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How long after swallowing a pill does it takes
for a drug to enter your bloodstream? How
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long does it take for hot molten glass to
cool? In this video, we'll see how the gradient
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helps us model molecular and thermal diffusion.
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This video is part of the Differential Equations
video series. Laws that govern a system's
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properties can be modeled using differential
equations.
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Hi, my name is Tom Peacock, and I'm a Professor
of Mechanical Engineering here at MIT. Today
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I'd like to talk to you a little bit about
the gradient.
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Partial differential equations describe the
world around us. And partial differential
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equations often contain grad, div, and/or
curl terms. In order to use these operations
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to describe physical phenomena, the first
step is to understand what each mathematical
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process means geometrically and how it behaves
in different examples.
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The gradient is an operation that takes in
a scalar function and outputs a vector field.
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Many scalar quantities such as temperature
and density have time derivatives that exhibit
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both a magnitude and a direction. Therefore
it makes sense that we would need an operation
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that turns scalar functions into vector fields.
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Before watching this video, you should be
familiar with the definition of the gradient,
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and its connection to the directional derivative.
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After watching this video, you will be able
to
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Recognize that the gradient vector points
in the direction of the maximum slope of a
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scalar function and has magnitude equal to
that slope.
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Describe the physicality of Fick's First Law
as it applies to concentration gradients.
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[Pause]
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Imagine what happens when you swallow a pill.
Usually the pill contains an active ingredient,
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or drug, and a mixture of other inactive ingredients,
such as binders, flavoring agents, etc. Some
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pills are coated to make the pill easier to
swallow and to control the release of the
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drug. When you swallow the pill, it starts
to dissolve.
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It is usually desired for there to be a constant
and predictable delivery rate of drug to the
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body, that is that the diffusion of drug reaches
steady state. We need to understand what this
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steady state amount is to ensure that we are
delivering the desired dose.
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The equation that describes diffusion is the
partial derivative of c with respect to time
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is equal to D del squared c.
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where c is concentration, and D is the diffusion
coefficient, which we will assume is constant.
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But where does this come from? In order to
understand this completely, we will need to
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combine the divergence and gradient to have
a full description of the del squared term.
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In this video, our goal is to understand how
flux is related to the gradient of the concentration.
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[Pause]
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Let's review the properties and meaning of
the gradient. The gradient is a local property
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of a function. That is, it depends only on
points that are near a point of interest.
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Given a function f(x,y) of two variables,
we can represent this function as a surface
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in 3-dimensions z=f(x,y)
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Or as a collection of level curves.
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The gradient at a point (x,y) can be determined
by finding a vector in the tangent plane to
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z=f(x,y) at (x,y) that points in the direction
of the steepest slope.
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The gradient vector is a vector in the x,y-plane.
The direction is found by projecting the vector
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in the tangent plane down onto the xy-plane.
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The magnitude of the gradient is the slope
of that vector in the tangent plane.
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This vector is always perpendicular to the
level curve because along the level curve,
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the function is constant.
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What is the 1-dimensional analogue of the
gradient?
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Take the tangent line to the graph of the
function. Point a vector up the hill, then
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project down. The direction is either positive
or negative. The magnitude is the slope of
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the graph. But 1-dimensional vectors are scalars.
So the gradient is simply the derivative.
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And we already know that the derivative is
a local property of a function: because it
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is a limit, it depends only on points in a
small region near the point at which we are
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looking for the derivative.
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What happens in 3-dimensions? It is somewhat
difficult to represent a 3-dimensional function.
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The best way to represent such a function
is through a collection of level surfaces.
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The gradient field can be computed at every
point on the level surface. We know that the
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gradient vector is a 3-dimensional vector
that is normal to this surface. The magnitude
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of the gradient vector measures the steepest
increase of a shape we can't imagine because
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it is 4-dimensional.
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[Pause]
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Let's get back to our tablet diffusion example.
We aren't going to attack the entire problem
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all at once. The first thing that we want
to try to understand is the movement of drug
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molecules through any given surface area per
unit time, i.e. we want to understand the
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flux from the pill into its surroundings.
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In order to better understand this process,
we begin with a demo. Here you see a tank
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of water and a drop of dye. Initially, the
dye is concentrated in a single droplet at
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the center of the tank. Over time, the dye
particles move away from the center, until
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a point in time when the process reaches steady
state.
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In order to model what is happening at the
atomic level in this demo, we are going to
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start by making a 1-dimensional discrete model.
This one-dimensional model will be simpler,
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and allow us to describe the flux of particles
more easily. Then we will extend the model
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to 2-dimensions, creating a discrete time
step simulation to determine the equation
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for flux. Then we will look at our 3 dimensional
demo and discuss the equation for flux.
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In the 1-dimensional model, we are going to
model the particles of dye as random walkers
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on a line. Each random walker has an equal
probability of moving one step of length Delta
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x to the right or to the left during a time
step Delta t. The walkers move independently
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of each other.
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We make an assumption that Delta x and Delta
t are both small.
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In order to understand how the particles are
moving, we want to understand the flux through
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any given point.
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Recall that flux is flow per unit "area" per
unit time. Our random walk model is one dimensional,
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so we will define the flow of particles through
a single point over a time step Delta t to
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be flux.
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While we can look at the flux through any
point, for mathematical convenience, let us
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determine the flux through the point x + Delta
x over 2 at time t. This point is half way
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between the point x and x+delta x. Because
of the hypotheses of our random walk, any
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particle that is within a step length Delta
x to the left or the right of x + Delta x
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over 2 has a ½ probability of flowing through
the point during the next time step. So in
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order to find the flux, the first step is
to determine many particles are within our
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step distance Delta x from the point x + Delta
x over 2.
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Let the concentration of particles be denoted
by the function c(x,t), which is the number
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of particles per unit length at a time t.
To find the number of particles to the left
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of x + Delta x over 2, we could integrate
the concentration function over the interval
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of length Delta x centered about the point
x. However, because we have assumed that Delta
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x is small, we can approximate the concentration
function by the value of the concentration
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at x for the whole interval. So the number
of particles on the interval of length centered
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about the point x can be approximated by c(x,t)
times Delta x . The number of particles on
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the interval of length Delta x centered about
the point x + Delta x is approximately c(x+Delta
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x, t) times Delta x particles.
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We assume that any particle has 1/2 probability
of moving one step to the left or the right.
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Thus the flux through our point is given by
one half times the number of particles to
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the left minus one half times the number of
particles to the right a time t. We divide
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the entire expression by the time step, which
is the unit of time over which we are looking
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at the motion of particles.
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To dig a little deeper into this equation,
we can take a Taylor expansion of our concentration
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function c(x + Delta x, t) about x, holding
t fixed. This gives us the following expression,
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which is a polynomial in Delta x with coefficients
given by multiples of sequentially higher
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partial derivatives of the concentration function
c.
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Our equation for flux becomes this seemingly
more complicated equation.
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However, if we make an assumption that Delta
x grows proportionally to the square root
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of Delta t, in other words that Delta x squared
is proportional to Delta t:
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This simplifies the expression for flux because
only the first term has a significant contribution,
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and we are left with the following expression
for flux:
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[pause]
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You can do a table top experiment by placing
a small drop of dye in a narrow test tube
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and measuring the change in height of dye
with respect to the change in time in order
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to verify that the assumption we made is valid.
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Rewriting the constant term in front as some
diffusion constant D, this equation is commonly
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written as flux is equal to negative D times
the partial derivative of c with respect to
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x.
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The negative sign in this equation says that
the direction of net flux goes from a region
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of high concentration to a region of low concentration,
in the opposite direction as the concentration
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gradient. Why is this? If there are more particles
on one side of a point than the other, we
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suspect half of them flow through the point
on either side, so the net flow through the
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point is away from the highest concentration.
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This behavior is consistent with what we saw
with the dye in the fish tank.
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Now we want to extend this to 2-dimensions.
Here we have modeled a system of 2000 particles
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walking randomly in the plane. Each particle
can move a unit distance away from its current
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location in any direction with equal probability.
A profile of the concentration at each time
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step is displayed to the right.
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We change the view of the concentration to
be contour lines, and add some more particles
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to increase the accuracy of our computation
in order to add in the flux vector. In 2D,
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the flux is a flow per length per unit time,
and is a vector quantity. Observe that the
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flux is everywhere perpendicular to the level
sets, or contours of the concentration map,
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and it points away from the highest concentration.
In other words, this simulation suggests that
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the flux points in the direction of the negative
gradient of the concentration.
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The equation that describes this says that
flux J is equal to some constant, which we
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will call D, times the negative gradient of
the concentration: Compare to the equation
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we had in the 1-dimensional example. Here
the derivative is replaced by the gradient
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because the derivative is the 1-dimensional
analogue of the gradient.
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Now let's look back to our 3-dimensional example.
The flow profile seems to follow the same
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basic principles.
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Experiments and observations have shown that
the flux of particles per unit area is determined
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by some constant times the negative gradient
of concentration, just as in our discrete
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2-dimensional model:
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This equation is one form of Fick's first
law. It says that flux points along the negative
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gradient of the concentration.
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It turns out that this equation describes
the flux of many familiar quantities. Let's
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consider some examples:
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When students exit a classroom when class
ends shows the flux of people through the
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doorway points away from the highest concentration
of students.
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The second law of thermodynamics says that
heat flows from high to low temperatures.
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This says that the
flux is proportional (perhaps non-uniformly)
to the negative temperature gradient.
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Be aware that this is just one form of Fick's
first law. The most general form says that
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flux is proportional to the negative gradient
of the chemical potential.
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You may see the equation in this form in later
courses. In all of the examples that we have
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considered in this video, the gradient of
the concentration and the gradient of the
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chemical potential pointed in the same direction.
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To review:
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The gradient is a vector quantity that points
in the direction of the maximum slope of a
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scalar function.
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Fick's first law says that flux points along
the negative gradient of concentration.
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In order to understand Fick's First Law, we
first considered models in 1-d and 2-d, before
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trying to understand the description in 3-d.