1 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 2 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 3 00:00:13,599 --> 00:00:17,110 helps us model molecular and thermal diffusion. 4 00:00:17,110 --> 00:00:21,270 This video is part of the Differential Equations video series. Laws that govern a system's 5 00:00:21,270 --> 00:00:24,570 properties can be modeled using differential equations. 6 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 7 00:00:31,689 --> 00:00:35,479 I'd like to talk to you a little bit about the gradient. 8 00:00:35,479 --> 00:00:40,120 Partial differential equations describe the world around us. And partial differential 9 00:00:40,120 --> 00:00:47,120 equations often contain grad, div, and/or curl terms. In order to use these operations 10 00:00:47,330 --> 00:00:52,400 to describe physical phenomena, the first step is to understand what each mathematical 11 00:00:52,400 --> 00:00:57,610 process means geometrically and how it behaves in different examples. 12 00:00:57,610 --> 00:01:04,250 The gradient is an operation that takes in a scalar function and outputs a vector field. 13 00:01:04,250 --> 00:01:09,850 Many scalar quantities such as temperature and density have time derivatives that exhibit 14 00:01:09,850 --> 00:01:15,789 both a magnitude and a direction. Therefore it makes sense that we would need an operation 15 00:01:15,789 --> 00:01:19,750 that turns scalar functions into vector fields. 16 00:01:19,750 --> 00:01:24,890 Before watching this video, you should be familiar with the definition of the gradient, 17 00:01:24,890 --> 00:01:28,030 and its connection to the directional derivative. 18 00:01:28,030 --> 00:01:30,810 After watching this video, you will be able to 19 00:01:30,810 --> 00:01:35,700 Recognize that the gradient vector points in the direction of the maximum slope of a 20 00:01:35,700 --> 00:01:40,310 scalar function and has magnitude equal to that slope. 21 00:01:40,310 --> 00:01:46,880 Describe the physicality of Fick's First Law as it applies to concentration gradients. 22 00:01:46,880 --> 00:01:49,190 [Pause] 23 00:01:49,190 --> 00:01:55,090 Imagine what happens when you swallow a pill. Usually the pill contains an active ingredient, 24 00:01:55,090 --> 00:02:02,090 or drug, and a mixture of other inactive ingredients, such as binders, flavoring agents, etc. Some 25 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 26 00:02:07,440 --> 00:02:12,380 drug. When you swallow the pill, it starts to dissolve. 27 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 28 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 29 00:02:24,540 --> 00:02:30,110 steady state amount is to ensure that we are delivering the desired dose. 30 00:02:30,110 --> 00:02:36,579 The equation that describes diffusion is the partial derivative of c with respect to time 31 00:02:36,579 --> 00:02:40,800 is equal to D del squared c. 32 00:02:40,800 --> 00:02:47,790 where c is concentration, and D is the diffusion coefficient, which we will assume is constant. 33 00:02:47,790 --> 00:02:52,459 But where does this come from? In order to understand this completely, we will need to 34 00:02:52,459 --> 00:02:59,170 combine the divergence and gradient to have a full description of the del squared term. 35 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. 36 00:03:06,170 --> 00:03:07,340 [Pause] 37 00:03:07,340 --> 00:03:12,610 Let's review the properties and meaning of the gradient. The gradient is a local property 38 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. 39 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 40 00:03:25,730 --> 00:03:31,900 in 3-dimensions z=f(x,y) 41 00:03:31,900 --> 00:03:34,930 Or as a collection of level curves. 42 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 43 00:03:41,959 --> 00:03:48,959 z=f(x,y) at (x,y) that points in the direction of the steepest slope. 44 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 45 00:03:56,190 --> 00:04:01,580 in the tangent plane down onto the xy-plane. 46 00:04:01,580 --> 00:04:07,849 The magnitude of the gradient is the slope of that vector in the tangent plane. 47 00:04:07,849 --> 00:04:13,160 This vector is always perpendicular to the level curve because along the level curve, 48 00:04:13,160 --> 00:04:19,659 the function is constant. 49 00:04:19,659 --> 00:04:23,559 What is the 1-dimensional analogue of the gradient? 50 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 51 00:04:30,300 --> 00:04:37,300 project down. The direction is either positive or negative. The magnitude is the slope of 52 00:04:37,960 --> 00:04:44,960 the graph. But 1-dimensional vectors are scalars. So the gradient is simply the derivative. 53 00:04:46,469 --> 00:04:53,139 And we already know that the derivative is a local property of a function: because it 54 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 55 00:04:58,659 --> 00:05:00,699 looking for the derivative. 56 00:05:00,699 --> 00:05:07,619 What happens in 3-dimensions? It is somewhat difficult to represent a 3-dimensional function. 57 00:05:07,619 --> 00:05:14,619 The best way to represent such a function is through a collection of level surfaces. 58 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 59 00:05:21,029 --> 00:05:26,520 gradient vector is a 3-dimensional vector that is normal to this surface. The magnitude 60 00:05:26,520 --> 00:05:32,558 of the gradient vector measures the steepest increase of a shape we can't imagine because 61 00:05:32,558 --> 00:05:33,099 it is 4-dimensional. 62 00:05:33,099 --> 00:05:33,169 [Pause] 63 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 64 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 65 00:05:35,449 --> 00:05:36,449 molecules through any given surface area per unit time, i.e. we want to understand the 66 00:05:36,449 --> 00:05:36,930 flux from the pill into its surroundings. 67 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 68 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 69 00:05:48,180 --> 00:05:53,169 the center of the tank. Over time, the dye particles move away from the center, until 70 00:05:53,169 --> 00:05:56,990 a point in time when the process reaches steady state. 71 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 72 00:06:01,259 --> 00:06:08,259 start by making a 1-dimensional discrete model. This one-dimensional model will be simpler, 73 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 74 00:06:15,399 --> 00:06:20,740 to 2-dimensions, creating a discrete time step simulation to determine the equation 75 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. 76 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 77 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 78 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 79 00:06:46,529 --> 00:06:48,159 of each other. 80 00:06:48,159 --> 00:06:54,929 We make an assumption that Delta x and Delta t are both small. 81 00:06:54,929 --> 00:07:00,110 In order to understand how the particles are moving, we want to understand the flux through 82 00:07:00,110 --> 00:07:01,800 any given point. 83 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, 84 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 85 00:07:17,809 --> 00:07:19,550 be flux. 86 00:07:19,550 --> 00:07:24,800 While we can look at the flux through any point, for mathematical convenience, let us 87 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 88 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 89 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 90 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 91 00:07:51,039 --> 00:07:56,800 order to find the flux, the first step is to determine many particles are within our 92 00:07:56,800 --> 00:08:03,050 step distance Delta x from the point x + Delta x over 2. 93 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 94 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 95 00:08:16,959 --> 00:08:23,509 of x + Delta x over 2, we could integrate the concentration function over the interval 96 00:08:23,509 --> 00:08:30,509 of length Delta x centered about the point x. However, because we have assumed that Delta 97 00:08:30,709 --> 00:08:37,709 x is small, we can approximate the concentration function by the value of the concentration 98 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 99 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 100 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 101 00:09:02,150 --> 00:09:08,560 x, t) times Delta x particles. 102 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. 103 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 104 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 105 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 106 00:09:33,990 --> 00:09:36,580 at the motion of particles. 107 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 108 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, 109 00:09:52,950 --> 00:09:58,930 which is a polynomial in Delta x with coefficients given by multiples of sequentially higher 110 00:09:58,930 --> 00:10:03,160 partial derivatives of the concentration function c. 111 00:10:03,160 --> 00:10:08,290 Our equation for flux becomes this seemingly more complicated equation. 112 00:10:08,290 --> 00:10:15,290 However, if we make an assumption that Delta x grows proportionally to the square root 113 00:10:18,500 --> 00:10:23,130 of Delta t, in other words that Delta x squared is proportional to Delta t: 114 00:10:23,130 --> 00:10:30,130 This simplifies the expression for flux because only the first term has a significant contribution, 115 00:10:32,860 --> 00:10:38,870 and we are left with the following expression for flux: 116 00:10:38,870 --> 00:10:39,530 [pause] 117 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 118 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 119 00:10:49,910 --> 00:10:54,250 to verify that the assumption we made is valid. 120 00:10:54,250 --> 00:11:00,990 Rewriting the constant term in front as some diffusion constant D, this equation is commonly 121 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 122 00:11:07,880 --> 00:11:09,100 x. 123 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 124 00:11:15,500 --> 00:11:22,500 of high concentration to a region of low concentration, in the opposite direction as the concentration 125 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 126 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 127 00:11:38,470 --> 00:11:40,700 point is away from the highest concentration. 128 00:11:40,700 --> 00:11:43,010 This behavior is consistent with what we saw with the dye in the fish tank. 129 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 130 00:11:51,340 --> 00:11:58,340 walking randomly in the plane. Each particle can move a unit distance away from its current 131 00:11:58,560 --> 00:12:05,560 location in any direction with equal probability. A profile of the concentration at each time 132 00:12:06,610 --> 00:12:08,690 step is displayed to the right. 133 00:12:08,690 --> 00:12:15,650 We change the view of the concentration to be contour lines, and add some more particles 134 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, 135 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 136 00:12:31,380 --> 00:12:38,380 flux is everywhere perpendicular to the level sets, or contours of the concentration map, 137 00:12:38,560 --> 00:12:45,150 and it points away from the highest concentration. In other words, this simulation suggests that 138 00:12:45,150 --> 00:12:52,150 the flux points in the direction of the negative gradient of the concentration. 139 00:12:54,180 --> 00:13:01,180 The equation that describes this says that flux J is equal to some constant, which we 140 00:13:02,550 --> 00:13:08,410 will call D, times the negative gradient of the concentration: Compare to the equation 141 00:13:08,410 --> 00:13:14,770 we had in the 1-dimensional example. Here the derivative is replaced by the gradient 142 00:13:14,770 --> 00:13:18,460 because the derivative is the 1-dimensional analogue of the gradient. 143 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 144 00:13:25,380 --> 00:13:26,490 basic principles. 145 00:13:26,490 --> 00:13:33,490 Experiments and observations have shown that the flux of particles per unit area is determined 146 00:13:39,270 --> 00:13:46,270 by some constant times the negative gradient of concentration, just as in our discrete 147 00:14:02,580 --> 00:14:09,040 2-dimensional model: 148 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 149 00:14:22,330 --> 00:14:23,820 gradient of the concentration. 150 00:14:23,820 --> 00:14:30,820 It turns out that this equation describes the flux of many familiar quantities. Let's 151 00:14:32,279 --> 00:14:34,260 consider some examples: 152 00:14:34,260 --> 00:14:41,260 When students exit a classroom when class ends shows the flux of people through the 153 00:14:44,110 --> 00:14:51,110 doorway points away from the highest concentration of students. 154 00:14:58,290 --> 00:15:03,390 The second law of thermodynamics says that heat flows from high to low temperatures. 155 00:15:03,390 --> 00:15:10,390 This says that the flux is proportional (perhaps non-uniformly) to the negative temperature gradient. 156 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 157 00:16:28,520 --> 00:16:34,800 flux is proportional to the negative gradient of the chemical potential. 158 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 159 00:16:43,690 --> 00:16:48,880 considered in this video, the gradient of the concentration and the gradient of the 160 00:16:48,880 --> 00:16:53,330 chemical potential pointed in the same direction. 161 00:16:53,330 --> 00:16:55,470 To review: 162 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 163 00:17:00,950 --> 00:17:02,850 scalar function. 164 00:17:02,850 --> 00:17:08,609 Fick's first law says that flux points along the negative gradient of concentration. 165 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 166 00:17:15,049 --> 00:17:17,439 trying to understand the description in 3-d.