 align:middle line:90% 00:33:07.140 --> 00:33:07.780 align:middle line:90% OK. 00:33:07.780 --> 00:33:12.780 align:middle line:84% The last main point I want to make here about viscoelasticity 00:33:12.780 --> 00:33:15.940 align:middle line:84% is this Boltzmann superposition principle. 00:33:15.940 --> 00:33:22.100 align:middle line:84% So we have discussed about some very simple formulogical models 00:33:22.100 --> 00:33:24.540 align:middle line:84% and how those elements, like Maxwell elements, 00:33:24.540 --> 00:33:29.120 align:middle line:84% would respond to simple forms of strain or stress variation-- 00:33:29.120 --> 00:33:32.360 align:middle line:84% oscillatory strains, constant strain, constant stress. 00:33:32.360 --> 00:33:35.680 align:middle line:84% But eventually, we may have to deal with more complex problem 00:33:35.680 --> 00:33:38.900 align:middle line:84% where you're given the viscoelastic solid material, 00:33:38.900 --> 00:33:41.880 align:middle line:84% you know it's viscoelastic characteristics, 00:33:41.880 --> 00:33:46.080 align:middle line:84% and then we may apply some very complex form of strain or stress 00:33:46.080 --> 00:33:47.200 align:middle line:90% on the material. 00:33:47.200 --> 00:33:49.920 align:middle line:84% And you will be asked to solve, what is the corresponding time 00:33:49.920 --> 00:33:52.920 align:middle line:90% domain response? 00:33:52.920 --> 00:33:56.068 align:middle line:90% OK? 00:33:56.068 --> 00:33:59.960 align:middle line:84% So the way to deal with that is to use the Boltzmann 00:33:59.960 --> 00:34:02.280 align:middle line:90% superposition principle, OK? 00:34:02.280 --> 00:34:06.200 align:middle line:84% So the starting point of the Boltzmann superposition 00:34:06.200 --> 00:34:10.040 align:middle line:84% principle, it lies on the linearity 00:34:10.040 --> 00:34:12.440 align:middle line:90% of the constitutive relations. 00:34:12.440 --> 00:34:14.199 align:middle line:84% If you think about the Hookean spring, 00:34:14.199 --> 00:34:19.280 align:middle line:84% think about the Newtonian viscous flow equations, 00:34:19.280 --> 00:34:23.600 align:middle line:84% both equations are completely linear. 00:34:23.600 --> 00:34:25.898 align:middle line:84% For Hookean springs, strain and stress, 00:34:25.898 --> 00:34:27.940 align:middle line:84% they are proportional with respect to each other. 00:34:27.940 --> 00:34:30.980 align:middle line:84% For the viscous flow, the strain rate always 00:34:30.980 --> 00:34:34.100 align:middle line:90% linearly scales with stress. 00:34:34.100 --> 00:34:36.500 align:middle line:84% So if you have equations that are linear, 00:34:36.500 --> 00:34:40.380 align:middle line:84% it means that if you have a composite stimuli, 00:34:40.380 --> 00:34:43.659 align:middle line:84% you have combined two stimuli, then 00:34:43.659 --> 00:34:45.580 align:middle line:84% the resulting response will also be 00:34:45.580 --> 00:34:49.900 align:middle line:84% the sum of the response from the individual stimulus. 00:34:49.900 --> 00:34:52.840 align:middle line:90% So that's what I'm showing here. 00:34:52.840 --> 00:34:54.400 align:middle line:84% If you think about, in this case, 00:34:54.400 --> 00:34:56.580 align:middle line:84% applying some kind of strain, this 00:34:56.580 --> 00:34:58.900 align:middle line:84% is a strain step at time t1, you suddenly 00:34:58.900 --> 00:35:01.660 align:middle line:84% apply this strain, delta epsilon 1 there. 00:35:01.660 --> 00:35:04.020 align:middle line:84% Then obviously, because it's viscoelastic, 00:35:04.020 --> 00:35:07.380 align:middle line:84% you're going to get instantaneous response 00:35:07.380 --> 00:35:09.980 align:middle line:90% in terms of the stress. 00:35:09.980 --> 00:35:11.660 align:middle line:90% And then it will all relax. 00:35:11.660 --> 00:35:15.280 align:middle line:84% So the blue part will then be the viscoelastic relaxation. 00:35:15.280 --> 00:35:20.220 align:middle line:84% So generically, we can write a stress response as two parts 00:35:20.220 --> 00:35:21.020 align:middle line:90% here. 00:35:21.020 --> 00:35:23.240 align:middle line:84% You have the instantaneous response 00:35:23.240 --> 00:35:28.640 align:middle line:84% given by the real part of the elastic constant. 00:35:28.640 --> 00:35:32.840 align:middle line:84% Then you have this time-dependent function here 00:35:32.840 --> 00:35:36.640 align:middle line:84% that gives you the kinetics of the relaxation. 00:35:36.640 --> 00:35:38.600 align:middle line:84% So in the most simple case, if you only 00:35:38.600 --> 00:35:42.000 align:middle line:84% have one microscopic mechanism responsible for relaxation, 00:35:42.000 --> 00:35:45.400 align:middle line:84% then this function would be a simple exponential decay 00:35:45.400 --> 00:35:46.600 align:middle line:90% function. 00:35:46.600 --> 00:35:50.373 align:middle line:90% But this is a more generic case. 00:35:50.373 --> 00:35:52.040 align:middle line:84% This can be a Prony theory, for example. 00:35:52.040 --> 00:35:53.320 align:middle line:84% It can be a stretched exponential, right? 00:35:53.320 --> 00:35:55.528 align:middle line:84% Or any arbitrary function you can define with respect 00:35:55.528 --> 00:35:56.840 align:middle line:90% to the material, OK? 00:35:56.840 --> 00:35:59.320 align:middle line:84% Now, this relaxation only takes place 00:35:59.320 --> 00:36:01.100 align:middle line:84% after you have applied the stimuli. 00:36:01.100 --> 00:36:05.680 align:middle line:84% That's why the argument here is t minus t1. 00:36:05.680 --> 00:36:06.460 align:middle line:90% Same thing. 00:36:06.460 --> 00:36:12.240 align:middle line:84% If you have a time t goes to t2, you apply the second stimulus, 00:36:12.240 --> 00:36:15.520 align:middle line:84% then you're going to get some stress response. 00:36:15.520 --> 00:36:18.000 align:middle line:84% In the end, the combined response 00:36:18.000 --> 00:36:23.180 align:middle line:84% would just be a sum of these two parts there, OK? 00:36:23.180 --> 00:36:25.100 align:middle line:84% So that's pretty straightforward. 00:36:25.100 --> 00:36:27.420 align:middle line:84% Now we can move on and discuss about now 00:36:27.420 --> 00:36:31.740 align:middle line:84% what happens if you apply an arbitrary profile of a strain. 00:36:31.740 --> 00:36:34.100 align:middle line:84% So for any arbitrary profile of a strain, 00:36:34.100 --> 00:36:36.940 align:middle line:84% you can always apply the trick of divide and conquer. 00:36:36.940 --> 00:36:39.420 align:middle line:84% You divide up the arbitrary strain profile 00:36:39.420 --> 00:36:41.420 align:middle line:84% into many, many small time steps. 00:36:41.420 --> 00:36:46.420 align:middle line:84% Within each time step, you apply a small variation of the strain. 00:36:46.420 --> 00:36:47.140 align:middle line:90% OK? 00:36:47.140 --> 00:36:48.540 align:middle line:84% You then look at the contribution 00:36:48.540 --> 00:36:51.540 align:middle line:84% from each individual steps and then sum it up together. 00:36:51.540 --> 00:36:57.140 align:middle line:84% So discretely, you can write the total strain 00:36:57.140 --> 00:37:01.100 align:middle line:84% as a sum of many individual, little strain steps. 00:37:01.100 --> 00:37:04.920 align:middle line:84% Then in terms of response, same thing. 00:37:04.920 --> 00:37:08.760 align:middle line:90% You just add up the response. 00:37:08.760 --> 00:37:09.572 align:middle line:90% OK? 00:37:09.572 --> 00:37:11.980 align:middle line:84% If you now go into the limit, where 00:37:11.980 --> 00:37:14.140 align:middle line:84% each step becomes very, very small, 00:37:14.140 --> 00:37:17.020 align:middle line:84% this sum then becomes the integral. 00:37:17.020 --> 00:37:19.580 align:middle line:84% Converting a sum to integral, you essentially 00:37:19.580 --> 00:37:24.400 align:middle line:84% rely on this relation here, where every step, delta epsilon, 00:37:24.400 --> 00:37:29.280 align:middle line:84% you would convert it into d epsilon, dt times dt. 00:37:29.280 --> 00:37:33.760 align:middle line:90% 00:37:33.760 --> 00:37:38.040 align:middle line:84% So you use that conversion, you convert your summation sign 00:37:38.040 --> 00:37:44.200 align:middle line:84% into an integral, and you will get, basically, 00:37:44.200 --> 00:37:47.000 align:middle line:84% the Boltzmann superposition principle that 00:37:47.000 --> 00:37:51.200 align:middle line:84% now allows you to infer the response 00:37:51.200 --> 00:37:57.680 align:middle line:84% of the viscoelastic solid to any arbitrary time-varying input 00:37:57.680 --> 00:37:59.280 align:middle line:90% stimulus. 00:37:59.280 --> 00:37:59.780 align:middle line:90% OK. 00:37:59.780 --> 00:38:02.600 align:middle line:90% 00:38:02.600 --> 00:38:04.600 align:middle line:84% So basically, what I'm saying here 00:38:04.600 --> 00:38:07.832 align:middle line:84% is that if you know the relaxation function, 00:38:07.832 --> 00:38:10.040 align:middle line:84% you know the kinetics of structural relaxation, which 00:38:10.040 --> 00:38:15.320 align:middle line:84% is this function psi here, if you know this function psi, 00:38:15.320 --> 00:38:20.930 align:middle line:84% you would then be able to derive the response of the solid using 00:38:20.930 --> 00:38:23.750 align:middle line:90% this superposition principle. 00:38:23.750 --> 00:38:24.250 align:middle line:90% OK? 00:38:24.250 --> 00:38:27.330 align:middle line:90% 00:38:27.330 --> 00:38:28.110 align:middle line:90% All right. 00:38:28.110 --> 00:38:32.770 align:middle line:90% So this is for viscoelasticity. 00:38:32.770 --> 00:38:35.270 align:middle line:84% Some of the key characteristics here-- obviously, as I said, 00:38:35.270 --> 00:38:37.010 align:middle line:90% this is history-dependent. 00:38:37.010 --> 00:38:40.153 align:middle line:84% You can quite clearly see from this superposition principle 00:38:40.153 --> 00:38:42.570 align:middle line:84% here, because you have this kind of exponentially decaying 00:38:42.570 --> 00:38:47.130 align:middle line:84% tail or this time-varying function part, 00:38:47.130 --> 00:38:50.170 align:middle line:84% at any time point, the stress response 00:38:50.170 --> 00:38:52.070 align:middle line:84% depends not only on the instantaneous trait, 00:38:52.070 --> 00:38:56.083 align:middle line:84% but also the previous history of the strain applied. 00:38:56.083 --> 00:38:57.750 align:middle line:84% And the time domain response, of course, 00:38:57.750 --> 00:39:02.290 align:middle line:84% is dictated by this function for psi there. 00:39:02.290 --> 00:39:03.850 align:middle line:84% So that's the mathematical formalism 00:39:03.850 --> 00:39:06.450 align:middle line:84% we will need in preparation for the discussion 00:39:06.450 --> 00:39:12.130 align:middle line:84% when we go to the glass structure relaxation. 00:39:12.130 --> 00:39:15.450 align:middle line:84% So what I want to do here, the focus of the remainder 00:39:15.450 --> 00:39:20.830 align:middle line:84% of the lecture is to derive the mathematical formalism that now 00:39:20.830 --> 00:39:24.830 align:middle line:84% allows us to predict the structural relaxation 00:39:24.830 --> 00:39:29.670 align:middle line:84% kinetics in the glass transition regime, OK? 00:39:29.670 --> 00:39:33.710 align:middle line:84% So for that, I will be drawing an analogy between these two 00:39:33.710 --> 00:39:36.030 align:middle line:90% kinds of relaxation here. 00:39:36.030 --> 00:39:39.910 align:middle line:84% You have a stress relaxation in a simple Maxwell 00:39:39.910 --> 00:39:43.470 align:middle line:84% element versus structural relaxation in glasses. 00:39:43.470 --> 00:39:46.470 align:middle line:84% Now, I want to mention that this mixture element, obviously, it's 00:39:46.470 --> 00:39:49.510 align:middle line:84% an abstraction of real structural relaxation 00:39:49.510 --> 00:39:50.670 align:middle line:90% in materials. 00:39:50.670 --> 00:39:54.550 align:middle line:84% So it's not surprising at all that these two kind of seemingly 00:39:54.550 --> 00:39:56.150 align:middle line:84% different phenomena, they will exhibit 00:39:56.150 --> 00:39:58.110 align:middle line:84% the same mathematical formalism there. 00:39:58.110 --> 00:40:01.950 align:middle line:84% So to look at the kinetics of any kind of a relaxation 00:40:01.950 --> 00:40:05.950 align:middle line:84% phenomena, we need to look at three things, OK? 00:40:05.950 --> 00:40:09.430 align:middle line:90% First is the equilibrium state. 00:40:09.430 --> 00:40:11.670 align:middle line:84% So the equilibrium state refers to the state 00:40:11.670 --> 00:40:13.830 align:middle line:84% when there's no more time evolution. 00:40:13.830 --> 00:40:15.550 align:middle line:90% It becomes static. 00:40:15.550 --> 00:40:18.490 align:middle line:84% For the Maxwell element, the equilibrium state 00:40:18.490 --> 00:40:20.550 align:middle line:84% is pretty clear if I have stressed relaxation. 00:40:20.550 --> 00:40:23.210 align:middle line:84% So when the stress completely dips down to zero stress, 00:40:23.210 --> 00:40:26.230 align:middle line:84% then there will be no more relaxation. 00:40:26.230 --> 00:40:30.010 align:middle line:84% So this is when you have the dashpot fully stretched 00:40:30.010 --> 00:40:32.290 align:middle line:84% and the spring is not elongated at all. 00:40:32.290 --> 00:40:33.582 align:middle line:90% The driving force. 00:40:33.582 --> 00:40:35.290 align:middle line:84% The second factor we need to look at here 00:40:35.290 --> 00:40:36.650 align:middle line:90% is the driving force. 00:40:36.650 --> 00:40:39.630 align:middle line:84% So I have already mentioned, for the stress relaxation, 00:40:39.630 --> 00:40:41.970 align:middle line:84% the driving force is basically a residual stress 00:40:41.970 --> 00:40:43.630 align:middle line:90% inside the material, OK? 00:40:43.630 --> 00:40:46.450 align:middle line:84% We just represent it by the stress exerted 00:40:46.450 --> 00:40:49.250 align:middle line:90% by the Hookean spring here. 00:40:49.250 --> 00:40:53.610 align:middle line:84% Now, finally is the realization kinetics. 00:40:53.610 --> 00:40:58.730 align:middle line:84% As a first-order approximation, what we have shown here 00:40:58.730 --> 00:41:02.930 align:middle line:84% is that the relaxation kinetics, characterized 00:41:02.930 --> 00:41:06.010 align:middle line:84% by the rate of change is linearly 00:41:06.010 --> 00:41:11.690 align:middle line:84% proportional to the driving force, with the proportionality 00:41:11.690 --> 00:41:14.990 align:middle line:84% constant being basically 1 over tau, 00:41:14.990 --> 00:41:16.930 align:middle line:84% where tau is the relaxation time. 00:41:16.930 --> 00:41:19.870 align:middle line:90% 00:41:19.870 --> 00:41:24.390 align:middle line:84% So now let's try to do the same thing for glass structure 00:41:24.390 --> 00:41:26.190 align:middle line:90% relaxation. 00:41:26.190 --> 00:41:29.510 align:middle line:84% Let's start out by defining what is equilibrium state. 00:41:29.510 --> 00:41:33.870 align:middle line:84% Now, by definition, equilibrium state 00:41:33.870 --> 00:41:35.530 align:middle line:90% is crystalline state, obviously. 00:41:35.530 --> 00:41:37.630 align:middle line:84% That's where it's thermodynamically stable. 00:41:37.630 --> 00:41:39.408 align:middle line:84% However, this is not the equilibrium state 00:41:39.408 --> 00:41:40.450 align:middle line:90% we are referring to here. 00:41:40.450 --> 00:41:43.130 align:middle line:84% That's why I put the quotation marks on "equilibrium state." 00:41:43.130 --> 00:41:43.630 align:middle line:90% OK? 00:41:43.630 --> 00:41:47.230 align:middle line:84% In the context of relaxation, the equilibrium state 00:41:47.230 --> 00:41:49.310 align:middle line:84% is basically the final state where 00:41:49.310 --> 00:41:50.790 align:middle line:84% the material is relaxing towards, 00:41:50.790 --> 00:41:52.910 align:middle line:90% where glass is relaxing towards. 00:41:52.910 --> 00:41:55.430 align:middle line:84% So if you look at the definition I give you 00:41:55.430 --> 00:41:57.190 align:middle line:84% in the very first lecture, it basically 00:41:57.190 --> 00:41:59.270 align:middle line:84% says that glass is always, constantly 00:41:59.270 --> 00:42:04.270 align:middle line:84% trying to relax towards its supercooled liquid state. 00:42:04.270 --> 00:42:06.830 align:middle line:84% So supercooled liquid state, obviously, 00:42:06.830 --> 00:42:11.110 align:middle line:84% when it's below melting point, it's not in thermal equilibrium. 00:42:11.110 --> 00:42:14.270 align:middle line:84% But from a structural relaxation perspective, 00:42:14.270 --> 00:42:16.850 align:middle line:84% that would be sort of the ultimate state 00:42:16.850 --> 00:42:19.650 align:middle line:84% where the glass is trying to relax towards. 00:42:19.650 --> 00:42:20.150 align:middle line:90% OK? 00:42:20.150 --> 00:42:22.570 align:middle line:84% So we define the supercooled liquid state 00:42:22.570 --> 00:42:25.690 align:middle line:90% as the equilibrium state. 00:42:25.690 --> 00:42:28.130 align:middle line:84% So now if we know the equilibrium state, 00:42:28.130 --> 00:42:30.410 align:middle line:84% then what about the driving force? 00:42:30.410 --> 00:42:34.450 align:middle line:84% The driving force would be the deviation of the state 00:42:34.450 --> 00:42:37.850 align:middle line:84% that the glass currently residing from the equilibrium 00:42:37.850 --> 00:42:38.350 align:middle line:90% state. 00:42:38.350 --> 00:42:42.330 align:middle line:90% 00:42:42.330 --> 00:42:45.150 align:middle line:84% For example, we can take volume relaxation as an example. 00:42:45.150 --> 00:42:47.850 align:middle line:84% But this applies to relaxation of other physical parameters 00:42:47.850 --> 00:42:48.610 align:middle line:90% as well. 00:42:48.610 --> 00:42:53.650 align:middle line:84% So for volume relaxation, let's consider volume 00:42:53.650 --> 00:42:57.370 align:middle line:84% as a function of temperature, OK? 00:42:57.370 --> 00:43:00.210 align:middle line:84% So for volume as a function of temperature, 00:43:00.210 --> 00:43:04.910 align:middle line:84% we know that we have basically this curve here. 00:43:04.910 --> 00:43:10.330 align:middle line:90% 00:43:10.330 --> 00:43:12.790 align:middle line:84% So this would be the supercooled liquid. 00:43:12.790 --> 00:43:15.670 align:middle line:90% This would be the glass. 00:43:15.670 --> 00:43:18.870 align:middle line:84% Now, we have mentioned that the supercooled liquid is basically 00:43:18.870 --> 00:43:21.910 align:middle line:84% the equilibrium state of the system. 00:43:21.910 --> 00:43:24.590 align:middle line:84% So if you look at the cooling curve here, 00:43:24.590 --> 00:43:29.430 align:middle line:84% initially, the system retraces this supercooled liquid. 00:43:29.430 --> 00:43:31.310 align:middle line:84% Means that it's actually in equilibrium. 00:43:31.310 --> 00:43:33.150 align:middle line:84% At this point here, it starts to deviate 00:43:33.150 --> 00:43:37.190 align:middle line:84% from the supercooled liquid state, OK? 00:43:37.190 --> 00:43:39.010 align:middle line:84% As soon as that deviation happens, 00:43:39.010 --> 00:43:43.110 align:middle line:84% you now have a driving force towards equilibrium state. 00:43:43.110 --> 00:43:47.370 align:middle line:84% So for the driving force, as I said, 00:43:47.370 --> 00:43:50.750 align:middle line:84% this is defined as the difference 00:43:50.750 --> 00:43:56.790 align:middle line:84% of the current volume versus the equilibrium volume, OK? 00:43:56.790 --> 00:43:58.470 align:middle line:84% So the current volume, this is what 00:43:58.470 --> 00:44:02.990 align:middle line:84% we label as V, the volume of the supercooled liquid, this 00:44:02.990 --> 00:44:08.990 align:middle line:84% is what we label as the V subscript e. 00:44:08.990 --> 00:44:13.372 align:middle line:84% So here e essentially refers to equilibrium, OK? 00:44:13.372 --> 00:44:15.330 align:middle line:84% This is the equilibrium state we're looking at. 00:44:15.330 --> 00:44:20.650 align:middle line:84% So the difference of the two, V minus Ve, this 00:44:20.650 --> 00:44:26.210 align:middle line:84% is the driving force of volume relaxation. 00:44:26.210 --> 00:44:28.210 align:middle line:84% Now, you may recall from the last lecture, 00:44:28.210 --> 00:44:30.930 align:middle line:84% I gave you the definition of the free volume. 00:44:30.930 --> 00:44:33.510 align:middle line:84% Free volume is exactly V minus Ve. 00:44:33.510 --> 00:44:36.850 align:middle line:84% This is the excess volume the glass possesses with respect 00:44:36.850 --> 00:44:42.210 align:middle line:84% to the supercooled liquid at the same temperature. 00:44:42.210 --> 00:44:49.850 align:middle line:84% So Vf defines how far away your system is away from equilibrium. 00:44:49.850 --> 00:44:53.430 align:middle line:84% So in relaxation process, the system will try to minimize Vf. 00:44:53.430 --> 00:44:56.570 align:middle line:84% It will try to essentially completely remove the Vf. 00:44:56.570 --> 00:44:57.210 align:middle line:90% OK? 00:44:57.210 --> 00:44:59.050 align:middle line:84% If you're able to stay at that temperature for a sufficiently 00:44:59.050 --> 00:45:02.030 align:middle line:84% long amount of time, it's going to relax towards that point. 00:45:02.030 --> 00:45:04.770 align:middle line:90% 00:45:04.770 --> 00:45:05.370 align:middle line:90% OK. 00:45:05.370 --> 00:45:08.890 align:middle line:90% Finally, kinetics. 00:45:08.890 --> 00:45:10.890 align:middle line:84% We are going to make exactly the same assumption 00:45:10.890 --> 00:45:13.550 align:middle line:84% here as in the case of the stress relaxation. 00:45:13.550 --> 00:45:18.950 align:middle line:84% We are going to assume that the relaxation rate scales linearly 00:45:18.950 --> 00:45:20.350 align:middle line:90% with the driving force. 00:45:20.350 --> 00:45:21.070 align:middle line:90% OK? 00:45:21.070 --> 00:45:25.870 align:middle line:84% So basically, if you look at how the volume is going 00:45:25.870 --> 00:45:33.750 align:middle line:84% to change over time, OK, so at a fixed temperature, 00:45:33.750 --> 00:45:36.990 align:middle line:84% then V is basically a constant, right? 00:45:36.990 --> 00:45:40.010 align:middle line:84% It's basically you have a well defined this line here 00:45:40.010 --> 00:45:41.270 align:middle line:84% that corresponds with the supercooled liquid line. 00:45:41.270 --> 00:45:42.645 align:middle line:84% So for every temperature, there's 00:45:42.645 --> 00:45:45.590 align:middle line:84% a well-defined volume for the supercooled liquid. 00:45:45.590 --> 00:45:47.750 align:middle line:84% So therefore, for fixed temperature, 00:45:47.750 --> 00:45:50.750 align:middle line:90% Ve is considered as a constant. 00:45:50.750 --> 00:45:52.670 align:middle line:84% So when you're taking the derivative of volume 00:45:52.670 --> 00:45:54.310 align:middle line:84% with respect to t, essentially, you 00:45:54.310 --> 00:45:58.590 align:middle line:84% are looking at the change of a free volume with respect 00:45:58.590 --> 00:46:01.470 align:middle line:90% to time. 00:46:01.470 --> 00:46:02.970 align:middle line:90% Right? 00:46:02.970 --> 00:46:05.832 align:middle line:90% 00:46:05.832 --> 00:46:07.790 align:middle line:84% So what we are going to assume here 00:46:07.790 --> 00:46:10.230 align:middle line:84% is the same mathematical formalism 00:46:10.230 --> 00:46:13.330 align:middle line:84% as what we have for stress relaxation. 00:46:13.330 --> 00:46:18.650 align:middle line:84% We have-- it scales inversely with 1 over tau, Vf. 00:46:18.650 --> 00:46:22.770 align:middle line:90% 00:46:22.770 --> 00:46:25.550 align:middle line:84% I used a different tau here, tau relaxation, 00:46:25.550 --> 00:46:26.790 align:middle line:90% just to characterize that. 00:46:26.790 --> 00:46:29.970 align:middle line:84% This is the characteristic relaxation time 00:46:29.970 --> 00:46:31.410 align:middle line:90% scale in the solid-- 00:46:31.410 --> 00:46:35.710 align:middle line:84% in the glass rather than in the Maxwell element, OK? 00:46:35.710 --> 00:46:41.170 align:middle line:90% 00:46:41.170 --> 00:46:45.410 align:middle line:84% Now, this gives you relaxation kinetics at one temperature. 00:46:45.410 --> 00:46:47.410 align:middle line:84% In the end, if you want to apply this theory 00:46:47.410 --> 00:46:49.130 align:middle line:84% to look at the structural relaxation 00:46:49.130 --> 00:46:51.410 align:middle line:84% during glass processing, typically, this 00:46:51.410 --> 00:46:53.970 align:middle line:84% involves changing the temperature over time. 00:46:53.970 --> 00:46:55.850 align:middle line:84% So we would need another equation 00:46:55.850 --> 00:46:59.690 align:middle line:84% that then correlates the relaxation 00:46:59.690 --> 00:47:01.650 align:middle line:90% rate with temperature, OK? 00:47:01.650 --> 00:47:04.610 align:middle line:84% So this can be done most straightforwardly 00:47:04.610 --> 00:47:07.010 align:middle line:84% by looking at the time dependence of the relaxation 00:47:07.010 --> 00:47:09.940 align:middle line:90% time scale. 00:47:09.940 --> 00:47:12.660 align:middle line:84% So relaxation involves, actually, 00:47:12.660 --> 00:47:14.840 align:middle line:84% motion of atoms or motion of molecules. 00:47:14.840 --> 00:47:20.320 align:middle line:84% So this motion of atoms or molecules, in a lot of cases, 00:47:20.320 --> 00:47:23.460 align:middle line:84% the kinetics can be described by using an Arrhenius equation 00:47:23.460 --> 00:47:25.420 align:middle line:90% as a first-order approximation. 00:47:25.420 --> 00:47:28.460 align:middle line:84% So if you follow Arrhenius equation, 00:47:28.460 --> 00:47:43.860 align:middle line:84% then the relaxation time can also 00:47:43.860 --> 00:47:46.780 align:middle line:84% be written with this exponential form. 00:47:46.780 --> 00:47:48.280 align:middle line:90% Right? 00:47:48.280 --> 00:47:53.585 align:middle line:90% 00:47:53.585 --> 00:47:57.120 align:middle line:84% You can always improve the theory. 00:47:57.120 --> 00:48:00.017 align:middle line:84% You can essentially assume a temperature-dependent Ea 00:48:00.017 --> 00:48:02.100 align:middle line:84% if you wanted to be a bit more accurate to account 00:48:02.100 --> 00:48:05.060 align:middle line:84% for, essentially, a fragile or liquid behavior. 00:48:05.060 --> 00:48:06.860 align:middle line:84% But this is an adequate, at least, 00:48:06.860 --> 00:48:10.760 align:middle line:84% for the purpose of this theory here. 00:48:10.760 --> 00:48:12.060 align:middle line:90% OK, that's all we need. 00:48:12.060 --> 00:48:14.640 align:middle line:90% 00:48:14.640 --> 00:48:18.760 align:middle line:84% This equation is really all we need 00:48:18.760 --> 00:48:21.840 align:middle line:84% to solve the glass response during any kind 00:48:21.840 --> 00:48:24.040 align:middle line:90% of thermal processing. 00:48:24.040 --> 00:48:28.880 align:middle line:84% So to see how this can be implemented, let's think about-- 00:48:28.880 --> 00:48:30.800 align:middle line:84% for example, you have some kind of process, 00:48:30.800 --> 00:48:33.540 align:middle line:90% time-dependent process. 00:48:33.540 --> 00:48:36.040 align:middle line:84% To implement the process, you need to know the initial state 00:48:36.040 --> 00:48:38.000 align:middle line:84% and you need to know the temperature history. 00:48:38.000 --> 00:48:39.300 align:middle line:84% For example, what's the heating rate? 00:48:39.300 --> 00:48:40.300 align:middle line:90% What's the cooling rate? 00:48:40.300 --> 00:48:41.560 align:middle line:90% And how long it lasts? 00:48:41.560 --> 00:48:45.760 align:middle line:84% So what you would know for this kind of a process 00:48:45.760 --> 00:48:50.220 align:middle line:84% is that you have to know that t equals to t1. 00:48:50.220 --> 00:48:53.200 align:middle line:84% You need to assign the boundary conditions, right? 00:48:53.200 --> 00:48:57.400 align:middle line:84% At t equals to t1, you know the starting volume. 00:48:57.400 --> 00:49:02.380 align:middle line:84% OK, volume, V, at t equals to t1, OK? 00:49:02.380 --> 00:49:07.380 align:middle line:90% 00:49:07.380 --> 00:49:11.180 align:middle line:84% OK, so this is what you would know as your starting boundary 00:49:11.180 --> 00:49:13.060 align:middle line:90% condition. 00:49:13.060 --> 00:49:16.220 align:middle line:84% Then we want to predict how this volume is going to evolve 00:49:16.220 --> 00:49:19.380 align:middle line:90% as a function of time. 00:49:19.380 --> 00:49:21.660 align:middle line:84% It will also be given about how the temperature is 00:49:21.660 --> 00:49:23.080 align:middle line:90% going to change over time. 00:49:23.080 --> 00:49:25.300 align:middle line:90% That defines your process. 00:49:25.300 --> 00:49:28.260 align:middle line:84% So then what you would do is you would apply, basically, 00:49:28.260 --> 00:49:32.220 align:middle line:84% a finite difference approach in the time domain, OK? 00:49:32.220 --> 00:49:35.360 align:middle line:84% You would look at a very small time step. 00:49:35.360 --> 00:49:39.660 align:middle line:84% Let's see what happens at t now equals to t1 plus 00:49:39.660 --> 00:49:42.420 align:middle line:90% the little time step dt. 00:49:42.420 --> 00:49:43.700 align:middle line:90% OK? 00:49:43.700 --> 00:49:48.900 align:middle line:84% So to get the volume at t1 equals to t1 plus dt, 00:49:48.900 --> 00:49:50.660 align:middle line:84% you then first need to calculate what 00:49:50.660 --> 00:49:54.380 align:middle line:84% is the dV over that time interval, dt. 00:49:54.380 --> 00:49:54.940 align:middle line:90% OK? 00:49:54.940 --> 00:49:57.520 align:middle line:90% So you plug into that equation. 00:49:57.520 --> 00:50:03.820 align:middle line:84% You have minus tau relaxation 1, and then Vf 00:50:03.820 --> 00:50:05.740 align:middle line:90% is the big V minus Ve. 00:50:05.740 --> 00:50:09.440 align:middle line:90% 00:50:09.440 --> 00:50:11.440 align:middle line:90% Now look at this equation here. 00:50:11.440 --> 00:50:13.920 align:middle line:84% All the quantities are actually known to you. 00:50:13.920 --> 00:50:17.900 align:middle line:84% Tau relaxation, you know the temperature at this time point. 00:50:17.900 --> 00:50:19.375 align:middle line:90% That's your thermal history. 00:50:19.375 --> 00:50:22.000 align:middle line:84% So if you know the temperature, you can plug into this equation 00:50:22.000 --> 00:50:24.520 align:middle line:84% here to derive the corresponding relaxation 00:50:24.520 --> 00:50:26.300 align:middle line:84% time for that particular instant of time. 00:50:26.300 --> 00:50:29.263 align:middle line:90% 00:50:29.263 --> 00:50:30.680 align:middle line:84% You know the instantaneous volume. 00:50:30.680 --> 00:50:31.460 align:middle line:90% That's given. 00:50:31.460 --> 00:50:32.840 align:middle line:90% That's your boundary condition. 00:50:32.840 --> 00:50:34.800 align:middle line:84% V is also something that's known, 00:50:34.800 --> 00:50:36.220 align:middle line:84% because you know the temperature. 00:50:36.220 --> 00:50:38.920 align:middle line:84% Once you know the temperature, the volume 00:50:38.920 --> 00:50:41.960 align:middle line:84% of the supercooled liquid is a single-value function 00:50:41.960 --> 00:50:43.320 align:middle line:90% of the temperature. 00:50:43.320 --> 00:50:46.280 align:middle line:84% So all the parameters on the right side of the equation 00:50:46.280 --> 00:50:47.020 align:middle line:90% is known. 00:50:47.020 --> 00:50:50.280 align:middle line:90% 00:50:50.280 --> 00:50:57.560 align:middle line:84% Therefore, your V at t1 plus dt would just simply 00:50:57.560 --> 00:51:01.680 align:middle line:90% be V at t1 plus dv. 00:51:01.680 --> 00:51:05.000 align:middle line:90% 00:51:05.000 --> 00:51:07.940 align:middle line:84% I should put the dt here. dV, dt. 00:51:07.940 --> 00:51:09.440 align:middle line:90% OK? 00:51:09.440 --> 00:51:15.376 align:middle line:90% 00:51:15.376 --> 00:51:17.680 align:middle line:84% Now you know the volume at the next time step. 00:51:17.680 --> 00:51:19.880 align:middle line:84% You just then need to repeat this process. 00:51:19.880 --> 00:51:24.620 align:middle line:90% 00:51:24.620 --> 00:51:27.780 align:middle line:84% And provided, as I said, you know the temperature 00:51:27.780 --> 00:51:30.940 align:middle line:84% at all the time points, then you would 00:51:30.940 --> 00:51:33.380 align:middle line:84% be able to derive how the volume is going to change 00:51:33.380 --> 00:51:35.040 align:middle line:90% as a function of time. 00:51:35.040 --> 00:51:37.660 align:middle line:90% 00:51:37.660 --> 00:51:38.680 align:middle line:90% Does that make sense? 00:51:38.680 --> 00:51:43.460 align:middle line:90% 00:51:43.460 --> 00:51:45.380 align:middle line:90% OK? 00:51:45.380 --> 00:51:49.580 align:middle line:84% This may look remarkably simple, but then, in fact, this 00:51:49.580 --> 00:51:51.680 align:middle line:90% is also very, very powerful. 00:51:51.680 --> 00:51:53.540 align:middle line:84% It allows you to make predictions 00:51:53.540 --> 00:51:56.300 align:middle line:84% that agrees with experiments and can predict some 00:51:56.300 --> 00:51:59.680 align:middle line:84% of the phenomena, as I mentioned before, about the relaxation, 00:51:59.680 --> 00:52:02.200 align:middle line:84% for example, during a reheating of glasses. 00:52:02.200 --> 00:52:03.380 align:middle line:90% OK? 00:52:03.380 --> 00:52:10.240 align:middle line:84% So I have actually put together a little MATLAB code myself. 00:52:10.240 --> 00:52:13.200 align:middle line:84% I can run the MATLAB code, define all kinds of temperature 00:52:13.200 --> 00:52:15.440 align:middle line:84% changes, and look at how the glass volume is going 00:52:15.440 --> 00:52:17.360 align:middle line:90% to vary as a function of time. 00:52:17.360 --> 00:52:18.200 align:middle line:90% OK. 00:52:18.200 --> 00:52:20.400 align:middle line:84% So now what I'm showing here, I'm 00:52:20.400 --> 00:52:22.480 align:middle line:84% showing you a few examples-- what's 00:52:22.480 --> 00:52:25.560 align:middle line:84% the results of the prediction coming from the code, OK? 00:52:25.560 --> 00:52:31.640 align:middle line:84% The first example here, we are considering the same cooling 00:52:31.640 --> 00:52:34.480 align:middle line:84% rate but then different reheating rate, OK? 00:52:34.480 --> 00:52:36.680 align:middle line:84% So the thermal history-- and so we're 00:52:36.680 --> 00:52:39.400 align:middle line:84% starting out from a supercooled liquid. 00:52:39.400 --> 00:52:42.910 align:middle line:84% We cool down with a constant cooling rate of 10 degrees 00:52:42.910 --> 00:52:44.880 align:middle line:90% Celsius per second. 00:52:44.880 --> 00:52:49.440 align:middle line:84% It then going to follow this blue line here. 00:52:49.440 --> 00:52:52.640 align:middle line:84% And then once it completely cools down, 00:52:52.640 --> 00:52:57.138 align:middle line:84% we then start to reheat it up with a different reheating rate. 00:52:57.138 --> 00:52:58.680 align:middle line:84% So what we're seeing here is that you 00:52:58.680 --> 00:53:00.795 align:middle line:84% start to see this kind of a funny shape 00:53:00.795 --> 00:53:02.920 align:middle line:84% that I have already mentioned in the very beginning 00:53:02.920 --> 00:53:04.260 align:middle line:90% of the semester. 00:53:04.260 --> 00:53:07.340 align:middle line:84% So now we explain why you have this kind of interesting shape 00:53:07.340 --> 00:53:07.840 align:middle line:90% here. 00:53:07.840 --> 00:53:11.380 align:middle line:84% Essentially, what happens is that right here, 00:53:11.380 --> 00:53:13.440 align:middle line:84% the glass material is very far from equilibrium. 00:53:13.440 --> 00:53:15.360 align:middle line:84% It has a very large excess free volume. 00:53:15.360 --> 00:53:16.820 align:middle line:84% So it wants to relax, but it cannot 00:53:16.820 --> 00:53:19.340 align:middle line:84% because it's, kinetically, very slow. 00:53:19.340 --> 00:53:21.420 align:middle line:84% So as you start to heat it up, the curve 00:53:21.420 --> 00:53:23.620 align:middle line:84% starts to bend downwards, indicating 00:53:23.620 --> 00:53:25.060 align:middle line:84% that the system is trying to relax 00:53:25.060 --> 00:53:27.860 align:middle line:84% towards the supercooled liquid line. 00:53:27.860 --> 00:53:30.740 align:middle line:84% Now, the slower the heating rate, the smaller the heating 00:53:30.740 --> 00:53:33.740 align:middle line:84% rate, the more relaxation you see. 00:53:33.740 --> 00:53:36.440 align:middle line:84% Basically, it just means that for smaller heating rate, 00:53:36.440 --> 00:53:39.820 align:middle line:84% you allow the glass material to dwell at the glass transition 00:53:39.820 --> 00:53:42.220 align:middle line:84% regime for a longer period of time. 00:53:42.220 --> 00:53:44.900 align:middle line:84% So if you have longer time, then more structural relaxation 00:53:44.900 --> 00:53:45.920 align:middle line:90% is going to take place. 00:53:45.920 --> 00:53:50.500 align:middle line:84% And that's why you're seeing the curve dips more. 00:53:50.500 --> 00:53:53.460 align:middle line:90% More significant relaxation. 00:53:53.460 --> 00:53:55.860 align:middle line:84% Now, once the curve dips, it's going 00:53:55.860 --> 00:53:58.380 align:middle line:84% to hit this supercooled liquid line. 00:53:58.380 --> 00:54:01.400 align:middle line:84% Once it starts to approach the supercooled liquid line, 00:54:01.400 --> 00:54:04.840 align:middle line:84% then it will try to trace the supercooled liquid line. 00:54:04.840 --> 00:54:06.167 align:middle line:90% Because that's equilibrium. 00:54:06.167 --> 00:54:07.500 align:middle line:90% That's at the equilibrium state. 00:54:07.500 --> 00:54:12.960 align:middle line:84% So as you slowly heat it up, the kinetics of relaxation 00:54:12.960 --> 00:54:14.480 align:middle line:90% start to accelerate. 00:54:14.480 --> 00:54:19.200 align:middle line:84% At some point, the kinetics will be sufficiently fast such 00:54:19.200 --> 00:54:23.780 align:middle line:84% that your system can now fully remain in the equilibrium state. 00:54:23.780 --> 00:54:24.940 align:middle line:90% So that's what happens. 00:54:24.940 --> 00:54:26.080 align:middle line:84% The curve starts to dip, and then 00:54:26.080 --> 00:54:27.560 align:middle line:84% eventually, you will start to trace 00:54:27.560 --> 00:54:29.018 align:middle line:84% this supercooled liquid line there. 00:54:29.018 --> 00:54:30.740 align:middle line:84% That means that the system, the kinetics 00:54:30.740 --> 00:54:32.800 align:middle line:84% is sufficiently fast to guarantee 00:54:32.800 --> 00:54:37.120 align:middle line:84% complete structural relaxation over the process. 00:54:37.120 --> 00:54:42.880 align:middle line:84% So that's why you see these different curves 00:54:42.880 --> 00:54:46.600 align:middle line:90% with different reheating rates. 00:54:46.600 --> 00:54:48.820 align:middle line:90% Now, this is another example. 00:54:48.820 --> 00:54:53.040 align:middle line:84% In this case, I'm looking at now different cooling rates. 00:54:53.040 --> 00:54:54.560 align:middle line:90% So same starting glass. 00:54:54.560 --> 00:54:56.480 align:middle line:84% I will cool down with different cooling rates 00:54:56.480 --> 00:54:57.920 align:middle line:90% here that I have labeled here. 00:54:57.920 --> 00:54:59.640 align:middle line:84% And then I use the constant heating rate 00:54:59.640 --> 00:55:02.410 align:middle line:90% of 1 degree Celsius per second. 00:55:02.410 --> 00:55:04.590 align:middle line:84% So there are two things we should talk about here. 00:55:04.590 --> 00:55:07.730 align:middle line:84% First is that when you are actually cooling it down, 00:55:07.730 --> 00:55:11.770 align:middle line:84% initially, the system will trace the supercooled liquid line 00:55:11.770 --> 00:55:12.990 align:middle line:90% because it's ergodic. 00:55:12.990 --> 00:55:15.170 align:middle line:84% The structural relaxation is fast enough. 00:55:15.170 --> 00:55:17.410 align:middle line:84% It can always stay in equilibrium. 00:55:17.410 --> 00:55:20.650 align:middle line:84% Now, at some point, the system will not 00:55:20.650 --> 00:55:22.970 align:middle line:84% be able to remain in the equilibrium state 00:55:22.970 --> 00:55:26.130 align:middle line:84% because the relaxation kinetics is not fast enough. 00:55:26.130 --> 00:55:29.770 align:middle line:84% And this happens first for faster cooling rates, 00:55:29.770 --> 00:55:33.730 align:middle line:84% because faster cooling rates, you give it a much narrower time 00:55:33.730 --> 00:55:35.810 align:middle line:90% window for it to relax. 00:55:35.810 --> 00:55:37.530 align:middle line:84% So therefore, the system will start 00:55:37.530 --> 00:55:40.690 align:middle line:84% to deviate from this supercooled liquid line at an earlier point 00:55:40.690 --> 00:55:42.710 align:middle line:90% for a faster cooling rate. 00:55:42.710 --> 00:55:46.370 align:middle line:84% So the gap here indicates that the system 00:55:46.370 --> 00:55:48.450 align:middle line:90% is no longer in equilibrium. 00:55:48.450 --> 00:55:51.690 align:middle line:90% Relaxation is not fast enough. 00:55:51.690 --> 00:55:56.250 align:middle line:84% So then once you reach the plateau during reheating here, 00:55:56.250 --> 00:55:58.830 align:middle line:84% you see that, for the 100-degree C per second, 00:55:58.830 --> 00:56:01.590 align:middle line:84% the faster cooling glass, it exhibits more stronger 00:56:01.590 --> 00:56:04.170 align:middle line:84% structural relaxation, which, again, 00:56:04.170 --> 00:56:07.990 align:middle line:84% is because it has a much larger driving force. 00:56:07.990 --> 00:56:11.410 align:middle line:84% Faster cooling glass has a higher volume, 00:56:11.410 --> 00:56:13.390 align:middle line:90% so higher excess free volume. 00:56:13.390 --> 00:56:15.890 align:middle line:84% Large free volume gives you a larger driving force. 00:56:15.890 --> 00:56:17.710 align:middle line:84% More driving force, the faster relaxation 00:56:17.710 --> 00:56:20.070 align:middle line:84% kinetics, according to our assumption there. 00:56:20.070 --> 00:56:24.670 align:middle line:84% And that's why you see a much significant dip compared 00:56:24.670 --> 00:56:26.490 align:middle line:90% to slower cooling glasses. 00:56:26.490 --> 00:56:33.858 align:middle line:90% 00:56:33.858 --> 00:56:36.950 align:middle line:84% So from this plot, you can automatically 00:56:36.950 --> 00:56:40.310 align:middle line:84% derive the dependence of a fictive temperature with respect 00:56:40.310 --> 00:56:42.510 align:middle line:84% to cooling rate here, just based on the theory. 00:56:42.510 --> 00:56:47.302 align:middle line:90% 00:56:47.302 --> 00:56:49.510 align:middle line:84% Now, one thing I also want to mention about this code 00:56:49.510 --> 00:56:53.630 align:middle line:84% here is that you may notice, in our typical schematic, 00:56:53.630 --> 00:56:57.350 align:middle line:84% the glass line actually has a finite slope. 00:56:57.350 --> 00:57:02.290 align:middle line:84% This is associated with thermal expansion of the glass material. 00:57:02.290 --> 00:57:06.150 align:middle line:84% In my code, I basically did not assume any thermal expansion. 00:57:06.150 --> 00:57:08.010 align:middle line:90% I basically assumed it's zero. 00:57:08.010 --> 00:57:11.210 align:middle line:84% So that's why this line becomes horizontal, 00:57:11.210 --> 00:57:13.430 align:middle line:84% but it's fairly easy to correct for it. 00:57:13.430 --> 00:57:15.630 align:middle line:84% You just add a linear term over temperature, 00:57:15.630 --> 00:57:17.310 align:middle line:84% so that it will go to a linear chain. 00:57:17.310 --> 00:57:21.450 align:middle line:84% So that's something you can do if you want for your own code. 00:57:21.450 --> 00:57:25.110 align:middle line:90% 00:57:25.110 --> 00:57:28.810 align:middle line:84% Any more questions about this part here? 00:57:28.810 --> 00:57:30.730 align:middle line:84% We are actually going to end a little bit 00:57:30.730 --> 00:57:32.330 align:middle line:90% early for this lecture. 00:57:32.330 --> 00:57:34.170 align:middle line:90% We'll cover up to this point. 00:57:34.170 --> 00:57:37.090 align:middle line:84% In the next lecture, we're going to discover the theory 00:57:37.090 --> 00:57:40.450 align:middle line:84% about fixed temperature, fictive temperature, and Tool's 00:57:40.450 --> 00:57:41.110 align:middle line:90% equations. 00:57:41.110 --> 00:57:42.930 align:middle line:90% So that's a different module. 00:57:42.930 --> 00:57:44.630 align:middle line:90% I want to wrap up here. 00:57:44.630 --> 00:57:47.770 align:middle line:84% But before that, any questions about structural relaxation? 00:57:47.770 --> 00:57:50.170 align:middle line:84% The key point really is this slide. 00:57:50.170 --> 00:57:53.770 align:middle line:84% I think this slide is the most important part that 00:57:53.770 --> 00:57:55.810 align:middle line:84% covers about the key elements you want 00:57:55.810 --> 00:57:59.110 align:middle line:84% to know how to mathematically model 00:57:59.110 --> 00:58:01.650 align:middle line:84% structural relaxation using first-order kinetic theory, 00:58:01.650 --> 00:58:02.390 align:middle line:90% essentially. 00:58:02.390 --> 00:58:05.390 align:middle line:90% 00:58:05.390 --> 00:58:08.790 align:middle line:84% AUDIENCE: Can you go back to the-- 00:58:08.790 --> 00:58:09.870 align:middle line:90% which slide was it-- 00:58:09.870 --> 00:58:14.890 align:middle line:84% the loss modulus, I think, slide, with all the math, 00:58:14.890 --> 00:58:18.350 align:middle line:84% breaking down the [INAUDIBLE] like prime? 00:58:18.350 --> 00:58:19.390 align:middle line:90% I didn't get it all. 00:58:19.390 --> 00:58:20.265 align:middle line:90% JUEJUN HU: Which one? 00:58:20.265 --> 00:58:21.070 align:middle line:90% This one? 00:58:21.070 --> 00:58:24.710 align:middle line:84% AUDIENCE: I think it was the one before. 00:58:24.710 --> 00:58:26.540 align:middle line:90% JUEJUN HU: Oh. 00:58:26.540 --> 00:58:27.050 align:middle line:90% This one? 00:58:27.050 --> 00:58:28.050 align:middle line:90% AUDIENCE: Yes, that one. 00:58:28.050 --> 00:58:28.633 align:middle line:90% JUEJUN HU: OK. 00:58:28.633 --> 00:58:31.270 align:middle line:90% 00:58:31.270 --> 00:58:31.830 align:middle line:90% OK. 00:58:31.830 --> 00:58:33.070 align:middle line:90% Thank you. 00:58:33.070 --> 00:58:35.328 align:middle line:84% Any questions about this particular slide or? 00:58:35.328 --> 00:58:35.870 align:middle line:90% AUDIENCE: No. 00:58:35.870 --> 00:58:39.150 align:middle line:84% I was listening and then didn't actually write it down. 00:58:39.150 --> 00:58:40.700 align:middle line:90% JUEJUN HU: Oh. 00:58:40.700 --> 00:58:58.000 align:middle line:90%