WEBVTT 00:00:03.330 --> 00:00:08.250 How long after swallowing a pill does it takes for a drug to enter your bloodstream? How 00:00:08.250 --> 00:00:13.599 long does it take for hot molten glass to cool? In this video, we'll see how the gradient 00:00:13.599 --> 00:00:17.110 helps us model molecular and thermal diffusion. 00:00:17.110 --> 00:00:21.270 This video is part of the Differential Equations video series. Laws that govern a system's 00:00:21.270 --> 00:00:24.570 properties can be modeled using differential equations. 00:00:24.570 --> 00:00:31.570 Hi, my name is Tom Peacock, and I'm a Professor of Mechanical Engineering here at MIT. Today 00:00:31.689 --> 00:00:35.479 I'd like to talk to you a little bit about the gradient. 00:00:35.479 --> 00:00:40.120 Partial differential equations describe the world around us. And partial differential 00:00:40.120 --> 00:00:47.120 equations often contain grad, div, and/or curl terms. In order to use these operations 00:00:47.330 --> 00:00:52.400 to describe physical phenomena, the first step is to understand what each mathematical 00:00:52.400 --> 00:00:57.610 process means geometrically and how it behaves in different examples. 00:00:57.610 --> 00:01:04.250 The gradient is an operation that takes in a scalar function and outputs a vector field. 00:01:04.250 --> 00:01:09.850 Many scalar quantities such as temperature and density have time derivatives that exhibit 00:01:09.850 --> 00:01:15.789 both a magnitude and a direction. Therefore it makes sense that we would need an operation 00:01:15.789 --> 00:01:19.750 that turns scalar functions into vector fields. 00:01:19.750 --> 00:01:24.890 Before watching this video, you should be familiar with the definition of the gradient, 00:01:24.890 --> 00:01:28.030 and its connection to the directional derivative. 00:01:28.030 --> 00:01:30.810 After watching this video, you will be able to 00:01:30.810 --> 00:01:35.700 Recognize that the gradient vector points in the direction of the maximum slope of a 00:01:35.700 --> 00:01:40.310 scalar function and has magnitude equal to that slope. 00:01:40.310 --> 00:01:46.880 Describe the physicality of Fick's First Law as it applies to concentration gradients. 00:01:46.880 --> 00:01:49.190 [Pause] 00:01:49.190 --> 00:01:55.090 Imagine what happens when you swallow a pill. Usually the pill contains an active ingredient, 00:01:55.090 --> 00:02:02.090 or drug, and a mixture of other inactive ingredients, such as binders, flavoring agents, etc. Some 00:02:03.360 --> 00:02:07.440 pills are coated to make the pill easier to swallow and to control the release of the 00:02:07.440 --> 00:02:12.380 drug. When you swallow the pill, it starts to dissolve. 00:02:12.380 --> 00:02:17.120 It is usually desired for there to be a constant and predictable delivery rate of drug to the 00:02:17.120 --> 00:02:24.120 body, that is that the diffusion of drug reaches steady state. We need to understand what this 00:02:24.540 --> 00:02:30.110 steady state amount is to ensure that we are delivering the desired dose. 00:02:30.110 --> 00:02:36.579 The equation that describes diffusion is the partial derivative of c with respect to time 00:02:36.579 --> 00:02:40.800 is equal to D del squared c. 00:02:40.800 --> 00:02:47.790 where c is concentration, and D is the diffusion coefficient, which we will assume is constant. 00:02:47.790 --> 00:02:52.459 But where does this come from? In order to understand this completely, we will need to 00:02:52.459 --> 00:02:59.170 combine the divergence and gradient to have a full description of the del squared term. 00:02:59.170 --> 00:03:06.170 In this video, our goal is to understand how flux is related to the gradient of the concentration. 00:03:06.170 --> 00:03:07.340 [Pause] 00:03:07.340 --> 00:03:12.610 Let's review the properties and meaning of the gradient. The gradient is a local property 00:03:12.610 --> 00:03:19.610 of a function. That is, it depends only on points that are near a point of interest. 00:03:19.630 --> 00:03:25.730 Given a function f(x,y) of two variables, we can represent this function as a surface 00:03:25.730 --> 00:03:31.900 in 3-dimensions z=f(x,y) 00:03:31.900 --> 00:03:34.930 Or as a collection of level curves. 00:03:34.930 --> 00:03:41.930 The gradient at a point (x,y) can be determined by finding a vector in the tangent plane to 00:03:41.959 --> 00:03:48.959 z=f(x,y) at (x,y) that points in the direction of the steepest slope. 00:03:49.579 --> 00:03:56.190 The gradient vector is a vector in the x,y-plane. The direction is found by projecting the vector 00:03:56.190 --> 00:04:01.580 in the tangent plane down onto the xy-plane. 00:04:01.580 --> 00:04:07.849 The magnitude of the gradient is the slope of that vector in the tangent plane. 00:04:07.849 --> 00:04:13.160 This vector is always perpendicular to the level curve because along the level curve, 00:04:13.160 --> 00:04:19.659 the function is constant. 00:04:19.659 --> 00:04:23.559 What is the 1-dimensional analogue of the gradient? 00:04:23.559 --> 00:04:30.300 Take the tangent line to the graph of the function. Point a vector up the hill, then 00:04:30.300 --> 00:04:37.300 project down. The direction is either positive or negative. The magnitude is the slope of 00:04:37.960 --> 00:04:44.960 the graph. But 1-dimensional vectors are scalars. So the gradient is simply the derivative. 00:04:46.469 --> 00:04:53.139 And we already know that the derivative is a local property of a function: because it 00:04:53.139 --> 00:04:58.659 is a limit, it depends only on points in a small region near the point at which we are 00:04:58.659 --> 00:05:00.699 looking for the derivative. 00:05:00.699 --> 00:05:07.619 What happens in 3-dimensions? It is somewhat difficult to represent a 3-dimensional function. 00:05:07.619 --> 00:05:14.619 The best way to represent such a function is through a collection of level surfaces. 00:05:14.759 --> 00:05:21.029 The gradient field can be computed at every point on the level surface. We know that the 00:05:21.029 --> 00:05:26.520 gradient vector is a 3-dimensional vector that is normal to this surface. The magnitude 00:05:26.520 --> 00:05:32.558 of the gradient vector measures the steepest increase of a shape we can't imagine because 00:05:32.558 --> 00:05:33.099 it is 4-dimensional. 00:05:33.099 --> 00:05:33.169 [Pause] 00:05:33.169 --> 00:05:34.240 Let's get back to our tablet diffusion example. We aren't going to attack the entire problem 00:05:34.240 --> 00:05:35.449 all at once. The first thing that we want to try to understand is the movement of drug 00:05:35.449 --> 00:05:36.449 molecules through any given surface area per unit time, i.e. we want to understand the 00:05:36.449 --> 00:05:36.930 flux from the pill into its surroundings. 00:05:36.930 --> 00:05:41.839 In order to better understand this process, we begin with a demo. Here you see a tank 00:05:41.839 --> 00:05:48.180 of water and a drop of dye. Initially, the dye is concentrated in a single droplet at 00:05:48.180 --> 00:05:53.169 the center of the tank. Over time, the dye particles move away from the center, until 00:05:53.169 --> 00:05:56.990 a point in time when the process reaches steady state. 00:05:56.990 --> 00:06:01.259 In order to model what is happening at the atomic level in this demo, we are going to 00:06:01.259 --> 00:06:08.259 start by making a 1-dimensional discrete model. This one-dimensional model will be simpler, 00:06:09.009 --> 00:06:15.399 and allow us to describe the flux of particles more easily. Then we will extend the model 00:06:15.399 --> 00:06:20.740 to 2-dimensions, creating a discrete time step simulation to determine the equation 00:06:20.740 --> 00:06:25.789 for flux. Then we will look at our 3 dimensional demo and discuss the equation for flux. 00:06:25.789 --> 00:06:31.389 In the 1-dimensional model, we are going to model the particles of dye as random walkers 00:06:31.389 --> 00:06:38.159 on a line. Each random walker has an equal probability of moving one step of length Delta 00:06:38.159 --> 00:06:45.159 x to the right or to the left during a time step Delta t. The walkers move independently 00:06:46.529 --> 00:06:48.159 of each other. 00:06:48.159 --> 00:06:54.929 We make an assumption that Delta x and Delta t are both small. 00:06:54.929 --> 00:07:00.110 In order to understand how the particles are moving, we want to understand the flux through 00:07:00.110 --> 00:07:01.800 any given point. 00:07:01.800 --> 00:07:08.800 Recall that flux is flow per unit "area" per unit time. Our random walk model is one dimensional, 00:07:11.710 --> 00:07:17.809 so we will define the flow of particles through a single point over a time step Delta t to 00:07:17.809 --> 00:07:19.550 be flux. 00:07:19.550 --> 00:07:24.800 While we can look at the flux through any point, for mathematical convenience, let us 00:07:24.800 --> 00:07:31.800 determine the flux through the point x + Delta x over 2 at time t. This point is half way 00:07:31.909 --> 00:07:38.159 between the point x and x+delta x. Because of the hypotheses of our random walk, any 00:07:38.159 --> 00:07:44.550 particle that is within a step length Delta x to the left or the right of x + Delta x 00:07:44.550 --> 00:07:51.039 over 2 has a ½ probability of flowing through the point during the next time step. So in 00:07:51.039 --> 00:07:56.800 order to find the flux, the first step is to determine many particles are within our 00:07:56.800 --> 00:08:03.050 step distance Delta x from the point x + Delta x over 2. 00:08:03.050 --> 00:08:10.050 Let the concentration of particles be denoted by the function c(x,t), which is the number 00:08:10.119 --> 00:08:16.959 of particles per unit length at a time t. To find the number of particles to the left 00:08:16.959 --> 00:08:23.509 of x + Delta x over 2, we could integrate the concentration function over the interval 00:08:23.509 --> 00:08:30.509 of length Delta x centered about the point x. However, because we have assumed that Delta 00:08:30.709 --> 00:08:37.709 x is small, we can approximate the concentration function by the value of the concentration 00:08:37.830 --> 00:08:44.810 at x for the whole interval. So the number of particles on the interval of length centered 00:08:44.810 --> 00:08:51.810 about the point x can be approximated by c(x,t) times Delta x . The number of particles on 00:08:53.820 --> 00:09:00.820 the interval of length Delta x centered about the point x + Delta x is approximately c(x+Delta 00:09:02.150 --> 00:09:08.560 x, t) times Delta x particles. 00:09:08.560 --> 00:09:15.190 We assume that any particle has 1/2 probability of moving one step to the left or the right. 00:09:15.190 --> 00:09:20.880 Thus the flux through our point is given by one half times the number of particles to 00:09:20.880 --> 00:09:27.880 the left minus one half times the number of particles to the right a time t. We divide 00:09:28.320 --> 00:09:33.990 the entire expression by the time step, which is the unit of time over which we are looking 00:09:33.990 --> 00:09:36.580 at the motion of particles. 00:09:36.580 --> 00:09:42.010 To dig a little deeper into this equation, we can take a Taylor expansion of our concentration 00:09:42.010 --> 00:09:49.010 function c(x + Delta x, t) about x, holding t fixed. This gives us the following expression, 00:09:52.950 --> 00:09:58.930 which is a polynomial in Delta x with coefficients given by multiples of sequentially higher 00:09:58.930 --> 00:10:03.160 partial derivatives of the concentration function c. 00:10:03.160 --> 00:10:08.290 Our equation for flux becomes this seemingly more complicated equation. 00:10:08.290 --> 00:10:15.290 However, if we make an assumption that Delta x grows proportionally to the square root 00:10:18.500 --> 00:10:23.130 of Delta t, in other words that Delta x squared is proportional to Delta t: 00:10:23.130 --> 00:10:30.130 This simplifies the expression for flux because only the first term has a significant contribution, 00:10:32.860 --> 00:10:38.870 and we are left with the following expression for flux: 00:10:38.870 --> 00:10:39.530 [pause] 00:10:39.530 --> 00:10:44.970 You can do a table top experiment by placing a small drop of dye in a narrow test tube 00:10:44.970 --> 00:10:49.910 and measuring the change in height of dye with respect to the change in time in order 00:10:49.910 --> 00:10:54.250 to verify that the assumption we made is valid. 00:10:54.250 --> 00:11:00.990 Rewriting the constant term in front as some diffusion constant D, this equation is commonly 00:11:00.990 --> 00:11:07.880 written as flux is equal to negative D times the partial derivative of c with respect to 00:11:07.880 --> 00:11:09.100 x. 00:11:09.100 --> 00:11:15.500 The negative sign in this equation says that the direction of net flux goes from a region 00:11:15.500 --> 00:11:22.500 of high concentration to a region of low concentration, in the opposite direction as the concentration 00:11:22.580 --> 00:11:29.580 gradient. Why is this? If there are more particles on one side of a point than the other, we 00:11:31.590 --> 00:11:38.470 suspect half of them flow through the point on either side, so the net flow through the 00:11:38.470 --> 00:11:40.700 point is away from the highest concentration. 00:11:40.700 --> 00:11:43.010 This behavior is consistent with what we saw with the dye in the fish tank. 00:11:43.010 --> 00:11:50.010 Now we want to extend this to 2-dimensions. Here we have modeled a system of 2000 particles 00:11:51.340 --> 00:11:58.340 walking randomly in the plane. Each particle can move a unit distance away from its current 00:11:58.560 --> 00:12:05.560 location in any direction with equal probability. A profile of the concentration at each time 00:12:06.610 --> 00:12:08.690 step is displayed to the right. 00:12:08.690 --> 00:12:15.650 We change the view of the concentration to be contour lines, and add some more particles 00:12:15.650 --> 00:12:22.650 to increase the accuracy of our computation in order to add in the flux vector. In 2D, 00:12:23.430 --> 00:12:30.430 the flux is a flow per length per unit time, and is a vector quantity. Observe that the 00:12:31.380 --> 00:12:38.380 flux is everywhere perpendicular to the level sets, or contours of the concentration map, 00:12:38.560 --> 00:12:45.150 and it points away from the highest concentration. In other words, this simulation suggests that 00:12:45.150 --> 00:12:52.150 the flux points in the direction of the negative gradient of the concentration. 00:12:54.180 --> 00:13:01.180 The equation that describes this says that flux J is equal to some constant, which we 00:13:02.550 --> 00:13:08.410 will call D, times the negative gradient of the concentration: Compare to the equation 00:13:08.410 --> 00:13:14.770 we had in the 1-dimensional example. Here the derivative is replaced by the gradient 00:13:14.770 --> 00:13:18.460 because the derivative is the 1-dimensional analogue of the gradient. 00:13:18.460 --> 00:13:25.380 Now let's look back to our 3-dimensional example. The flow profile seems to follow the same 00:13:25.380 --> 00:13:26.490 basic principles. 00:13:26.490 --> 00:13:33.490 Experiments and observations have shown that the flux of particles per unit area is determined 00:13:39.270 --> 00:13:46.270 by some constant times the negative gradient of concentration, just as in our discrete 00:14:02.580 --> 00:14:09.040 2-dimensional model: 00:14:09.040 --> 00:14:16.040 This equation is one form of Fick's first law. It says that flux points along the negative 00:14:22.330 --> 00:14:23.820 gradient of the concentration. 00:14:23.820 --> 00:14:30.820 It turns out that this equation describes the flux of many familiar quantities. Let's 00:14:32.279 --> 00:14:34.260 consider some examples: 00:14:34.260 --> 00:14:41.260 When students exit a classroom when class ends shows the flux of people through the 00:14:44.110 --> 00:14:51.110 doorway points away from the highest concentration of students. 00:14:58.290 --> 00:15:03.390 The second law of thermodynamics says that heat flows from high to low temperatures. 00:15:03.390 --> 00:15:10.390 This says that the flux is proportional (perhaps non-uniformly) to the negative temperature gradient. 00:16:21.790 --> 00:16:28.520 Be aware that this is just one form of Fick's first law. The most general form says that 00:16:28.520 --> 00:16:34.800 flux is proportional to the negative gradient of the chemical potential. 00:16:34.800 --> 00:16:41.800 You may see the equation in this form in later courses. In all of the examples that we have 00:16:43.690 --> 00:16:48.880 considered in this video, the gradient of the concentration and the gradient of the 00:16:48.880 --> 00:16:53.330 chemical potential pointed in the same direction. 00:16:53.330 --> 00:16:55.470 To review: 00:16:55.470 --> 00:17:00.950 The gradient is a vector quantity that points in the direction of the maximum slope of a 00:17:00.950 --> 00:17:02.850 scalar function. 00:17:02.850 --> 00:17:08.609 Fick's first law says that flux points along the negative gradient of concentration. 00:17:08.609 --> 00:17:15.049 In order to understand Fick's First Law, we first considered models in 1-d and 2-d, before 00:17:15.049 --> 00:17:17.439 trying to understand the description in 3-d.