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AP®︎/College Biology

Course: ap®︎/college biology   >   unit 2, facilitated diffusion, diffusion and passive transport.

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Diffusion I: An Introduction

by Heather MacNeill Falconer, M.A./M.S., Gina Battaglia, Ph.D., Anthony Carpi, Ph.D.

Listen to this reading

Did you know that the process of diffusion is responsible for the way smells travel from the kitchen throughout the house? In diffusion, particles move randomly, beginning in an area of higher concentration and ending in an area of lower concentration. This principle is fundamental throughout science and is very important to how the human body and other living things function.

Diffusion is the process by which molecules move through a substance, seemingly down a concentration gradient, because of the random molecular motion and collision between particles.

Many factors influence the rate at which diffusion takes place, including the medium through with a substance is diffusing, the size of molecules diffusing, the temperature of the materials, and the distance molecules travel between collisions.

The diffusion coefficient, or diffusivity, provides a relative measure at specific conditions of the speed at which two substances will diffuse into one another.

If you’ve ever made cookies and left the kitchen door open, you’re probably aware that the aroma spreads throughout the house. It is strongest in the kitchen, where the cookies are baking, a little less in the dining or living room, and least in the upstairs corner bedroom. And if the door is closed in the corner bedroom, the cookie scent is even weaker.

This is a delicious example of diffusion , or the movement of matter from a region of high concentration (the cookie pan in the kitchen) to a region of low concentration (the corner bedroom). This principle of diffusion is fundamental throughout science, from gas exchange in the lungs to the spread of carbon dioxide in the atmosphere to the movement of water from one side of a cell’s plasma membrane to the other. However, the concept of diffusion is rarely as simple as molecules moving from one place to another. Temperature, the size of the molecules involved, the distance molecules need to travel, the barriers they may encounter along the way, and other factors all influence the rate at which diffusion takes place.

  • Random walk: Molecular movement through a given space

The universe is in constant motion: from the orbiting of planets around the sun, to the movement of particles from one area to another. And while on a grand scale it may appear that there is a rationale to this movement – for example, the planets in our solar system have regular revolutions that can be predicted – in truth there is a great deal of motion that occurs randomly.

When we learn about diffusion , we often hear about the movement of particles from an area of high concentration to an area of low concentration, as if the particles themselves are somehow motivated to move in this direction. But this movement is in fact a by-product of what scientists refer to as the “random walk” of particles. Molecules do not move in straight paths from Point A to Point B. Instead, they interact with their environment , bumping into other molecules and barriers encountered along their way, as well as interacting with the medium through which they are moving.

The observation of the spontaneous, random movement of small particles was first recorded in the first century BCE . Lucretius, a Roman poet and philosopher, described the dust seen in sunbeams coming through a window (Figure 1):

You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves [i.e., spontaneously]… So the movement mounts up from the atoms and gradually emerges to the level of our senses, so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible.

Dust Particles

While Lucretius’s “dancing” particles were likely dust particles or pollen grains that are affected by air currents and other phenomenon, his description is a wonderfully accurate assessment of what goes on at the molecular level. Many scientists have explored this random molecular motion in a variety of contexts, most famously by the Scottish botanist Robert Brown in the 19 th century.

In 1828, while observing pollen granules suspended in water under a microscope, Brown discovered that the motion of the granules were “neither from currents in the fluid , nor from its gradual evaporation , but belonged to the particle itself.” After suspending various organic and inorganic substances in water and seeing this same inherent , random movement, he concluded that this random walk of particles – later termed Brownian motion in his honor – was a general property of matter that is suspended in a liquid medium. However, it would take nearly a century for scientists to mathematically quantify Brownian motion and demonstrate that this random movement of molecules dictates diffusion .

Comprehension Checkpoint

  • What causes random molecular movement?

About the same time that Brown was making his observations , a group of scientists including the French engineer Sadi Carnot and German physicist Rudolph Clausius were establishing a whole new field of scientific study: the field of Thermodynamics (see our Thermodynamics I module for more information). Clausius’s work in particular led to the development of the kinetic theory of heat – the idea that atoms and molecules are in motion and the speed of that motion is related to a number of things, including the heat of the substance. The molecules of a solid are generally considered to be locked in place (though they vibrate); however, the molecules of a liquid or a gas are free to move around, and they do: bumping in to one another or the walls of their container like balls on a pool table.

As molecules in a liquid or gas move through space, they bump into one another and follow random paths – moving in a straight line until something blocks their way and then bouncing off of that thing. This random molecular movement is constantly occurring and can be measured, giving a molecule’s mean free path – or, the average distance a particle moves between impacts with other particles.

It is this spontaneous and random motion that leads to diffusion . For example, as the scent molecules from baking cookies move into the air, they interact with air molecules – crashing into them and changing direction. Over time, these random processes will cause the scent molecules to disperse throughout the room. Diffusion is presented as a process in which a substance moves down a concentration gradient – from an area of high concentration to an area of low concentration. However, it is important to recognize that there is no directional force at play – the scent molecules are not pushed to the edge of the room because the concentration is lower there. It is the random movement of these molecules within the roomful of moving air molecules that causes them to evenly spread out throughout the entire space – bouncing off walls, moving through doors, and eventually moving through the whole house. In this way, it appears to move along a concentration gradient – from the kitchen oven to the most distant rooms of the house.

  • How concentration gradients work

It may sound like a paradox – the movement of molecules are random, yet at the same time appear to occur along a gradient – but in practice, it’s actually quite logical. A simple illustration of this process can be seen using a glass of water and food coloring. When a drop of food coloring enters the water, the food coloring molecules are highly concentrated at the location where the dye molecules meet the water molecules, giving the water in that area a very dark color (Figure 2). The bottom of the glass initially has few or no food coloring molecules and so remains clear. As the food coloring molecules begin to interact with the water molecules, molecular collisions cause them to move randomly around the glass. As collisions continue, the molecules spread out, or diffuse , over space.

Figure 2: Diffusion of a purple dye in a liquid.

Figure 2 : Diffusion of a purple dye in a liquid.

Eventually, the molecules spread throughout the entire glass, becoming evenly distributed and filling the space. At this point, the molecules have reached a state of equilibrium in which no net diffusion is taking place and the concentration gradient no longer exists. In this state, the molecules are still moving haphazardly and colliding with each other; we just can’t see that motion because the water and color molecules are evenly dispersed throughout the space. Once equilibrium has been reached, the probability that a molecule will move from the top to the bottom is equal to the probability a molecule will move from the bottom to the top.

  • Temperature and other factors influencing the rate of diffusion

We know that diffusion involves the movement of particles from one place to another; thus, the speed at which those particles move affects diffusion. Since molecular motion can be measured by the heat of an object, it follows that the hotter a substance is the faster diffusion will take place in that substance. (Click the animation below to see how temperature affects diffusion.) If you were to repeat your food coloring and water experiment comparing a glass of cold to a glass of hot water, you would see that the color disperses much more quickly in the hot water. But what other factors influence the speed, or rate, at which diffusion takes place?

The Effect of Temperature on Diffusion

Interactive Animation: The Effect of Temperature on Diffusion

  • Size matters

In 1829, the Scottish physical chemist Thomas Graham first quantified diffusion behavior before the idea of atoms and molecules was widely established. Basing his observations on real-life “substances,” Graham measured the diffusion rates of gases through plaster plugs, fine tubes, and small orifices that were meant to slow down the diffusion process so that he could quantify it. One of his experiments , detailed in Figure 3, used an apparatus with the open end of a tube sitting in a beaker of water and the other end sealed with a plaster stopper containing holes large enough for gases to enter and leave the tube. Graham filled the open end of the tube with various gases (as indicated by the red tube in Figure 3), and observed the rate at which the gases effused , or escaped through the plaster plug. If the gas effused from the tube faster than the air outside of the tube moved in, the water level in the tube would rise. On the other hand, if the outside air moved through the plaster faster than the gas in the tube escaped to the outside, the water level in the tube would go down. He used the rate of change in the water level to determine the relative rate at which the different gases diffused into air.

Figure 3: Thomas Graham's experiment to measure the diffusion rates of gases.

Figure 3 : Thomas Graham's experiment to measure the diffusion rates of gases.

Graham experimented with many combinations of different gases and published his findings in an 1829 publication of the Quarterly Journal of Science, Literature, and Art titled “A Short Account of Experimental Researches on the Diffusion of Gases Through Each Other, and Their Separation by Mechanical Means.” He stated that when gases come into contact with each other, “indefinitely minute volumes” of the gases spontaneously intermix with each other until they reach equilibrium (Graham, 1829). However, he discovered that different types of gases did not mix at the same rate – rather, the rates at which two gases diffuse is inversely proportional to the square root of their densities, a relationship now known as Graham’s law . Although Graham’s original relationship used density , or mass per unit volume , the modern form of the equation uses molar mass, or the mass of one mole of a substance.

What Graham showed was that the molecular weight of a molecule directly affects the speed at which that molecule can move. Graham’s work actually helped lay the foundations of kinetic molecular theory because it recognized that at a given temperature, a heavy molecule would move more slowly than a light molecule. In other words, more kinetic energy is needed to move a large molecule at the same speed as a small molecule. You can think of it this way: A small push will get a tennis ball rolling quickly; however, it takes a much harder push to move a bowling ball at the same speed. At a given temperature, small molecules move faster, and will diffuse more quickly than large ones. View the animation below to see how atomic mass affects diffusion .

The Effect of Atomic Mass on Diffusion

Interactive Animation: The Effect of Atomic Mass on Diffusion

  • Solution properties

Graham later studied the diffusion of salts into liquids and discovered that the diffusion rate in liquids is several thousand times slower than in gases. This seems relatively obvious to us today, as we know that the molecules of a gas move faster and are more spread out than molecules in a liquid. Therefore, the movement of one substance within a gas occurs more freely than in a liquid. Diffusion in liquids is proportional to temperature, as it is in gases, as well as to the viscosity of the specific liquid into which the material is diffusing. (View the animation below to compare diffusion in gases and liquids.) Diffusion, in fact, can even take place in solids . While this is a very slow process , Sir William Chandler Roberts-Austen, a British metallurgist, fused gold plates to the end of cylindrical rods made of lead. He analyzed the lead rods after a period of 31 days and actually found that gold atoms had “flowed” into the solid rods.

The Effect of State on Diffusion: Gases versus Liquids

Interactive Animation: The Effect of State on Diffusion: Gases versus Liquids

  • Concentration and the diffusion coefficient

While we have talked extensively about diffusion and concentration gradients, it was not until the mid-1800s when a German-born physicist and physiologist named Adolf Fick built upon Graham’s work and introduced the notion of a diffusion coefficient, or diffusivity, to characterize how fast molecules diffuse .

In his 1855 publication “On Diffusion” in Annalen der Physik , Fick described an experimental setup in which he connected cylindrical and conical tubes with solid salt crystals at the bottom to an “infinitely large” reservoir filled with freshwater (Figure 4). The solid salt crystals dissolved into the water in the tubes and diffused toward the water reservoir. A stream of freshwater swept the saltwater out of the reservoir. This stream of water kept the salt concentration at the very top of the tubes (the point where the salt solution met the water reservoir) close to zero. The dissolving salt at the bottom of the tube maintained a high salt concentration in the water at that end of the tube. Because the tubes had a different shape (conical versus cylindrical), the concentration gradient in the tubes differed, setting up a system in which diffusion could be compared in relation to a concentration gradient.

Figure 4: Fick's experimental setup in which he connected cylindrical and conical tubes to a reservoir filled with freshwater. (Image from the 1903 publication, Collected Works, I.  Stahel’sche Verlags-Anstalt, Würzburg: Germany.)

Figure 4 : Fick's experimental setup in which he connected cylindrical and conical tubes to a reservoir filled with freshwater. (Image from the 1903 publication, Collected Works, I . Stahel’sche Verlags-Anstalt, Würzburg: Germany.)

Fick then calculated the diffusion rate of the salt by measuring the amount of salt that passed through the top of the respective tubes (just before they met the freshwater in the reservoir) within a given time period. He discovered that the movement rate of the salt solution into the water reservoir depended on the concentration difference between the solution at the bottom of the tube and the concentration of the solution leaving the tube and entering the reservoir. In other words – the higher the concentration of salt at the top of the tube, the faster it diffused into the water reservoir. You can see how concentration affects diffusion in the animation below.

The Effect of Concentration on Diffusion

Interactive Animation: The Effect of Concentration on Diffusion

After studying the phenomenon, Fick hypothesized that the relationship between the concentration gradient and the diffusion rate was similar to what Joseph Fourier, a French mathematician and physicist, found in his study of heat conduction in 1822. Fourier had described the rate of heat transfer through a substance as proportional to the difference in temperature between two regions. Heat moves from warmer to cooler objects, and the greater the temperature difference between the two objects, the faster the heat moves. (This is why your mug of hot coffee cools off much faster outside on a cold morning than when you leave it in your heated apartment). Using Fourier’s law of thermal conduction as a model , Fick created a mathematical framework for the movement of salt into the water, proposing that the diffusion rate of a substance is proportional to the difference in concentration between the two regions. What this means for diffusion of a substance is that if the concentration of a given substance is high in relation to the substance it is diffusing into (e.g., food coloring into water), it will diffuse faster than if the concentration difference is low (e.g., food coloring into food coloring). The application of a successful principle from one branch of science to another is not uncommon, and Fick was a classic example of this process . Fick knew of Fourier’s work because he had modeled his experimental apparatus on that of Fourier. Thus it was natural for him to apply Fourier’s law to diffusion. While he had no way to know that the underlying mechanism of heat conduction and diffusion were both based on atomic collisions (in fact, some researchers at the time still doubted the existence of atoms), he had a feeling. That feeling, and the existence of atoms themselves, would be mathematically proven some 50 years later when Albert Einstein published his seminal work, Investigations on the Theory of the Brownian Movement (Einstein, 1905).

The diffusion coefficient, or diffusivity D , defined by Fick is a proportionality constant between the diffusion rate and the concentration gradient. The diffusion coefficient is defined for a specific solute-solvent pair, and the higher the value for the coefficient, the faster two substances will diffuse into one another. For example, at 25°C the diffusivity of gaseous air into gaseous water is 0.282 cm 2 /sec (Cussler, 1997). At the same temperature, the diffusivity of dissolved air into liquid water is 2.00 x 10 -5 cm 2 /sec, a much lower number than that for the two gases, representing the much slower diffusion rate in liquids compared to gases. And the diffusivity of dissolved helium into liquid water at 25°C is 6.28 x 10 -5 cm 2 /sec – higher than that of dissolved air, representing the smaller size of helium atoms compared to the nitrogen and oxygen molecules in air.

  • Distance molecules travel

Yet another factor that influences the rate at which diffusion occurs is the distance a molecule travels before bumping into something (referred to as a molecule’s mean free path). Imagine taking a container filled with a gas and putting it under pressure so that the molecules in the gas are squeezed together. This would slow the rate of diffusion through the gas because the molecules travel a shorter distance before colliding with something else and changing direction. (The animation below shows the effect of pressure on diffusion.)

The Effect of Pressure on Diffusion

Interactive Animation: The Effect of Pressure on Diffusion

This is an important factor affecting the difference in diffusion rates in gases versus liquids versus solids ; because gas particles are the most spread out of the three, molecules travel the furthest between collisions and diffusion occurs most rapidly in this state (Figure 5).

Figure 5 : The three states of matter at the atomic level: gas, liquid, and solid.

To fully understand why we can smell the cookies baking in the kitchen from the bedroom we also have to consider another process at work here – advection . Advection involves the transfer of a material or heat due to the movement of a fluid . So, because people walk through the rooms of your house and because heat rises from your radiators, the air is constantly moving, and that movement carries and mixes the scent molecules in your house. In many situations (such as your house), the effects of advection exceed those of diffusion , but these processes work in tandem to bring you the cookie smell.

From the traveling smells of cookies to the dissolving of salt into water, diffusion is a process happening around (and within!) us every second of every day. It is a process that is critical to moving oxygen across the membranes of our lungs, moving nutrients through soil to be taken up by plants, dispersing pollutants that are released into the atmosphere , and a whole host of other events that are necessary for life to exist.

Table of Contents

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Learning Objectives

After completing the lab, the student will be able to:

  • Explain or define the term diffusion.
  • Explain how different media affect the rate of diffusion.

Activity 1: Pre-Assessment

  • What happens when an air freshener is sprayed in a corner? What is the name of the process that causes the molecules to move?
  • Do you think that the rate of the air freshener molecules moving would change if the room temperature was warmer or colder? Why or why not?
  • Discuss the answers to questions 1 and 2 with the class.

Activity 1: Diffusion

The movement of molecules from a higher concentrated area to a wider and less concentrated area is referred to as diffusion . For example, you can smell the aroma of food flowing through the atmosphere as you walk towards a cafeteria. Molecules collide with each other and are in constant motion because of their kinetic energy. This activity propels molecules to move where there is a less concentrated area. Therefore, the net movement of molecules is always from a tightly concentrated area to a less tightly packed area. Osmosis is the process of water diffusion through a selectively permeable membrane. In body systems, various constituents such as gases, liquids, and solids are dissolved in water when they flow through the cell membrane from a highly concentrated place to a less concentrated area in bodily systems. In a solution, the dissolved substance is called the solute and the substance in which the solute is dissolved is called the solvent.

Diffusion is the movement of molecules from an area where the molecule is highly concentrated to an area of low concentration, as illustrated in Figure 6.1. The rate of diffusion is dependent upon the temperature of a system, molecular size, and the medium through which diffusion is occurring (i.e., semi-solid, liquid, air). In this activity, we will be observing the diffusion of a dye through a beaker of water and through agar (a gelatinous substance), diffusion as a function of temperature, and diffusion as a function of molecular weight.

Illustration of the movement of molecules in two beakers of liquid showing how the molecules more from areas of higher concentration, where they are closer together, to areas of lower concentration, where they are more spread out.

Safety Precautions

  • Inform your teacher immediately of any broken glassware, as it could cause injuries.
  • Clean up any spilled water or other fluids to prevent other people from slipping.
  • Be careful with the dye as it can stain your clothes, and it should not be ingested.
  • Wash your hands with soap and water after completion of the activity.

For this activity, you will need the following:

  • Three 250 mL beakers
  • Food coloring
  • Agar plates
  • Potassium permanganate
  • Methylene blue
  • Thermometer
  • Refrigerator
  • Clock or timer

For this activity, you will work in groups of four .

Structured Inquiry

Step 1: Measure 200 mL of room temperature water in a beaker. Put three drops of food coloring into the water. Time how long it takes for the dye to completely diffuse throughout the water. Record the time and describe in your notebook what you observe. Create a data table for your observations.

Step 2: Hypothesize/Predict: Predict what would happen to the rate of diffusion if you had beakers with both very hot and very cold water in them. Add your predictions to the data table you created in step 1.

Step 3: Student-led Planning: Determine how diffusion of the food color would be affected when the water is either very hot or very cold. Use a thermometer and record the temperature for each. Use a timer to measure how long it takes for complete diffusion to occur in all scenarios.

Step 4: Critical Analysis: Create a graph that shows how the diffusion rate is affected because of temperature change. Are the predictions you made in step 2 supported by your data? Why or why not? What methods could you use to improve your results? Discuss with your group and then write your answers in your notebook.

Guided Inquiry

Step 1: Gather four agar plates and the three dyes, provided by your instructor, that differ in molecular size: Congo red (mol. wt. 696.66 g/mol), methylene blue (319.85 g/mol), and potassium permanganate (mol. wt. 158.03).

Step 2: Hypothesize/Predict: How would the rate of diffusion of a molecule through a gel compare to its rate of diffusion through water? How would the rate of diffusion differ between molecules of different molecular sizes? Write your ideas in your notebook.

Step 3: Student-led planning: Use 1 plate for determining how molecular size affects diffusion using the 3 dyes. Determine how best to measure movement of the dye in an agar plate. Be sure to keep the dyes far enough apart so that they do not touch once they start diffusing. Get your instructor’s approval before proceeding with the experiment. Measure the distance that the dye spreads in 20-minute intervals for 1 hour.

Step 4: Examine the effect of temperature on the rate of diffusion for 1 dye of your choosing. With your group, determine 3 temperatures that would be appropriate. Measure the diameter of the dye spread for each. Write the results in your notebook.

Step 5: Critical Analysis: Rank all 3 dyes in terms of diffusion rate. What was the relationship between diffusion rate and molecular size? What is the relationship between temperature and diffusion rate? Discuss your answers with your group and write them in your notebook.

Assessments

  • In a system, there is a concentration of molecules. However, on the outside, there is little to no concentration of this particular molecule. In which direction would the molecules be moving more so than the other direction?
  • Diffusion is affected by what factors?
  • Dye tends to move faster in warmer temperatures. Why is this?

Lab Manual for Biology 2e Part I, 2nd edition Copyright © 2022 by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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March 29, 2021

New research provides insights into the process of diffusion in living systems

by Erica K. Brockmeier, University of Pennsylvania

New research provides insights into the process of diffusion in living systems

Adrop of food coloring slowly spreading in a glass of water is driven by a process known as diffusion. While the mathematics of diffusion have been known for many years, how this process works in living organisms is not as well understood.

Now, a study published in Nature Communications provides new insights on the process of diffusion in complex systems . The result of a collaboration between physicists at Penn, the University of Chile, and Heinrich Heine University Düsseldorf, this new theoretical framework has broad implications for active surfaces, such as ones found in biofilms, active coatings, and even mechanisms for pathogen clearance.

Diffusion is described by Fick's laws: Particles, atoms, or molecules will always move from a region of high to low concentration. Diffusion is one of the most important ways that molecules move within the body. However, for the transport of big objects over large distances, standard diffusion becomes too slow to keep up.

"That's when you need active components to help transport things around," says study co-author Arnold Mathijssen. In biology, these actuators include cytoskeletal motors that move cargo vesicles in cells, or cilia that pump liquid out of human lungs. When many actuators accumulate on a surface, they are known as "active carpets." Together, they can inject energy into a system in order to help make diffusion more efficient.

Mathijssen, whose research group studies the physics of pathogens, first became interested in this topic while studying biofilms with Francisca Guzmán-Lastra, an expert on the physics of active matter, and theoretical physicist Hartmut Löwen. Biofilms are another example of active carpets since they use their flagella to create "flows" that pump liquid and nutrients from their environment. Specifically, the researchers were interested in understanding how biofilms are able to sustain themselves when access to nutrients is limited. "They can increase their food uptake by creating flows, but this also costs energy. So, the question was: How much energy do you put in to get energy out?" says Mathijssen.

But studying active carpets is difficult because they don't align neatly with Fick's laws, so the researchers needed to develop a way to understand diffusion in these non-equilibrium systems, or ones that have added energy. "We thought that we could generalize these laws for enhanced diffusion, when you have systems that do not follow Fick's laws but may still follow a simple formula that is widely applicable to many of these active systems," Mathijssen says.

After figuring out how to connect the math needed to understand both bacterial dynamics and Fick's laws, the researchers developed a model similar to the Stokes–Einstein equation, which describes the relationship with temperature and diffusion, and found that microscopic fluctuations could explain the changes they saw in particle diffusion. Using their new model, the researchers also found that the diffusion generated by these small movements is incredibly efficient, allowing bacteria to use just a small amount of energy to gain a large amount of food.

"We've now derived a theory that predicts the transport of molecules inside cells or close to active surfaces. My dream would be that these theories would be applied in different biophysical settings," says Mathijssen. His new research lab at Penn will start working on follow-up experiments to test out these new models. They plan to study active diffusion both in biological and engineered microscopic systems.

Mathijssen, who is also involved on a project related to the spread of COVID-19 in food-processing facilities, says that the cilia in lungs are another important example of active carpets in biology, especially since they serve as the first line of defense against pathogens like COVID-19. He says, "That would be another very important thing to test, whether this theory of active carpets may be linked to the theory of pathogen clearance in the airways."

Journal information: Nature Communications

Provided by University of Pennsylvania

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Module 4: Diffusion and Osmosis

Diffusion and osmosis.

The cell membrane plays the dual roles of protecting the living cell by acting as a barrier to the outside world, yet at the same time it must allow the passage of food and waste products into and out of the cell for metabolism to proceed. How does the cell carry out these seemingly paradoxical roles? To understand this process you need to understand the makeup of the cell membrane and an important phenomenon known as diffusion.

Diffusion is the movement of a substance from an area of high concentration to an area of low concentration due to random molecular motion. All atoms and molecules possess kinetic energy, which is the energy of movement. It is this kinetic energy that makes each atom or molecule vibrate and move around. (In fact, you can quantify the kinetic energy of the atoms/molecules in a substance by measuring its temperature.) The moving atoms bounce off each other, like bumper cars in a carnival ride. The movement of particles due to this energy is called Brownian motion. As these atoms/molecules bounce off each other, the result is the movement of these particles from an area of high concentration to an area of low concentration. This  is diffusion. The rate of diffusion is influenced by both temperature (how fast the particles move) and size (how big they are).

Screen Shot 2015-07-09 at 1.39.48 PM

Part 1: Brownian Motion

In this part of the lab, you will use a microscope to observe Brownian motion in carmine red powder, which is a dye obtained from the pulverized guts of female cochineal beetles.

  • Glass slide
  • Carmine red powder
  • Obtain a microscope slide and place a drop of tap water on it.
  • Using a toothpick, carefully add a very minuscule quantity of carmine red powder to the drop of water and add a coverslip.
  • Observe under scanning, low, and then high power.

Lab Questions

  • Describe the activity of the carmine red particles in water.
  • If the slide were warmed up, would the rate of motion of the molecules speed up, slow down, or remain the same? Why?

Part 2: Diffusion across a Semipermeable Membrane

Because of its structure, the cell membrane is a semipermeable membrane. This means that SOME substances can easily diffuse through it, like oxygen, or carbon dioxide. Other substances, like glucose or sodium ions, are unable to pass through the cell membrane unless they are specifically transported via proteins embedded in the membrane itself. Whether or not a substance is able to diffuse through a cell membrane depends on the characteristics of the substance and characteristics of the membrane. In this lab, we will make dialysis tubing “cells” and explore the effect of size  on a molecule’s ability to diffuse through a “cell membrane.”

Screen Shot 2015-07-09 at 2.10.48 PM

The following information might be useful in understanding and interpreting your results in this lab:

  • Atomic formula: C 20 H 14 O 4
  • Atomic mass: 318.32 g/mol
  • Color in acidic solution : Clear
  • Color in basic solution: Pink
  • Atomic formula: I or I2
  • Atomic mass: 126 g/mol
  • Atomic formula: (C 6 H 10 O 5 )n
  • Atomic mass: HUGE!
  • Color in Iodine: Bluish
  • Atomic formula: NaOH
  • Atomic mass: 40.1 g/mol
  • Acid/Base: Base
  • 2 pieces of dialysis tubing
  • Phenolphthalein
  • Starch solution
  • Using a wax pencil, label one beaker #1. Label the other beaker #2.
  • Fill beaker #1 with 300 ml of tap water, then add 10 drops of 1 M NaOH. Do not spill the NaOH—it is very caustic!
  • Fill beaker #2 with 300 ml of tap water, then add iodine drops drop by drop until the solution is bright yellow.
  • Now prepare your 2 dialysis tubing “bags.” Seal one end of each dialysis tube by carefully folding the end “hotdog style” 2 times, then “hamburger style” 1 time. Tie the folded portion of the tube securely with string. It is critical that your tubing is tightly sealed, to prevent leaks.
  • Add 10 ml of water and three drops of phenolphthalein to one of your dialysis tube bags. Seal the other end of the bag by carefully folding and tying as before.
  • Thoroughly rinse the bag containing phenolphthalein, then place it in into the beaker containing the NaOH.
  • Add 10 ml of starch solution to the other dialysis tube. Again seal the bag tightly and rinse as above. Place this bag containing the starch solution into beaker #2.
  • Let diffusion occur between the bags and the solutions in the beakers.

Screen Shot 2015-07-09 at 2.12.02 PM

Record the colors (below) and label contents inside and outside the bags (above):

  • Which substance diffused across the membrane in beaker #1? How do you know?
  • Which substance diffused across the membrane in beaker #2? How do you know?
  • Why might some ions and molecules pass through the dialysis bag while others might not?

Part 3: Osmosis and the Cell Membrane

Osmosis is the movement of water across a semipermeable membrane (such as the cell membrane). The tonicity of a solution involves comparing the concentration of a cell’s cytoplasm to the concentration of its environment. Ultimately, the tonicity of a solution can be determined by examining the effect a solution has on a cell within the solution.

By definition, a hypertonic solution is one that causes a cell to shrink. Though it certainly is more complex than this, for our purposes in this class, we can assume that a hypertonic solution is more concentrated  with solutes than the cytoplasm. This will cause water from the cytoplasm to leave the cell, causing the cell to shrink. If a cell shrinks when placed in a solution, then the solution is hypertonic to the cell.

If a solution is hypotonic to a cell, then the cell will swell when placed in the hypotonic solution. In this case, you can imagine that the solution is less concentrated  than the cell’s cytoplasm, causing water from the solution to flow into  the cell. The cell swells!

Finally, an isotonic solution is one that causes no change in the cell. You can imagine that the solution and the cell have equal concentrations, so there is no net movement of water molecules into or out of the cell.

In this exercise, you will observe osmosis by exposing a plant cell to salt water.

What do you think will happen to the cell in this environment? Draw a picture of your hypothesis.

  • Elodea leaf
  • Microscope slide
  • 5% NaCl solution
  • Remove a leaf from an Elodea plant using the forceps.
  • Make a wet mount of the leaf. Use the pond water to make your wet mount.
  • Observe the Elodea cells under the compound microscope at high power (400 X) and draw a typical cell below.
  • Next, add several drops of 5% salt solution to the edge of the coverslip to allow the salt to diffuse under the coverslip. Observe what happens to the cells (this may require you to search around along the edges of the leaf). Look for cells that have been visibly altered.

Draw a typical cell in both pond and salt water and label the cell membrane and the cell wall.

  • What do you see occurring to the cell membrane when the cell was exposed to salt water? Why does this happen?
  • Describe the terms hypertonic, hypotonic and isotonic.
  • How would your observations change if NaCl could easily pass through the cell membrane and into the cell?

Part 4: Experimental Design

You and your group will design an experiment to determine the relative molecular weights of methylene blue and potassium permanganate. You may use a petri dish of agar, which is a jello-like medium made from a polysaccharide found in the cell walls of red algae. You will also have access to a cork borer and a small plastic ruler.

  • 1 petri dish of agar
  • Methlylene blue
  • Potassium permanganate

Your experiment design should include all of the following portions:

  • Experimental design
  • Conclusions
  • Further questions/other comments
  • Biology Labs. Authored by : Wendy Riggs . Provided by : College of the Redwoods. Located at : http://www.redwoods.edu . License : CC BY: Attribution
  • Osmotic pressure on blood cells diagram. Authored by : LadyofHats. Located at : https://commons.wikimedia.org/wiki/File:Osmotic_pressure_on_blood_cells_diagram.svg . License : Public Domain: No Known Copyright
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Diffusion of Innovations Theory: Definition and Examples

Clay Halton is a Business Editor at Investopedia and has been working in the finance publishing field for more than five years. He also writes and edits personal finance content, with a focus on LGBTQ+ finance.

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What Is the Diffusion of Innovations Theory?

The diffusion of innovations theory is a hypothesis outlining how new technological and other advancements spread throughout societies and cultures, from introduction to widespread adoption. The diffusion of innovations theory seeks to explain how and why new ideas and practices are adopted, including why the adoption of new ideas can be spread out over long periods.

The way in which innovations are communicated to different parts of society and the subjective opinions associated with the innovations are important factors in how quickly diffusion—or spreading—occurs. This theory is frequently referred to when companies are developing a marketing strategy for new products and developing market share ,

Key Takeaways

  • The diffusion of innovations theory describes the pattern and speed at which new ideas, practices, or products spread through a population.
  • The main players in the theory are innovators, early adopters, early majority, late majority, and laggards.
  • In marketing, this diffusion of innovations theory is often applied to help understand and promote the adoption of new products.
  • The diffusion of innovations theory can also be used in areas such as public health to encourage populations to adopt new, healthy behaviors.

Understanding the Diffusion of Innovations Theory

The diffusion of innovations theory was developed by E.M. Rogers, a communication theorist at the University of New Mexico, in 1962. The theory explains the passage of a new idea through stages of adoption by different people who participate in or begin using the new idea. The main people in the diffusion of innovations theory are:

  • Innovators: Those who are open to risks and the first to try new ideas
  • Early adopters: People who are interested in trying new technologies and establishing their utility in society
  • Early majority: Those who pave the way for the use of an innovation within mainstream society and are part of the general population
  • Late majority: People who follow the early majority into adopting the innovation as part of their daily life and are also part of the general population
  • Laggards: People who lag behind the general population in adopting innovative products and new ideas

In general, innovators and early adopters are open to the possibility of risk that comes with trying our new innovations, technologies, or ideas. Laggards, on the other hand, are risk-averse and set in their ways of doing things. Eventually, the integration of an innovation into mainstream society makes it impossible for them to conduct their daily life (and work) without it. As a result, they are forced to begin using it.

The "new ideas" in the diffusion of innovations theory can be things like ideas, technologies, goods, services, or behaviors.

The diffusion of innovations theory was developed in part by integrating previous sociological theories of behavioral change. Factors that affect the rate of innovation diffusion include the mix of rural to urban within a society's population, the society's level of education, and the extent of industrialization and development. Different societies are likely to have different adoption rates —the rate at which members of a society accept a new innovation.

Adoption rates for different types of innovation vary. For example, a society may have adopted the internet faster than it adopted the automobile due to cost, accessibility, and familiarity with technological change.

Examples of the Diffusion of Innovations Theory

While the diffusion of innovations theory was developed during the mid-1900s, most new technologies in human progress, whether it is the printing press during the 16th century or the internet in the 20th century, have followed a similar path to widespread adoption.

The diffusion of innovations theory is extensively used by marketers to promote the adoption of their products. For example, marketers can find an set of people passionate about the product to receive it for free in return for sharing it widely. These early adopters are responsible for evangelizing its utility to mainstream audiences.

A recent example of this method is Facebook. It started off as a product targeted at students and professionals in educational institutions. As students' use increased beyond school, the social media site spread to mainstream society and across borders.

Influencer marketing is another use of the diffusion of innovations theory. Social media influencers are often contacted by brands with new products or services. The influencers become early adopters who use and post about the new product, normalizing it for a mainstream audience and causing its use to spread.

The diffusion of innovations theory is also used to design public health programs. Again, a set of people are chosen as early adopters of a new technology or practice and spread awareness about it to others. However, cultural limitations or people's access to resources and social support for the new behavior can impede these kinds of health programs from being successful. In the context of public health, this type of behavior diffusion is also more useful for encouraging people to adopt positive behaviors, rather than limit or cease negative behaviors.

What Are the Steps In the Diffusion of Innovations?

Diffusion happens through a five-step process of decision-making. The five steps are awareness, interest, evaluation, trial, and adoption. Rogers renamed these knowledge, persuasion, decision, implementation, and confirmation in later editions of his book.

What Are Barriers to Adoption in the Diffusion of Innovations?

At any point in the decision-making process, an individual might decide against adopting an innovation, usually due to some kind of barrier. These barriers are usually the usage or value of the innovation, the risk associated with adopting something new, or psychological factors such as cultural stigma.

What Are Some Areas Where Diffusion of Innovations Theory Applies?

In addition to marketing and public health, other areas where the diffusion of innovations theory has been used include agriculture, social work, communication, and criminal justice.

The diffusion of innovations theory describes how new ideas, behaviors, technologies, or goods spread through a population gradually, rather than all at once. Adoption starts with innovators and early adopters, then spreads through the population to the early majority and late majority. Laggards are the last ones to adopt a new innovation.

The diffusion of innovations theory can be applied to marketing strategies for new products, for example through influencer marketing. It can also be applied to fields such as public health, criminal justice, and communications. Though it has limitations in how it can be used in those areas, it can still be a helpful way to understand how technologies, goods, services, ideas, and behaviors spread through a population.

Boston University School of Public Health. " Behavioral Change Models ."

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Diffusion Decision Model: Current Issues and History

Roger ratcliff.

The Ohio State University, University of Melbourne, University of Newcastle, and The Ohio State University

Department of Psychology, 1835 Neil Avenue, Columbus, OH, 43210

Philip L. Smith

Melbourne School of Psychological Sciences, Level 12, Redmond Barry Building 115, University of Melbourne, Parkville, VIC 3010

Scott D. Brown

School of Psychology, University of Newcastle, Australia, Aviation Building, Callaghan, NSW 2308

Gail McKoon

There is growing interest in diffusion models to represent the cognitive and neural processes of speeded decision making. Sequential-sampling models like the diffusion model have a long history in psychology. They view decision making as a process of noisy accumulation of evidence from a stimulus. The standard model assumes that evidence accumulates at a constant rate during the second or two it takes to make a decision. This process can be linked to the behaviors of populations of neurons and to theories of optimality. Diffusion models have been used successfully in a range of cognitive tasks and as psychometric tools in clinical research to examine individual differences. In this article, we relate the models to both earlier and more recent research in psychology.

Modeling Simple Decision-Making

Decision-making is intimately involved in all of our everyday activities. Many decisions are made rapidly and at a low level cognitively, for example, deciding whether to drive left or right round a car in front. Others, such as deciding which candidate to vote for or which car to buy, are made at a higher level with prolonged deliberation. The diffusion models we discuss are of the former kind. In the real world, they involve a rapid matching of a perceptual representation to stored knowledge in memory, which allows us to identify things in our immediate surroundings and determine how we should respond to them. Much of what we have learned about such decisions comes from laboratory tasks in which people are asked to make fast two-choice decisions. The measures of performance are typically response times (RTs) and the probabilities of making the two choices. Researchers are usually interested in how and why RTs and choice probabilities change across experimental conditions, for example, whether a person tries to respond as quickly as possible or as accurately as possible.

There have been a moderate number of models for these tasks and most assume accumulation of noisy evidence to decision criteria representing each of the two choices. The models can include one versus two accumulators , decision rules that are relative or absolute, models with drift rate constant or varying over time, discrete or continuous time evidence, stochastic versus deterministic, and models with inhibition and decay. [ 1 ] showed the relationships between the models along with a detailed evaluation of the models ( Figure 1 , Key Figure).

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The standard model that we will discuss was developed by Ratcliff in the 1970's [ 2 ] and has only changed in assuming a single diffusion process instead of racing processes [ 3 ] and in adding across trial variability in starting point [ 4 - 5 ] and nondecision time [ 6 ]. In this model ( Figure 2A ), evidence about a stimulus from perception or memory accumulates from a starting point to a boundary or threshold (i.e., a criterion), one boundary for each choice. The boundaries represent the amount of evidence that must be accumulated before a response is made. The accumulation process is noisy; at each moment in time, the evidence might point to one or the other of the two boundaries, but more often to the correct than the incorrect one.

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A shows two (irregular) simulated paths in the diffusion model (green). The blue curves represent RT distributions for correct responses (top) and errors (bottom). The red lines represent the fastest, medium, and slowest responses.

B shows the effect of lowering drift rate by a fixed amount. The black double arrows show the effect on fast, medium, and slow average drift rates and the magenta arrows show the effect on the fastest and slowest responses from the blue RT distributions. There is a small change in the leading edge of the distribution and a large change in the tail.

C shows the effect of moving a boundary away from the starting point (a-speed to a-accuracy, the blue dotted arrow) to represent a speed-accuracy manipulation (both boundaries would move in a real experiment). The magenta arrows show the effect on the fastest and slowest responses from the blue RT distributions. There is a moderate change in the leading edge of the distribution and a large change in the tail. The difference in effects between B and C discriminates manipulations that change boundaries from manipulations that change drift rates.

D shows how a bias toward the A response can be modeled by a change in the starting point (blue dotted arrow with the starting point moving from the black line to the red line). RT distributions change as in C.

E shows how a bias toward the A response can be modeled by a change in the zero point of drift rate (blue dotted arrow with the zero point moving from the black line to the red line).

F shows the effect of a change in the zero point of drift rate (from E ). Drift rate is first symmetric (black arrows) and then biased toward A (the red arrows). RT distributions change as in B.

The parameters of the model are boundary separation ( a ), starting point ( z ), drift rate ( v , one of each condition), nondecision time ( T er ) which is the duration of encoding and response output processes and the transformation from the stimulus representation to a decision-relevant representation. Parameters of the model are assumed to vary from trial to trial, drift rate is normally distributed with standard deviation h, starting point and nondecision time are assumed to have rectangular distributions with ranges s z and s t respectively.

The main components of the model for the decision process represent the rate of accumulation and the settings of the boundaries. In the figure, the boundaries are set at 0 and a with starting point z . Evidence accumulates in a noisy fashion, and the average rate of accumulation is called the “drift rate”. In addition, there are nondecision components: encoding the evidence from a stimulus that will drive the decision process, extracting the dimension(s) of the stimulus that form the basis of the decision from the stimulus or memory, and executing a response. These nondecision components are combined and labeled the “nondecision” component, which has a mean time of T er . To set the measurement scale, one parameter of the model must be fixed (otherwise, e.g., doubling all the rates of evidence accumulation while also doubling the boundary separation would not change the model's predictions). In theory, any model parameter could be fixed; in practice, usually the parameter governing moment-by-moment variability in evidence accumulation is fixed. Concise introductions to this model, which we will refer to as “the diffusion model” in this article, are available elsewhere [ 7 ], as are more general comparative studies of the large class of sequential sampling models [ Figure 1 , Key Figure].

The diffusion model and others like it have become increasingly influential over the past 10 to 15 years as models of the psychological and neural processes involved in decision making. Box 1 gives a comprehensive list of advantages of the standard model and Box 2 presents a list of the paradigms and areas of research to which it has been applied. There are three main reasons for this success. First, the models account for all the behavioral data, namely accuracy and the shapes and locations of the distributions of RTs for correct responses and for incorrect responses. Second, they have been linked to neural processing for single cells and populations of neurons and they have been linked to aggregate behavior as measured by electroencephalography (EEG), functional magnetic resonance imaging (fMRI), and other imaging methods (Box 3 discusses explicit links between neural models and diffusion models). Third, they have been successful in explaining decision making across wide domains of psychology such as aging, child development, various clinical populations, and animal species, often providing new interpretations of data. For example, as age increases in adults response time increases. In many tasks, fits of the diffusion model show that the quality of the evidence encoded from a stimulus (drift rate) does not decrease; instead, the slowing occurs because the boundaries are set to increase the amount of evidence required for a response and nondecision times are longer.

Current diffusion models are the culmination of 50 years of theoretical and empirical research [ 8 - 11 , 2 ], which has identified the key features of experimental data that a model must explain and the key properties by which a model can do so. Many recent studies have focused on new phenomena and new areas of application but neglected findings in the older literature. This neglect is potentially detrimental because the older literature contains modeling and experimental work that speaks directly to current issues. Our aims in this review are to redress this neglect and to highlight findings in the older literature that present challenges to currently accepted interpretations of data and currently unresolved issues.

The Two-Choice Diffusion Model

Figure 2 shows simulated paths that represent the accumulation of evidence on individual trials (A) and it shows the effects of changes in drift rate (B) and boundary settings (C) on RT distributions. Because there is a minimum on RTs but no maximum, the model automatically produces right-skewed distributions that have the same shape as those found in most simple two-choice tasks. That the model predicts RT distributions that are the same as those found experimentally is one of the most important properties of the model and one of the strongest tests of it [ 12 ].

Drift rates are determined by the quality of evidence extracted from the stimulus or memory (often a different value for each condition of the experiment). Speed-accuracy tradeoff effects and the effects of bias toward one of the boundaries over the other are ubiquitous in the experimental literature and they are explained naturally by the structure of the model. Usually ( Figure 2C ), speed-accuracy effects are explained by changes in the boundary settings (e.g., [ 4 , 13 - 16 ]) and much smaller changes in nondecision time [ 17 - 18 ], although instructions that extremely stress speed can reduce drift rates [ 19 ; see also 20 ].

Figures 2D-F illustrate ways in which the model can accommodate bias towards one alternative or the other [ 21 ]. If the probability that one alternative is tested is made higher than the other, then the starting point moves towards the higher probability boundary ( Figure 2D ) [ 5 , 22 - 26 ]. Some recent investigations have suggested that the effects of this manipulation for both humans and monkeys can be accounted for better by a bias in drift rate, not starting point [ 27 ]. However, this account makes key predictions about the behavior of RT distributions, which were not examined. Bias can also be manipulated by payoffs, for example, paying more for correct responses to one of the alternatives than to the other. In this case, the result is a combination of bias in starting point ( Figure 2D ) and drift rate ( Figures 2E, F ) [ 23 - 25 ].

Changes in starting point and drift rates can also explain sequential effects [ 21 , 28 - 33 ] in which trial-by-trial variations in RT are partly determined by the prior stimulus and the prior response. With a rapid rate of presentation for easy stimuli, sequential effects can extend for several trials, but for slower presentation rates and more difficult stimuli, they are found only for the immediately preceding trial. Adaptive regulatory mechanisms to account for trial-by-trial effects have been proposed by a number of investigators [ 34 - 36 ]. In the diffusion model, sequential effects can be modeled by making the starting point and drift rate functions of prior trials [ 5 ].

In many experimental reports, plots of either accuracy or mean RT alone are used to describe data. However, the two dependent variables must be considered simultaneously ( Figures 3A-D . The data come from a motion discrimination task in which participants see a display of dots and decide whether a subset of them is moving right or left; the fewer the dots moving coherently, the more difficult the decision (Experiment 1, [ 7 ]). The effects are typical-- more difficult conditions have lower accuracy ( Figure 3A ) and slower responses ( Figure 3B ), but it is the relation between them (the latency-probability function in Figure 3C ) and the shapes and locations of the full RT distributions (the quantile-probability function in Figure 3D ) that must be the targets for models. In Figure 3D , the RT distribution is represented by the.1,.3,.5,.7, and.9 quantile RTs. The numbers are the data and the circles with lines between them are the values predicted by the diffusion model. The model fits the data well, and it does so quantitatively as well as qualitatively. The model is falsifiable in that it must predict the right-skewed shape of RT distributions [ 37 ].

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A-D . Plots of data from a motion discrimination experiment, Experiment 1 [ 7 ].

A Response proportion plotted against motion coherence.

B Mean RT plotted against motion coherence.

C Mean RT plotted against response proportion.

D RT quantiles plotted against response proportion for data (digits) and model predictions (o's and lines). The quantiles used were the .1, .3, .5, .7, and .9 quantiles and these represent, respectively, the fastest 10% of responses, the fastest 30%, the median RT, the slowest 30%, and the slowest 10%. The quantiles are stacked vertically and the small inset to the right shows equal-area rectangles drawn between quantiles to illustrate what RT distributions derived from the quantiles would look like.

E-F. Fits to a data set that has large differences in the leading edge of the RT distribution, changes too large for a diffusion model with only drift rate changing over conditions to fit (from Experiment 1, [ 38 ]).

E Data and predictions for fits to RT distribution quantiles. The .1 quantile (magenta) and median (red) miss the data.

F Data and predictions for RT quantiles for fits to accuracy and median RT. The median (red) fits well, but the other quantiles miss badly.

Some researchers question the need to fit RT distributions; some simply omit any consideration of them. The importance of distributions is illustrated in Figures 3E-F , which show data from a letter discrimination task with dynamic random pixel noise [ 38 , Experiment 1]. These data cannot be fit with only drift rate changing (in Figure 3E the magenta line should go through the “1” symbols, the red line through the “3” symbols, and the blue line through the “5” symbols). Figure 3F shows how the median RT alone can give misleading results. The red line goes through the “3” (median) symbols which indicates a good fit, but the RT distributions that would be predicted miss the data badly (the predicted.1 and.9 quantile RTs miss the data by several hundred ms). Thus, we strongly recommend that predictions for RT distributions be examined in any application of the model to data. Diffusion models for multialternative decision making and confidence are discussed in Box 4.

Across-Trial Variability in Model Components

A problem with early random walk models , which were discrete-time precursors of diffusion models, was that they predicted identical response time distributions for correct and incorrect responses (when the starting point is equidistant from the boundaries), which is never observed empirically. Several different approaches to this problem have been investigated, including: dynamically changing decision boundaries; nonlinear evidence accumulation processes; and non-normal random walk increments. The most extensively-evaluated approach to the problem has been the assumption of trial-to-trial variability in model parameters [ 2 , 7 , 9 ]. In the standard model, drift rate is normally distributed across trials with standard deviation (SD) η, the starting point is uniformly distributed with range s z (starting point variability is equivalent to variability in the boundaries), and nondecision time is uniformly distributed with range s t ([ 39 ] examined these parametric forms). Recently, it has been argued that with assumptions of across-trial variability, any pattern of data can be accommodated, but the argument only applies to deterministic models (see Box 5).

The assumption that model parameters vary from trial to trial is made by most current models that successfully account for experimental data in psychological applications [ 1 , 40 ]. This assumption has a long history [ 2 , 4 , 9 ] of extensive testing and it allows models to explain the relative speeds of correct and incorrect responses. When decisions are difficult and decision-makers are cautious, incorrect responses are reliably slower than correct responses (see below in relation to collapsing boundary models). When decisions are easier and decision-makers are hurrying, incorrect responses are reliably faster than correct responses (see [ 32 ]).

Across-trial variability in drift rate produces slow errors (relative to correct responses) because trials with randomly higher drift rates are associated with fast responses, but very few errors. By contrast, trials with randomly lower drift rates are associated with slow responses, many of which are also incorrect. From this mixture ( fast errors with low probability and slow errors with higher probability), error responses are slower than correct responses (see [ 7 ], Figure 4 ). For similar reasons, across-trial variability in starting point gives fast errors. We have seen few, if any, patterns of incorrect RTs vs. correct RTs that cannot be accounted for with the across-trial variability assumption, although many such possibilities exist. Note that there are other ways of producing fast or slow errors (relative to correct responses) such as collapsing bounds (discussed later), but few of these have received extensive testing [ 41 ].

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Modeling the response signal procedure and collapsing boundaries.

Another source of support for across-trial variability in drift rate comes from a face/car perceptual discrimination task in which EEG signals are used to sort responses into two groups, those that are more face-like and those that are more car-like [ 42 ; see also 43 ]. When the diffusion model was applied to the RT and accuracy data for the two groups separately, robust differences in drift rates were produced, showing that the EEG signals indexed the trial-to-trial differences in evidence entering the decision process.

The assumption that drift rate varies from trial to trial [ 2 ] has been controversial in some circles [ 44 - 46 ], but across-trial variability in drift rate is no different than variability in signal strength in signal detection theory and, ironically, the latter assumption is almost universally accepted. Alternatives to the across-trial variability assumption have been proposed including the assumption that drift rates ramp up over time (an “ urgency ” signal) and the assumption that boundaries collapse over time. Although these can predict slow errors, they account quantitatively for the full range of data and we discuss this shortly.

Response Signal and Go/No-Go Tasks: Implicit Boundaries

In the response signal task , a stimulus is presented and then after some amount of time, a signal is given. Participants are asked to choose between two alternatives just as in the usual two-choice procedure except that they are asked to respond as quickly as possible after the signal (in, say, 200-300 ms). The stimulus-to-signal time varies from trial to trial [ 47 - 50 ], which means that processing can be assumed to be the same for all signal times up to each specific signal time [ 2 ]. A response signal task is often used in animal studies in which responses are made following a cue (e.g., [ 51 - 53 ]).

In application of the diffusion model to response signal data [ 17 , 54 - 55 ], there are two response boundaries just as for the usual two-choice task. When a decision is made at some signal lag, responses come from a mixture of processes: those that have terminated at a boundary and those that have not ( Figure 4A ). As the stimulus-to-signal time increases, a larger and larger proportion of processes will have terminated before the signal. For non-terminated processes, there are two possible hypotheses: that decisions are made on the basis of the partial information that has already been accumulated (which has low accuracy relative to terminated processes [ 55 ]) or that they are guesses. In fits of the diffusion model, they could not be discriminated [ 17 ].

Response signal studies from the 1980's show that drift rate can change from stimulus-to-signal intervals that are about the mean RTs in the usual procedure (600 ms or less) up to intervals of around 2 s [ 56 - 59 ]. Changes in drift rate occur when two sources of information are pitted against each other. For example, early in processing, responses to “a bird is a robin” are mainly “true,” reflecting the strong association between birds and robins, but later in processing they are “false.” This differential availability of information over time might be thought similar to data from mouse-tracking paradigms in which, for example, tastiness information becomes available earlier than healthfulness information in a dietary choice task [ 60 ]. However, the smooth mouse tracks obtained on single trials in such studies do not match the highly irregular paths of the diffusion model ( Figure 2A ). The link between smooth mouse tracks or arm-reaching trajectories and the underlying process of evidence accumulation is unlikely to be as simplistic as the one-to-one mapping commonly assumed (e.g., [ 61 ]).

In a go/no-go task, participants are to respond to one of the two types of stimuli (e.g., dots moving left) but withhold a response to the other (e.g., dots moving right). In neuropsychological and clinical research, a pervasive view is that the task measures inhibitory control (e.g., [ 62 - 64 ]). However, in the diffusion model, when it is assumed that there are two boundaries, one implicit, the shorter RTs and lower accuracy for “go” stimuli are explained as a bias of the starting point toward the “go” boundary [ 65 ] and [ 66 ]. What has been assumed to be an ability to suppress responses is interpreted simply as a bias in processing (sometimes a combination of biases, as shown in Figures 2D, E, F ).

Another domain that concerns stationarity in processing is how changes in evidence might be detected. Diffusion and related model analyses of tasks in which stimulus information varies from moment to moment provide a theoretical account of change detection [ 67 - 68 ] in which evidence accumulation has to be balanced against changes in the stimulus environment.

Conflict Tasks

Another class of paradigms that appear to require dynamic changes in diffusion model parameters over time are conflict paradigms. In a reinforcement learning paradigm, subjects had to choose one of a pair of letters and feedback indicated which one was “correct” on that trial. Feedback was probabilistic with one letter of the pair being reinforced more often than the other. Later in the session, conflict conditions were created by pairing the letters from different pairs that had low probability of reinforcement. Responses to these conflict pairs had shifts in RT distributions relative to the training pairs and other pairs in which high probably letters were paired with other letters. The conflict conditions were modeled with collapsing decision boundaries that accounted for the shifts in RT distributions [ 69 ]. (This pattern of shifts in the RT distributions could also be modeled by shifts in nondecision time.)

A second conflict paradigm is the Eriksen flanker paradigm. In this, “>” symbols are used to indicate the direction of the response (“>” right and “<” left). The flanker manipulation involves placing other symbols indicating the same direction or the other direction around the central target that indicates the response. Two diffusion models have been developed, one that assumes dual stages, and one that assumes that evidence driving the process changed continuously over time as the result of attention gradually focussing on the central target [ 70 - 71 ]. The key for testing these models is the behavior of RT distributions: the behavior of error vs. correct RT distributions provides the critical tests of the models.

Considerations of optimality have played a significant role in the theoretical and experimental analysis of human and animal decision making. Theories of optimality prescribe how the available evidence should be used to produce a best decision, in some specified sense; experimental studies of optimality have investigated whether actual behavior approximates the theoretical ideal. In simple decision-making, two different senses of optimality have been promoted (Box 6). One of these is based on Wald's sequential probability ratio test (SPRT) from statistics; the other is based on reward rate maximization. Wald showed that a random-walk decision process that accumulates the log-likelihood ratios of the observed evidence sequence, given the two decision alternatives, is optimal in the sense of needing the smallest number of evidence samples to reach a prescribed level of accuracy. Optimality in Wald's sense was influential in the development of early random walk models of human decision making in psychology [ 8 - 9 , 22 ]. A pure (Wiener) diffusion process, with no across-trial variability, can be viewed as a continuous-time log-likelihood ratio accumulator, and is optimal in the same sense [ 72 - 73 ].

A significant limitation of Wald SPRT test is that restricts its applicability to real world decision tasks with constant evidence and boundaries and so it applies only to decisions between pairs of alternatives whose properties are known. It is not applicable, for example, to tasks in which stimuli of varying discriminability are presented in random order in a block of trials unless the boundary settings are set differently for each different difficulty level (and optimally, for each one). In most decision environments, this is not possible because setting the appropriate boundary requires advanced knowledge of the upcoming decision difficulty. However, in the few cases where it is possible to set different boundaries for each difficulty level, human decision-makers seem quite efficient at setting those many boundaries in an optimal fashion [ 74 - 75 ].

An alternative definition of optimality is reward rate maximization [ 76 ] defined as maximizing the number of correct decisions (and the associated reward) per unit time [ 72 , 77 - 81 ]. This definition of optimality has been promoted, especially in animal studies that use water-deprived animals and liquid rewards, as a biologically-principled theory of optimality. In these tasks, it seems likely that animals will be motivated to maximize their reward rate because it also minimizes the time until the next reinforcement. Although reward rate maximization has been promoted as a general definition of optimality with equal applicability to animals and humans, it is not clear that human decision makers are motivated in the same way. Rather than seeking to maximize the returns per unit time, humans seem to be motivated to maximize the returns in the available time. For example, if two students take a two-hour exam and one obtains 60% correct in one hour while the other obtains 80% correct in two hours, the second student will perform better on the course. There are some situations in which reward rate is explicitly set as a goal (e.g. “speed tests” in schools) but even there, there is little evidence that people actually attend optimally to this goal. Indeed, when we have investigated this hypothesis, it fails (Box 6).

Collapsing Bounds

The link with optimality theory, on the one hand, and neural studies of decision making, on the other, has led to models in which decision bounds collapse over time. In the collapsing-bound model, less evidence is required to trigger a decision as time passes, that is, the boundaries collapse from initially wide spacing toward the center ( Figure 4B ). Another assumption with much the same effect is that fixed boundaries are maintained, but an “urgency signal” is added to the accumulated evidence [ 82 - 83 ]. This signal is like a gain that magnifies evidence by larger and larger amounts as time passes. Models with collapsing bounds have been identified with urgency gating signals in some recent theoretical accounts of optimal coding in neural populations and in empirical single-cell recording studies [ 41 , 78 , 82 - 85 ].

Another issue concerns the relative speed of correct and incorrect decisions. In many paradigms, particularly those with difficult decisions and an emphasis on decision caution, incorrect responses are systematically slower than correct responses, on average. Standard evidence accumulation models account for this effect by assuming variability in decision difficulty across trials [ 2 ]. Collapsing boundaries, or increasing urgency signals provide an alternative way to predict that incorrect responses are slower than correct responses.

Because many of the predictions of fixed-bound and collapsing-bound models are very similar, a test between them is only likely to be successful if it uses large samples and is carried out at the distribution level. A large-scale investigation of data from hundreds of participants in 10 different studies, from three different laboratories addressed these questions empirically [ 86 ]. The data of the great majority of participants were better described by the regular, fixed-bounds diffusion model than by either of the collapsing boundaries or urgency signal variants. What support did exist for the new variants was mostly confined to experiments involving non-human primate participants, or using experimental procedures optimized for non-human primates (but with human participants). Other studies which have identified support for non-stationary models have mostly employed unusual decision-making tasks, e.g. with very long decision times, or slowly-changing stimulus properties [ 85 , 87 ].

Figures 4C and 4D illustrate, using example experimental data, that the fixed-bounds model out-performed the collapsing boundaries and urgency signal models. While it is true, in theory, that the collapsing boundaries can help the model to predict slow errors, in practice its predictions did not match data because the model systematically over-predicts the slowing of incorrect responses relative to correct responses. The lower-right panel of Figure 4D shows the best fit of a model with an urgency signal to decision data from monkeys [ 88 ] - these data are actually some of the most favorable for the collapsing bounds and urgency signal accounts). Incorrect responses (red crosses) in these data are slower than correct responses (green crosses). However, the urgency signal model over-predicts this effect, with incorrect responses becoming slower from right to left across the plot. This effect is almost never observed in data. Instead, especially in human data, there is almost always a characteristic inverted-U shape to these plots, which is accommodated well by the standard fixed-bounds model, but not by the collapsing bounds model (see top row of Figure 4D ).

Expanded Judgment Tasks

Most recent applications of diffusion models have been to experimental tasks in which a single stimulus is presented and the noise in the evidence accumulation process arises from moment-by-moment variability in the cognitive representation of the stimulus. However, some of the earliest applications of random walk models [ 8 , 22 ] were to expanded judgment tasks in which a noisy sequence of stimulus elements has to be integrated to make a decision. Studying such tasks was motivated by Wald's SPRT test [ 89 ], which provided an optimality theory for decisions about discrete sequences.

There has been a recent resurgence of interest in the application of diffusion and random walk models to expanded judgment tasks, especially in neuroscience, motivated in part by a renewed interest in optimality [ 76 , 90 - 91 ]. Many researchers have used a coherent-motion discrimination task (e.g., the moving dot paradigm mentioned earlier), originally developed in vision science as a pure motion stimulus, as a fast-paced expanded judgment task, assuming that successive states of the motion signal are accumulated directly by the decision process.

At present it is an open question whether the decision process treats variability in a sequence of stimulus elements as equivalent to moment-by-moment internal noise in the cognitive representation of a single stimulus. The hypothesis that external stimulus noise and internal noise are equivalent is an attractive one, but there are enough differences in the perceptual and memory demands between single-stimulus decision tasks and expanded judgment tasks to make the equivalence questionable. Only one stimulus representation is required in single-stimulus tasks, whereas expanded judgments tasks require a new representation of every element in the sequence, which must be integrated with the memory representation of the elements that precede it.

Currently, little is known about how this memory updating process might take place, how long it might require, or how it might depend on the complexity of the individual stimulus elements. Expanded judgment tasks vary widely in the kinds of stimuli they use and the way the stimuli are presented. The stimuli have included random dots [ 44 ], colored lights [ 92 ], and patches [ 93 ], line segments [ 94 ], sinusoidal gratings [ 95 ], clicks and visual pulses [ 52 ], and geometric shapes [ 96 ], with inter-element intervals ranging from a few tens of milliseconds to several hundred milliseconds or longer. Unless memory updating is rapid and effortless, one would expect working memory capacity limitations to have a significant effect on performance.

Consistent with this expectation, the picture of the decision process that emerges from expanded judgment studies is more complex than the one that has come from single-stimulus studies, and some of the findings have no obvious counterpart in single-stimulus studies. For example, one study [ 52 ] reported that the sole source of variability in decision-making by human and animal subjects was external noise in the stimulus; they found no effect of internal noise, or leakage, that is, decay of the memory representation of the stimulus sequence with the passage of time.

Other studies [ 93 , 95 ] found there was a strong recency weighting of stimuli: stimulus elements occurring later in the sequence were weighted more heavily in the decision than were earlier elements. Recency weighting has been observed in other studies [ 97 ] using longer stimulus sequences, but so has the opposite effect [ 45 ]. Recency weighting has been explained by various mechanisms, including a gain control process [ 95 ] and a working memory capacity limitation [ 97 ].

Most studies using expanded judgment tasks have not analyzed RT distributions. When they have been analyzed [ 94 ], the decision time distributions were much more variable than those found in single-stimulus tasks and could not be predicted by a random walk model with normally distributed increments, which is the discrete-time counterpart of the diffusion model with no across-trial variability in drift rates or starting points. Instead, they were better described by a version of the Vickers accumulator model [ 98 ], in which only large stimulus elements are accumulated and small ones are ignored.

Studies using stimuli that are perturbed by external dynamic noise cast further doubt on the assumption that the decision process treats internal noise and external noise as equivalent. Many studies have assumed that the decision system accumulates the noisy output of a motion discrimination system, in which the noise arises from variability in the stimulus sequence (following [ 88 , 99 ]). These effects are typically modeled using a constant drift diffusion model in which the drift rate is proportional to the mean strength of the motion signal. Other work [ 38 ] has found that the effect of dynamic noise on letter discrimination was to shift the entire RT distribution to the right, delaying all responses by a constant amount. They attributed the delay to the time needed to compute a stable stimulus representation, which determines the drift rate of the diffusion process. RT distributions from the dynamic noise task were modeled with a diffusion model with a time-varying drift, in which the growth of the drift rate depended on external noise [ 100 ].

These examples highlight the fact that expanded judgment tasks and tasks in which stimuli are presented in external noise differ in significant ways from tasks in which the only source of noise is internal to the decision process. These differences caution against any a priori equating of the effects of internal and external noise and suggest that further work is needed to understand the relationship between them.

Brief Stimulus Presentation

There is a growing interest in whether there is nonstationarity in processing that reflects changing stimulus information over time. This is often studied with stimuli flashed for very brief times (e.g., 50 ms) and is also partly motivated by the thought that stimulus quality can change during the time course of an expanded judgment task. In a highly relevant example from about 15 years ago, the question was whether drift rate tracks the stimulus or not. In other words, whether drift rate begins at some value reflecting the stimulus information and then drops to zero when the stimulus turns off, or instead, whether information from the stimulus is integrated over time into a short-term representation that provides a constant drift rate to drive the decision process [ 101 ]. Drift rate tracking stimulus availability would be equivalent to the starting point moving nearer the boundary for correct responses, which predicts errors much slower than correct responses because errors would have much further to travel to the incorrect boundary. However, the distributions of RTs for errors and correct responses were similar, supporting a model in which drift rate is driven by a constant representation of the stimulus (see also [ 14 , 16 , 37 , 102 ]).

Later work [ 103 ] examined manipulations of contrast, stimulus presentation duration, and attention with Gabor patch stimuli and culminated in a model that integrated the processes that construct a representation of stimulus information in short-term memory with the diffusion model's decision process [ 104 ]. Constructing a representation is necessary because stimuli are multidimensional (e.g., size, color, correspondence with entities in memory) and so there must be processes to pick out the dimension relevant to a decision. There are neurophysiological data that support this view. For example, in motion discrimination tasks that use a brief pulse of motion [ 105 ], there is a long-lasting effect (up to 800 ms) on decision-related firing rates in the lateral intraparietal cortex. Other studies in animals and humans have found that accuracy does not increase after some stimulus duration [ 37 , 106 - 110 ], suggesting a time by which the relevant stimulus information has been constructed. Integration time, as assessed by a change in accuracy thresholds as a function of stimulus duration, was shown to be the same (around 400 ms) for high and low coherence stimuli [ 107 ]. This suggests that the exposure duration effects in this paradigm may reflect perceptual integration processes involved in computation of the drift rate rather than evidence accumulation by a decision process.

Figure 5 illustrates how a nonstationary model mispredicts data when drift rate turns on at a relatively high value for 80 ms and then returns to zero ( Figure 5A ). Figure 5B shows the average position of the accumulation process rising over the first 80 ms and then falling back toward the starting point. Figures 5C and 5D show the difference in the RT distributions for correct and error responses. The red circles are the 1,.3,.5,.7, and.9 quantile RTs for correct responses and the blue are the quantiles for errors. Overall, error responses are slower than correct responses, not the result found by [ 14 , 16 , 37 , 102 ]. This illustration also demonstrates that RT distributions are the critical test between stationary and nonstationary processes. Some investigations have supported nonstationarity but have not generated the critical predictions for correct and error RT distributions [ 52 , 87 , 111 - 113 , see also [ 45 , 114 - 115 ].

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A shows drift rate as a function of time for the model in which drift tracks the stimulus (here turning on at zero and off at 80 ms).

B shows the mean and plus and minus 1 standard deviation average paths from simulations (using the random walk method, [ 116 ]). The red arrow is the point that drift rate turns off and the horizontal red lines are the decision boundaries.

C shows RT distributions for correct responses for the simulations with the red circles showing the .1, .3, .5, .7, and .9 quantile RTs.

D shows RT distributions for error responses for the simulations with the blue circles showing the .1, .3, .5, .7, and .9 quantile RTs (the red circles are the quantiles for the correct responses from C).

This discussion raises a related question, whether the stimulus evidence driving the decision process turns on abruptly. In a modest simulation, drift rate was ramped linearly over 50 ms from zero to a constant level [ 37 ]. The standard constant-drift model fit the simulated data almost perfectly (with different parameters than those used to generate the simulated data - increases in across trial variability in nondecision time, starting point, and nondecision time), suggesting that ramped-drift and constant-drift models mimic each other and that constant-drift models are a good approximation even if drift rate does ramp up over time.

Concluding Remarks

Current research in modeling decision processes has used diffusion models extensively. They are being applied in clinical and educational domains, economic decision making (Box 7), and the neuroscience of decision making. Here we have examined current issues that have a history in psychology and we have discussed the earlier research and how it complements new research. In some cases, the earlier research provides an answer to new research questions. Although we have separated the issues, many of them are related. Collapsing decision bounds have been argued to be optimal and able to replace assumptions of across-trial variability in drift rate. Likewise, nonstationary drift rates are related to collapsing bounds and are one hypothesis of what would occur with brief stimulus presentation. These basic questions are important as the field uses this modeling approach in clinical and neuroscience domains.

Box 1 Advantages of Diffusion Model Analyses

The model relates speed and accuracy to the same underlying components of processing for fast (under 1-2 s) two-choice decisions and it explains why speed and accuracy are sometimes correlated and sometimes not (especially across individuals).

An individual can decide to respond as quickly or as accurately as possible. The diffusion model factors out speed-accuracy settings and so provides better estimates of the quality of the evidence entering the decision process than is available from response time or accuracy data.

The model provides a fit to data which allows us to know whether the model is an adequate description of the data.

When there are limited numbers of materials, fitting the model to fillers and critical items increases the power for the critical items. This is because fillers are weighted heavily in determining model parameters common across conditions. [ 117 - 120 ]

When accuracy is at ceiling, it is still possible to estimate drift rates if some of the conditions have lower accuracy. These conditions are weighted most heavily in determining some model parameters and then RTs alone are sufficient to determine drift rates. [ 120 - 121 ]

The variability in model parameters is usually smaller than the variability between subjects, which means that the effects of individual-difference measures on performance can be measured, e.g., IQ and working memory. [ 122 - 125 ]

Fitting packages are available (but subject to misuse if not understood): fast-dm, [ 126 ]; DMAT, [ 127 ]; HDDM, [ 128 ] as well as alternative fitting methods [ 129 ]. See evaluations by [ 130 ] and [ 131 ].

Other current sequential-sampling models usually offer similar explanations of phenomena in terms of the behavior of model components. (parameters). [ 132 - 133 ]

In addition to standard two-choice tasks, the model has been successfully applied to go/no-go, response signal, and deadline tasks. Related diffusion models have been applied to multi-alternative decision making and confidence judgments.

The model can address model parameters changing of the time course of the decision [ 54 , 56 - 59 ]. The distribution of non-terminated processes is well known (derived from the Fokker-Planck forward equation) and this can be used as a new starting point distribution in a second phase of evidence accumulation after a change in drift rate. Changes in other model parameters can be examined using simulation methods.

Box 2 Domains of Application of Diffusion Model Analyses

Diffusion and other models have been applied to many basic perceptual and memory tasks such as item and associative recognition [ 2 , 15 , 123 - 124 , 134 ], lexical decision [ 123 , 135 ], perceptual tasks including brightness, letter, motion, visual search, contrast, orientation discrimination tasks [ 7 , 14 , 16 , 44 , 103 - 104 , 136 - 137 ], numeracy judgments [ 13 , 123 , 125 ], categorization [ 134 , 138 ], and text processing and priming [ 118 - 119 ]. Other tasks that have more interdisciplinary relationships include stop-signal tasks [ 139 - 140 ], conflict tasks [ 70 - 71 ], reinforcement learning [ 69 , 141 ], preferential-choice and value-based decisions [Box 7], and social decisions [ 142 - 143 and Box 7].

Relationships have been established between diffusion model analyses and behavioral measures such as eye tracking [ 130 , 139 ] and pupil dilation [ 144 ]. Many studies have established relationships between diffusion models and neurophysiological measures such as single cell recordings in rodents and monkeys [ 52 , 88 , 105 , 145 - 150 ], electroencephalography (EEG) [ 42 - 43 , 151 , 153 ], functional magnetic resonance imaging (fMRI) [ 42 - 43 , 141 , 152 , 154 - 157 ], transcranial magnetic stimulation (TMS) [ 158 ], transcranial alternating current stimulation (tACS) [ 159 ]. Diffusion model analyses have also been applied to bees and animal swarms [ 160 - 161 ] and even to slime moulds [ 162 ].

Diffusion model analyses have been used to study manipulations of state such as sleep deprivation [ 163 ], hypoglycemia [ 164 ], and alcohol [ 165 ]. They have been used to study individual differences in IQ, working memory, and reading measures [ 122 - 125 ], and to examine deficits in populations such as aphasics [ 166 ], older adults and children [ 13 - 14 , 123 - 124 ], children [ 167 ], low literacy adults [ 168 ], dyslexics [ 169 ], ADHD [ 170 - 171 ], schizophrenia [ 172 ], and in depressed and anxious individuals [ 116 , 173 ].

Box 3. Linking Neural Firing Rates to Diffusion Processes

Qualitative links have been made between neural firing rates and diffusion processes [ 145 - 146 , 174 ]. More explicit modeling has attempted to link the dynamics of attractor networks to decision-making by identifying decision making with a neural network entering an attractor state. This uses linear approximations of the network equations and constraints on the network parameters to reduce its dimensionality.

One theoretical challenge is to show that these network models can reproduce the structure found in families of RT distributions for correct responses and errors for real decision-making data

The attractor model developed by Wang assumes two pools of excitatory neurons coupled via a third pool of inhibitory neurons [ 175 - 176 ]. A two-component linear diffusion approximation to the network dynamics was proposed and the model was reduced to a much simpler representation consistent with current diffusion models. One attractive aspect of this modeling approach is that neurally plausible assumptions are made about inputs based on currents, neurotransmitters etc.

A different approach assumed two pools with a nonlinear diffusion approximation containing both linear and cubic terms [ 177 ]. The cubic term was designed to mimic the attractor dynamics of [ 175 ]. It is an open question whether the cubic diffusion equation (in which can also reproduce the detailed features of accuracy and RT distributions found in human behavioral data.

The attractor network model, the Ising Decision Model [ 178 ], is based on a stochastic Hopfield network . Stimulus information, represented as the drift of an approximating diffusion process, is identified with (minus) the gradient of a potential field that sets the attractor states of the network. A decision is made when the network first enters one of two neighborhoods surrounding the attractor states. The model successfully reproduces the behavior of RT distributions found in human data and the model makes an explicit theoretical connection between the physics of diffusion and the properties of the potential field that determine that network's attractor states.

Diffusive noise in a decision process can be derived from a Poisson shot noise model of stimulus representations [ 179 - 180 ]. The Poisson shot noise process represents the variability in the postsynaptic potential across a neural population that is induced by volley of action potentials modeled as a Poisson process. In this model, stimulus information is represented by the difference between excitatory and inhibitory shot noise pairs. The model produced the families of RT distributions predicted by the standard diffusion model.

Box 4. Multi-Alternative Decision Making Including Confidence Judgments

There is no simple generalization of the two-choice diffusion model to multiple alternatives [ 9 , 181 ]. Instead, the usual approach assumes independent racing single-boundary diffusion processes. This generalizes easily to any number of alternatives, but differs from the standard model because the racing diffusions model does not includes response competition as in the standard model. Response competition can be easily added to the model so that movement towards one boundary entails movement away from the others.

Recent studies have examined models for multi-alternative decision making that use diffusion processes [ 24 , 41 , 72 , 79 - 80 , 182 - 190 ] and a number of algorithms have been used:

  • Independent racing accumulators with termination when one reaches its decision criterion.
  • Independent racing accumulators with a relative stopping rule (termination occurs when one accumulator beats the maximum of the others by some amount).
  • Accumulators with dependence between accumulators: inhibition between accumulators that depends on the amount of accumulated evidence.
  • Accumulators with dependence: evidence for one alternative is evidence against the others so that the total evidence is constant. When one accumulator is incremented, the others are decremented (termed constant summed evidence or feedforward inhibition). This can be seen as a generalization of the two-choice model in which evidence for one choice is evidence against the other.
  • Other architectural choices include whether evidence can fall below zero, whether there is decay, and whether parameters vary across trials. The models that implement these choices are difficult to discriminate (see [ 187 ]).

Confidence judgment tasks are also multi-alternative tasks. In them, subjects are often asked to rate their confidence in decision about two alternatives. In the majority of applications, the proportion of responses at each level of confidence are the primary data and signal detection theory has been the dominant model. These tasks have had a long history in psychology [ 191 - 193 ].

Previously, little attention has been paid to RTs [ 194 - 195 ] and even less attention to modeling them. However, recent work in psychology has accounted jointly for response proportions and RT distributions in confidence judgment tasks [ 187 , 196 - 197 ].

In animal work, it seems impossible to get animals to respond on a scale (e.g., 6 confidence choices). An opt-out procedure has been used to examine confidence in animals [ 148 - 149 ]. In recent modeling in neuroscience, time has been used as a measure of decision confidence. However, in many studies, RT distributions for different levels of confidence overlap, and so RT cannot uniquely determine the confidence levels.

Box 5. Is the Diffusion Model Identifiable or Too Flexible?

If the forms of across-trial variability distributions are unconstrained, then [ 198 ] argued that the diffusion model and other evidence accumulation models can exactly match any data (response probabilities and RT distributions), rendering the models unfalsifiable. Below are considerations about this argument [ 199 ].

  • The most important point is that the proofs in [ 198 ] apply only to completely different deterministic or near-deterministic models, which the diffusion model is not.
  • In a deterministic model, if every process travels the same fixed distance (starting point to the boundary), then every RT can be converted into a velocity (drift rate) by velocity=distance/time (drift rate is a constant times 1/ RT) thus producing a one-to-one mapping between drift rates and RTs ( Figure IA ).
  • In such a deterministic model, to account for errors, complex, bimodal distributions of drift, consisting of two unequally sized, asymmetric lobes, must be assumed. The probability mass in each corresponds to the proportion of responses of each type, ( Figure IB ). Every different RT arises from a different value of drift (because it is a constant times 1/RT) and the drifts for correct responses and errors have opposite signs because errors can only occur when the sign of the drift is wrong.
  • Because response probabilities and RT distributions vary from condition to condition, a different bimodal distribution of drift is needed for every condition of an experiment. In the standard model, if the distributions of drift rate or starting point are changed modestly [ 2 , 39 ], the model produces similar predictions (within trial noise washes out effects of distribution shape as in the central limit theorem).
  • The resulting model is complex with highly unintuitive properties. As stimuli become more discriminable and easier, the asymmetry of the distribution of drift and the separation between its positive and negative lobes increases ( Figure IB ).
  • To account for the finding that repeated presentation of the same stimulus can lead to different responses, the model must assume that the sign of the drift can vary from presentation to presentation and that the magnitude of the difference between drifts leading to correct responses and errors increases as the task becomes easier (i.e., error drift rates become more strongly negative).
  • In a few cases, neurophysiological data can address the distributions of drift rates. For example, single-trial EEG measures are consistent with unimodal distributions rather than bimodal distributions [ 198 ] ([ 42 ], Figures 2D and ​ and2E 2E ).

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Box 6. Optimality: Do Subjects Adopt Optimal Boundary Settings?

There are two senses in common use:

  • A diffusion process is optimal in that for a single drift rate, the process produces the minimum time on average to produce a given level of accuracy (determined by boundary settings).
  • For any experiment with any number of conditions, a value of boundary settings (with bounds constant over time) can be computed that makes the number correct per unit time a maximum (reward rate optimality).

For 1: If parameters vary from trial to trial and/or there are multiple conditions in an experiment, sense 1 of optimality no longer holds.

It is possible to compute the optimal boundary shape, and this varies as a function of time.

It is hard to see, especially in the first trials of a task, how enough information could be gathered trial by trial to allow the shape of this optimal bound to be computed. The stimulus condition is almost always not known and the response choice and response time are stochastic so reliable information needed to compute the bound are not available. In the derivations of optimality, there is no term in the computations (e.g., based on feedback) representing the duration or effort needed to compute and update the optimal boundary.

For 2, when experiments have tested whether boundary settings are optimal, generally they are not. Only with feedback does performance of young adults approach reward rate optimal [ 200 ]. But older adults rarely moved more than a few percent away from asymptotic accuracy

In experiments in which blocks of trials are difficult or easy, there is a fixed time for the block, and subjects are instructed to get as many correct regardless of errors, for difficult relative to easy blocks, subjects slow down when it is reward rate optimal for them to speed up [ 201 ].

Box 7. Preferential Choice and Value Based Decision Making

Diffusion model have been applied to complex preferential choice applications from judgment and decision making that apply to economics and consumer behavior. Wiener and Ornstein-Uhlenbeck diffusion models have been applied to preferential choice, value based decisions, and economic decisions for over 25 years. The tasks in these applications are not simple speeded decisions, but rather choices between risky gambles or multi-attribute consumer products.

Initial work on preferential choice was carried out using what is called decision field theory (DFT, [ 202 ]). In this model, a decision maker's attention switches from one attribute to another over time, and the advantages and disadvantages of each alternative are accumulated into a preference state. When one option reaches a decision threshold, that choice is taken. DFT has been used to fit choice and response time data from choices between risky and uncertain gambles [ 202 ]. Later, a multi alternative version of DFT was used to account for the context effects (similarity, attraction, compromise effects) on choice found by consumer researchers [ 182 ]. In related research, [ 203 ] used the leaky competing accumulator diffusion model to predict context effects on choice. DFT has also been used to account for puzzling reversals in preference between choice and price of gambles [ 204 ] and to explain choice and RTs for inter-temporal choice [ 205 ].

Recent research on value based decision making has used the attention drift diffusion model [ 206 - 207 ]. The model assumes that attention to an option changes the drift rate during preference accumulation. Novel eye tracking methods have been used to track attention to options across time and use these measurements to moderate the drift rate across time. This model has achieved impressive success to account for eye fixation data as well as both choice and response time distributions for choices between food items.

This and other diffusion models have been used to account for, among other things, value-based decision, social choice, and purchasing decisions [ 153 - 155 , 159 , 208 - 211 ]

Outstanding Questions

  • What is the best architecture for multiple choice decisions and confidence judgments? There are a number of models but they are complicated and difficult to test as well as difficult to test against data and against one another.
  • What is the signal that initiates the decision process? One assumption is that there is a release from inhibition in the accumulation process, but this needs to be formally modeled. A change detector based on the perceptual signal has been used to model this.
  • How are decision boundaries set (criterion settings in general such as the criterion in signal detection theory and the drift criterion in the diffusion model)? Optimality theory attempts to do this but no model is completely satisfactory.
  • The issue of criterion setting is even more a puzzle because humans can set criteria to verbal instructions in one or two trials. In other words, there is no chance for feedback (in some experiments with human subjects, no feedback is given) to provide information with which to adjust criteria.
  • Much of the neurophysiological study of decision making uses tasks with responses in different locations in a retinotopic map for eye movement responses or a motor map for motor responses. However, in humans, it is easy to make the alternative responses two arbitrarily chosen words (“one”/”two”, “case”/”plumb”, “cricket”/”football”). A diffusion model could represent the decision process, but the process by which the categorical output is mapped onto the verbal response (such as by using a dynamical speech production model) is unexplored.
  • The relationship of criterion settings to control structures in basal ganglia and working memory processes in frontal cortex are beginning to be explored.
  • We are a long way away from truly integrated models of motor processes in decision making in motor cortex and oculomotor system.
  • Applications to economic decision-making tasks are beginning to penetrate the field of economics.
  • Applications that bring this kind of cognitive modeling to neuropsychological and educational testing is just beginning.
  • Diffusion models with drift and boundaries constant over time account for accuracy and correct and error response time distributions for many kinds of two-choice tasks in many populations of participants.
  • Collapsing decision bounds implement optimal decision making in certain cases, but fits to data show humans use constant boundaries.
  • Brief stimulus presentation produces time varying input, but data suggest that evidence is integrated to produce constant drift in the decision process. (Other tasks produce nonstationary evidence).
  • Evidence is assumed to vary from trial to trial, as in signal detection theory. This explains why incorrect decisions are often slower than correct decisions.
  • It is not clear if variability in a sequence of stimulus elements in expanded judgment tasks is equivalent to moment-by-moment internal noise in tasks with a single stationary stimulus.

Acknowledgments

We would like to thank Jerome Busemeyer, Josh Gold, and Tim Hanks for comments on the paper. Preparation of this article was supported by NIA grant R01-AG041176 and DOEd/IES grant R305A120189.

Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Contributor Information

Roger Ratcliff, The Ohio State University, University of Melbourne, University of Newcastle, and The Ohio State University. Department of Psychology, 1835 Neil Avenue, Columbus, OH, 43210.

Philip L. Smith, The Ohio State University, University of Melbourne, University of Newcastle, and The Ohio State University. Melbourne School of Psychological Sciences, Level 12, Redmond Barry Building 115, University of Melbourne, Parkville, VIC 3010.

Scott D. Brown, The Ohio State University, University of Melbourne, University of Newcastle, and The Ohio State University. School of Psychology, University of Newcastle, Australia, Aviation Building, Callaghan, NSW 2308.

Gail McKoon, The Ohio State University, University of Melbourne, University of Newcastle, and The Ohio State University. Department of Psychology, 1835 Neil Avenue, Columbus, OH, 43210.

Diffusion and Osmosis

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hypothesis on diffusion

  • Diffusion in Agar
  • Movement of Molecules Across a Semi-Permeable Membrane
  • NOTE: In this exercise, you will be given agar containing an indicator chemical called phenolphthalein. When phenolphthalein is exposed to the normal alkaline conditions in the agar, it will look pink. But when it is exposed to neutral or acidic conditions, it changes from pink to clear. You will make different size and shaped agar cubes as a model for cells to study the impact of cell size and shape on diffusion rate.
  • To make the first set of cells, measure out and cut a small cube of agar where each side measures one centimeter.
  • Next, measure and cut out a medium cell cube with sides of 2 cm and a large cell of 3 cm on each side.
  • Knowing the length of the sides of your cube cells, calculate their surface area using this equation, where lower case a represents the length of the sides: Surface Area = 6a^2
  • Record these values in the appropriate column in Table 1. Click Here to download Table 1
  • Then, use the same length value and the equation below to calculate the volume of each cube and add these to the table: Volume = a^3 HYPOTHESES: The experimental hypothesis might be that the acid will diffuse completely to the center of the small cell faster than the medium and large cells. The null hypothesis could be that the acid will diffuse to the center of the small and two larger cubes at around the same time.
  • Add 100 mL of 0.1 M HCl to each of the three 400 mL beakers to make the diffusion baths.
  • Working in a team, have one experimenter ready with the timer and the second and third experimenters ready to drop each cube into one of the beakers.
  • When the first experimenter says go, simultaneously drop all three cubes into their respective beakers and start the timer.
  • Observe carefully until one of the cubes becomes completely clear or 10 min have passed.
  • Stop the timer, remove the agar cubes from the beakers and place the cubes into a Petri dish.
  • Make a note of which of the three cells became clear or had the smallest remaining pink area. Then, also note which cell had the most remaining pink agar.
  • Next, in Table 1, calculate the surface area to volume ratio for each cell. Surface Area: Volume Ratio = (surface area)/volume
  • As the cell size increases, note whether the surface area to volume ratio increases or decreases. Also consider whether this correlates with your observation of the depth of diffusion into the agar cells. If cells rely on diffusion to deliver essential nutrients and molecules to the whole cell, discuss with your group if it would be better to have a smaller or larger surface area to volume.
  • Now, with the remaining agar, cut three rectangular shaped blocks of different sizes and record their length, width, and height. This will test what happens when the shapes of cells are different.
  • Calculate the surface area of your rectangular cells using the formula below, where length is l, width is w, and height is h. Surface Area = 2lw + 2lh + 2wh
  • Then, calculate the volume of your rectangles using this formula: Volume = l * w * h
  • Repeat the experiment by dropping the new shapes into the hydrochloric acid solution for 10 min or until one cube becomes completely clear.
  • Remove the cell shapes from the solutions and observe the depth that the hydrochloric acid diffused into each of these cells, and which shapes have the smallest and largest remaining pink areas not reached by the solute.
  • Using the surface area and volume data you recorded for your rectangular shapes, calculate the surface area to volume ratio of these cells. Surface Area: Volume Ratio = (surface area)/volume
  • Consider whether these values correlate to which cells had the most and least complete diffusion. Additionally, discuss with the group whether these rectangular cells displayed a similar or different pattern of diffusion to that observed with the cube shaped cells, and what this might mean.
  • Before beginning the experiment, add 250 mL of distilled water to each of four 1 L beakers.
  • Then, label the beakers from 1-4, and add 0.5 mL of iodine to the first beaker. HYPOTHESES: In this experiment, the experimental hypothesis is that some of the solutes will be able to pass through the dialysis tubing membrane and others will not. The null hypothesis is that there will be no difference in the ability to diffuse through the dialysis tubing membrane between the solutes.
  • To prepare the dialysis tubing, remove the pieces one at a time from the distilled water bath and tie a tight knot at one end of each tube. These tubes, when filled, will act as model cells with the dialysis tubing acting like the semipermeable membrane.
  • Add 10 mL of starch solution to the first tube and tie off the open end, making sure to leave space in case the tubing expands during the experiment.
  • Then add 10 mL of the NaCl and dextrose solutions to the second and third pieces of tubing, respectively, and tie off both tubes, again, leaving space in case of expansion.
  • After adding 10 mL of distilled water and tying off the fourth tube, weigh each of your model cells.
  • Record the initial weight values in grams and the colors of the starting solution in each tube in the appropriate columns of Table 2. Click Here to download Table 2
  • After quickly rinsing the outside with tap water, place each piece of tubing in its corresponding beaker for 1 h at room temperature. NOTE: For example, the starch solution tube should be placed into the beaker containing the iodine.
  • At the end of the diffusion period, weigh the tubes again.
  • Then, observe the tubes carefully, noting any color changes.
  • Record all of these data in Table 2.
  • Next, to perform a Benedict's Reagent test for simple sugars, make a water bath by adding 250 mL of water to a 600 mL beaker and placing it onto a hot plate.
  • Set the plate to high, to boil the water.
  • Label two new glass test tubes as H 2 O and dextrose, respectively.
  • Use a graduated cylinder to transfer 1 mL of solution from the water and dextrose beakers into the corresponding test tubes.
  • Then, add 2 mL of Benedict's Reagent to each tube.
  • Once the water is boiling, place each test tube into the water bath for 3-5 minutes.
  • After this time, note the color of the solution in each tube.
  • Then use this key to assess whether the test is positive or negative and record these data in the appropriate column in Table 2. Click Here to download Figure 1
  • First, look at the mass of your four dialysis tube cells at the beginning versus the end of the experiment. Calculate the change in mass for each of the four cells and plot it onto a bar chart.
  • Note which cells demonstrated the most change, and whether any of the cells appeared visibly different in size.
  • For the experiment with the starch and iodine indicator, note whether there was a color change in the fluid in the artificial cell. Also consider whether there was a color change in the water in the beaker, and what both of these observations say about the properties of the dialysis tubing membrane.
  • Finally, in the Benedict's Reagent test for dextrose, note whether this simple sugar was able to pass through the semipermeable membrane of the “cell” into the water in the beaker. Discuss with the class which of the molecules you think could and could not pass through the semipermeable membrane.

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HYPOTHESES OF DIFFUSION

2. theoretical part, 2.2. diffusion theory, 2.2.1 hypotheses of diffusion.

Some hypotheses have been progressed to explain how a bond is formed in the solid state.

There are currently 6 hypotheses that can be used to explain the diffusion process. They are film hypothesis, recrystallization hypothesis, energy hypothesis, dislocation hypothesis, electron hypothesis and diffusion hypothesis.

By the film hypothesis, all metals and alloys possess the same property to seize, when clean surfaces are brought together within the range of interatomic forces. The observed differences in weldability among various metals are clarified by the presence of surface films.

The oxide films which are bad for joining can be hard, brittle, viscous, or plastic. When the metals being joined are subjected to cold plastic deformation, the hard and brittle films are broken up to reveal clean metal layers which, on being closed together within the range of inter-atomic forces, form a strong bond. In all the cases oxide film was played the minor role in bonding. [4]

The recrystallization hypothesis is put important on recrystallization as the principal factor in bond formation. By this hypothesis, deformation and the following strain hardening, coupled with exposure to relatively high temperatures at the interface, because the atoms in the lattices of the materials are being joined to flow to other sites so that there appear, at their boundaries, grains common to both pieces with the result that a bond is formed.

Recrystallization formed the new grain in welding area, from the strong bond. [4]

The energy hypothesis, for a diffusion bond to form the atoms of the metals being joined, should be raised to what may be called the energy threshold of adhesion. At this threshold, the formation of atomic bonds is not an important factor, metallic bonds come into being between the atoms at the surfaces, and the interface between the two pieces disappears.

The combination of plastic deformation important for the onset force applied to the metal decreases with increase in the energy of an atom of the metal. The energy hypothesis fails to derive which properties of the metals being joined are responsible for the degree of force to make bonding. [5,6]

By the dislocation hypothesis due to J. Friedel, E. I. Astrov, and some others says that

“the joint plasti eformation auses islo ations to move to the surfa e”. One body of opinion is that the introduced of dislocations at the contact surface minimise resistance to plastic deformation and aids in joining the metals. Bond formation is a result of the plastic flow of the metal within the contact zone/welding zone. [2,5]

The electron hypothesis has been advanced by G. V. Samsonov et al. In their opinion, “the surface pressure results in the formation of stable electron configurations involving the

Sanjeeb Samal 18 atoms of the metals in onta t”. If the ele tron onfiguration of t o metals having a high weight in statistically the bond strength or adhesion must be a lower strength. The electrical configuration of metal and element factor gives on sight into their weld ability, wettability, diffusion processes, etc. [1,5]

By the diffusion hypothesis, the formation of a good bond between the surfaces in contact based on the inter-diffusion of atoms into the dimension of the specimens. The surface atoms of a metal have free, unfilled bonds (vacancies) which capture any atoms moving within the range of inter-atomic forces. A high concentration alloy joint with low concentration alloy after the diffusion with the help of inter atomic force both alloy having equal concentration. [5]

2.3. The activation energy of diffusion

Activation energy is important for atom movements inside the lattice structure. During interstitial diffusion, there is a chance that the neighboring sites are vacant. However, in substitutional diffusion, it is bit complicated, since vacancy should be present in the next neighbor position and then only there will be a possibility of the jump. Here is talked about activation energy required for interstitial self diffusion. [3]

Atom from the ground state (marked as G g ) jumps to another ground state but go through an activated state (marked as G a ) in Fig. 6. where it has to move its neighboring atoms elastically and then it should move to one more place as a plastically and energy level should be again G g . [3]

Fig. 6. Free energy vs atom reversibly move distance [3]

Activation energy can be evaluated from the diffusion coefficients, calculated at different temperatures according to the eq n .(3), respectively according to the eq n . (4).

Sanjeeb Samal 19 (3) (4) Where:

D - Diffusion coefficient (m 2 .s -1 ) D 0 - Pre expontial factor (m 2 .s -1 )

R - Gas constant 8,3144598 (8.314 J/mol k) Q - Activation energy for diffusion (J)

T - Temperature (°C)

The result is plotted in graph D vs. 1/T as is shown in Fig. 7. From graph (slope) is possible to determine the activation energy which is necessary for diffusion, Q. [3]

Fig.7. Diffusion coefficient vs temperature [8]

2.4. Bonding mechanism

Generally, different types of bonding mechanisms are used for the diffusion process. These are according to the sequence of occurrence: (1) plastic yielding resulting in deformation of original surface asperities, (2) next surface diffusion from surface source to a neck. Then follows volume diffusion from surface source to a neck (3) and disappear from a surface source to condensation at the neck (4). Next steps are grain boundary diffusion from an inter -facial source to a neck and volume diffusion from interfacial source to a neck (6). The last one is a creep at a modest temperature (7). [9,10]

All these mechanisms are separated into three main parts:

Sanjeeb Samal 20 Stage 1 - Plastic deformation. The contact area between two surfaces is very small, having asperity, initially small when applies pressure and temperature quickly grow the contact surface area, which means local stress below the yield strength of the material. Some factors are very important for the first step of bonding such as surface roughness, yield strength, hardening after machining, temperature, and pressure. Stage 1 called as high pressure diffusion welding phase is schematically explained in Fig. 8. [11,14]

Fig.8. Surface structure before and after welding [14]

Stage 2: During the second stage, creep and diffusion role more than deformation and many of the voids shrink and some of the voids are disappeared as grain boundary diffusion of atoms continues. [9,14]

Stage 3: In the third stage, the remaining voids are removed by volume diffusion of atoms to the void surface.

For good bonding required a proper combination of flatness and smoothness of the surface.

A certain minimum degree of flatness and smoothness is required to guarantee uniform contact. Recrystallization plays the important role in surface diffusion, which increases the speed of the diffusion. [9,14]

The various routes for diffusion are contained in Fig. 9 and these mechanisms are divided into two main stages. Stages of deformation and diffusion and power law creep. The atoms start to migration provides the basic class of mechanisms by applies pressure and temperature simultaneously. The mechanisms are in Fig.9: (1) plastic yielding resulting in deformation of original surface asperities; (2) surface diffusion from surface source to a neck; (3) volume diffusion from a surface to a neck; (4) evaporation from a surface source condensation at a neck; (5) grain boundary diffusion from an interfacial source to a neck; (6) volume diffusion from an interfacial source to a neck; and (7) diffusional creep under the action of capillary force. [9,10,14] In other words, no fundamental distinction needs be made between stress induced matter transport (coble creep) that giving from the presence of a curved interface. [13]

Sanjeeb Samal 21 Fig 9 various mechanisms of materials transfer [13]

The Fig (9) explains that diffusion of the atom through the surface source, Fig (9.b) material transfer through interface source and Fig (9.c) the bulk deformation of mechanism [13].

2.5. Fundamental process parameter

Diffusion welding depends upon the certain parameters and parameters assembled into six categories: (1) surface preparation, (2) temperature, (3) time, (4) pressure, (5) special metallurgical effect and (6) using of the interlayer. Highly important and main process parameters are temperature, time and pressure. [8]

The influence of temperature on the diffusion process should be characterized in the following points:

1) Temperature is possible promptly changed and easy to measure and control.

2) Temperature has an impact on plasticity, diffusivity, oxide solubility, etc.

3) Temperature has an influence on allotropic transformation, recrystallization and other actions in materials.

4) Increasing temperature increases a diffusion rate and it allows decreasing of welding cycles. It has an influence on economic of the operation.

5) The temperature should be higher than 0.5 melting temperature T m . It correctness in between 0.6 to 0.8 melting point.

Also, the influence of pressure on the diffusion process should be characterized in the following points:

1) Pressure affects several of the diffusion welding mechanisms. The initial deformation phase of bond directly affects the intensity of pressure applied.

Sanjeeb Samal 22 2) Higher pressure means greater interface deformation and lower localized

recrystallization temperature.

3) Pressure should be kept up constant during all welding process (holding time).

4) The level of pressure heavily depends on the used temperature and on the mechanical properties of the welded materials.

5) Press is for a given temperature level evaluating with help of cascade (RAMP) test.

The influence of time on the diffusion process should be characterized in the following points:

1) Time depends on temperature and pressure because diffusional reaction linearly or parabolic related to time. An increase in temperature compresses the amount of time required to complete a diffusion process. [6]

2) In the system with thermal and mechanical inertia, diffusion time is longer due to the unreasonably of a suddenly changing variable. If there is no inertia, the problem may be a welding time reduction.

3) For conservative point of view, it should be reduced welding time factor so than it can increase the production rate. thick oxides prior to bonding is also crucial. [6,8,12]

The initial surface finish is simply obtained by machining, grinding or polishing. An accurately prepared surface is flat. Flatness and smoothness are essential in order to assure that the interface can achieve the necessary compliance without an excessive level of deformation at welding zone. [8]

Machine finishes, grinding or abrasive polishing is usually adequate as long as an appropriate precaution is exercised to minimise warpage and distortion. [5]

The secondary effect of the initial machining or abrading, not always recognized, is the deformation established into the surface during machining. [5,6]

In diffusion bonding, the oxide film is also a big issue to obtain high quality joining. There have been reported several solutions to this issue. They are: inserted inter layer, surface treatment, carried out diffusion bonding in a vacuum or an inert gas such as Ar. The surface treatment, especially grit blast, is the most famous method in this solution. Before diffusion

Sanjeeb Samal 23 bonding, grit blast treatment is performed to expel the surface oxides and supply adequate rugged surface. [5]

In Fig. 10 there is shown the effect of roughness on weld strength by diffusion welding of steels 12 060 and 19 463. In this case, was used at temperature 950 °C, press 20 MPa and welding time 300 sec.. From graph his clear that the high strength was achieved in the range of roughness R a from 1,6 to 3,2 µm.

Fig. 10. Effect of roughness weld strength for welding steels 12 060 and 19 463 [5]

2.7. Method of heating material during diffusion welding

The various processes of heating the work pieces during diffusion welding can be divided into two groups. In the first group, the heat is transferred to the work piece through conduction or radiation thanks to an external heat source. The other one used the heat which is produced in the work pieces themselves by the conversion of electricity into thermal energy. It is especially at the contact point of the materials where the greatest transient resistance. [5]

2.7.1. Radiation heating

A heat source may be situated inside or outside the work or vacuum chamber. The highest acceptable temperature for radiation heating depends on the thermal stability of the chamber material. In diagram form, several postioning using radiation heating are shown in Fig. 11.

The workpiece, 1, is set up on the amount, 2, inside the vacuum chamber, 3, and is heated by radiation from a heater,4, placed outside (Fig. 11a) or inside (Fig. 11b). The heating rate

Sanjeeb Samal 24 can be controlled by varying the voltage applied to the heater. In practice, the heaters, 4, are usually placed inside the chamber. [5]

The above process some demerits also there, where heater material might vapour and stick on the surface of the workpiece. This can be eliminated by (Fig. 11.c), where S 0 that the work piece could not melt and be welded to the heater if they were in direct contact, the latter is given a thin coat of aluminium oxide, 4, which separates them.

Fig. 11 Heating parts by radiation and conduction [5]

In the arrangement shown in (Fig. 11 d), the workpiece, 1, is placed inside the chamber, 2, and heat is supplied by an electric furnace, 3, situated outside. Now heats usually supplied by thermal conduction, moreover, radiation from the surface of the hot chamber also plays a crucial part.

2.7.2. Resistance heating

During resistance heating, the essential heat is supplied by the passage of an electric current through the work pieces themselves. The pieces are in direct contact with the current-source. The rate of heating determines the resistance of the specimen R s , and the rms value of the current I rms , passing through the specimen. The amount of heat, Q (J), generated by the movement of current can be found by Joule's law - eq n .(5):

Q= (5) Where

Q - Amount of heat (J)

t - time (sec)

I rms - Value of current (A)

R s - Rate of heating (W)

Sanjeeb Samal 25 In resistance heating, the higher temperature of the work depends upon only its melting point. An important requirement for resistance heating is an arrangement of a reliable physical contact between the work and the electrodes conveying the current. [3-6]

Fig. 12. Resistance heating [5]

An arrangement for resistance heating is shown in Fig. 12. The workpiece, 1, placed inside a vacuum chamber, 2, is clamped in jaws, 3 and 4. Terminal 3 is made fast to an electrode, 5, whereas terminal 4 is connected to a second electrode, 5, by a copper pig-tail, 7, and a copper block, 8. Provision of a flexible pig-tail is important as it avoids straining the work due to the volumetric changes occurring in the course of heating and cooling. [2,5]

Apart from copper, the electrodes can be also made of graphite and tungsten, in which case the materials to be heated may broadly vary in thermal conductivity and resistance. A further advantage is that heat can be provided to hard-to-reach spots. The best type of electrode for a given material can be established by experiment, using various combinations and various geometry. With graphite or carbon electrodes which are generally soft, a minimum pressing load should be used than in the case of electrodes made of high-temperature alloys and steels. [5]

2.7.3. Induction heating

In this case, the workpiece to be heated is put in the high-frequency electromagnetic field set up around an inductor by a source of high frequency current. A refinement of induction heating is that electric energy is transferred from the inductor to work over a distance of a few centimetres, without any definite contact between them. Heat is generated within the work by the circulating eddy currents induced by the applied magnetic field. [5] The current thus induced is possible express by eq n .(6):

Sanjeeb Samal 26 I= E/R (6) Where,

R - total apparent resistance (Ω) E - Electro motive force (V)

I - Current (A)

2.8. Thermal-stress simulator Gleeble 3500

The temperature-stress simulator Gleeble 3500 is a product of the American company Dynamic System Inc. and it is used to test material response during various mechanical and metallurgical conditions. Physical simulations of technological processes are being used nowadays more often.These simulations serve to simulate the thermal-mechanical processes which correspond to real conditions, but they are performed in laboratories.

Simulator Gleeble 3500 was purchased by the Technical University of Liberec in 2013 and is used especially for the testing of forming and welding processes under different temperature-stresses conditions. The Gleeble device will be used to solve the practical part of this thesis. Fig. 13 shows the basic Gleeble 3500 simulator assembly.

Fig. 13. Basic Gleeble 3500 simulator assembly. 1- Console; 2- Load unit; 3 – Pocket Jaw

Sanjeeb Samal 27

2.8.1. Basic information

The Gleeble 3500 is the most commonly used temperature-stress simulator. This dynamic system can be used to identify almost all of the happenings run in metals at high temperatures. The device library was created over 50 years and therefore contains a huge amount of information about the machine. Device Gleeble can simulate almost any thermal-mechanical load that occurs both during processing and during subsequent operation.

The Gleeble system is capable of testing samples with a maximum diameter of 20 mm or

The direct resistance heating system in Gleeble can heat specimens at rates of up to 10,000

°C. S -1 or can hold steady state equilibrium temperatures. High thermal conductivity grips hold the specimen, making Gleeble capable of high cooling rates. The clamping jaws are used both to heat and to cool with the test sample. When combined with other cooling devices, it is possible to cool the surface of the sample up to 6,000 °C. S -1 . This way of heating can do keep the required heating and storage temperatures accurate to ± 1 °C. The Gleeble heating rate is many times higher than the pressure load.

There are two ways of controlling and monitoring the temperature of the specimen during heating and cooling. One is by thermocouples and the other one by an optical pyrometer.

There are four thermal channels available in a standard Gleeble system, with either four thermocouples channels or one pyrometer channel and three thermocouples channels.

Different thermocouples can be used depending on the temperature range of the test. It is possible to choose from types B, E, K, R and S. One of the most widely used types of thermocouples is type K because it has a wide operating range from -180 °C to +1250 °C.

For very high temperature is using an R-type thermocouple. It is operating range up to 1450

°C. Thermocouples are welded to the test sample in most cases with help of capacitor welder. Before welding are important samples to clean and degrease properly. Fig. 14.

shows connection points for measurement of temperature.

Sanjeeb Samal 28 Fig.14. Channels of four thermocouples

2.8.3. Temperature gradients in the sample during simulation

The Gleeble device allows to control the temperature gradient patterns in the samples during the simulation. In cases like welding processes to which, of course also includes diffusion welding, working with a steep temperature gradient to degrade material in the vicinity of the joint as small as possible. In other cases, it is required for uniform temperature effects and flat temperature gradients.

The temperature gradients of the test sample are affected by the type of test material, its electrical and temperature resistance, the free length between jaws, cross section and the ratio of the surface to the total volume of the component. Free sample length represents the distance between the edges of the high temperature clamping jaws. It also applies that the longer it is contact between the sample and clamping jaws, the steeper the temperature gradient inside of the test sample. Extending the distance between the specimen and the jaws increases differences between the maximum and minimum temperatures achieved in the sample.

In practice, jaws with partial or full contact are used and theoretically it is possible to use any material to produce them. The most used materials for production these jaws are copper (containing Cu 99%) or austenitic high - alloy X5CrNi18-8 steel. There are considerable differences in the thermal conductivity of both materials; therefore, they have very different temperature gradients. Fig. 15. shows a temperature gradient on a sample of S355J2 steel

  • HYPOTHESES OF DIFFUSION (You are here)
  • THERMAL-STRESS SIMULATOR GLEEBLE 3500
  • DESIGN OF A DIFFUSION WELDING EXPERIMENT

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Title: diffusion models with learned adaptive noise.

Abstract: Diffusion models have gained traction as powerful algorithms for synthesizing high-quality images. Central to these algorithms is the diffusion process, which maps data to noise according to equations inspired by thermodynamics and can significantly impact performance. A widely held assumption is that the ELBO objective of a diffusion model is invariant to the noise process (Kingma et al.,2021). In this work, we dispel this assumption -- we propose multivariate learned adaptive noise (MuLAN), a learned diffusion process that applies Gaussian noise at different rates across an image. Our method consists of three components -- a multivariate noise schedule, instance-conditional diffusion, and auxiliary variables -- which ensure that the learning objective is no longer invariant to the choice of the noise schedule as in previous works. Our work is grounded in Bayesian inference and casts the learned diffusion process as an approximate variational posterior that yields a tighter lower bound on marginal likelihood. Empirically, MuLAN sets a new state-of-the-art in density estimation on CIFAR-10 and ImageNet compared to classical diffusion. Code is available at this https URL

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