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Lecture 3. Microscopic dynamics and Macroscopic D We can see that if we want to understand the diffusion constant measured in any material a knowledge of how the microscopic structure and dynamics of the material determine the diffusion coefficient is required.
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Microscopic dynamics and Macroscopic D We can see that if we want to understand the diffusion constant measured in any material a knowledge of how the microscopic structure and dynamics of the material determine the diffusion coefficient is required. Consider a particle moving on a 1-D lattice For a large number (n ) of random hops of distance on a 1-D lattice the mean displacement, , will be zero because moves left or right (±x) are equally probable.
Microscopic dynamics and Macroscopic D We can see that if we want to understand the diffusion constant measured in any material a knowledge of how the microscopic structure and dynamics of the material determine the diffusion coefficient is required. Consider a particle moving on a 1-D lattice Large # hops, n For a large number (n ) of random hops of distance on a 1-D lattice the mean displacement, , will be zero because moves left or right (±x) are equally probable.
Microscopic dynamics and Macroscopic D We can see that if we want to understand the diffusion constant measured in any material a knowledge of how the microscopic structure and dynamics of the material determine the diffusion coefficient is required. Consider a particle moving on a 1-D lattice Large # hops, n with the same individual hop distance For a large number (n ) of random hops of distance on a 1-D lattice the mean displacement, , will be zero because moves left or right (±x) are equally probable.
Microscopic dynamics and Macroscopic D We can see that if we want to understand the diffusion constant measured in any material a knowledge of how the microscopic structure and dynamics of the material determine the diffusion coefficient is required. Consider a particle moving on a 1-D lattice Large # hops, n with the same individual hop distance On average, distance moved is zero. For a large number (n ) of random hops of distance on a 1-D lattice the mean displacement, , will be zero because moves left or right (±x) are equally probable.
Microscopic dynamics and Macroscopic D We can see that if we want to understand the diffusion constant measured in any material a knowledge of how the microscopic structure and dynamics of the material determine the diffusion coefficient is required. Consider a particle moving on a 1-D lattice Large # hops, n with the same individual hop distance On average, distance moved is zero. However, any individual particle may have moved a long way. For a large number (n ) of random hops of distance on a 1-D lattice the mean displacement, , will be zero because moves left or right (±x) are equally probable.
The mean of the squared displacements, however, will not be zero
The mean of the squared displacements, however, will not be zero Use this to quantify extent of diffusion/particle mobility
The mean of the squared displacements, however, will not be zero Use this to quantify extent of diffusion/particle mobility Squared displacement
The mean of the squared displacements, however, will not be zero Use this to quantify extent of diffusion/particle mobility Squared displacement n hops results in an nxn matrix
The mean of the squared displacements, however, will not be zero Use this to quantify extent of diffusion/particle mobility Squared displacement n hops results in an nxn matrix All diagonal elements are positive
The mean of the squared displacements, however, will not be zero Use this to quantify extent of diffusion/particle mobility Squared displacement n hops results in an nxn matrix All diagonal elements are positive Off-diagonal elements can be positive or negative, on average sum to zero
This is true because when squaring i, the off-diagonal terms will sum to zero for large n We have then If t is the time taken to acquire an mean square displacement, and is the time for a single elementary hop then:
This is true because when squaring i, the off-diagonal terms will sum to zero for large n Recall, We have then If t is the time taken to acquire an mean square displacement, and is the time for a single elementary hop then:
This is true because when squaring i, the off-diagonal terms will sum to zero for large n Recall, We have then Average distance a particle has moved is given by: If t is the time taken to acquire an mean square displacement, and is the time for a single elementary hop then:
This is true because when squaring i, the off-diagonal terms will sum to zero for large n Recall, We have then Average distance a particle has moved is given by: Elementary hop distance x square root of the number of hops If t is the time taken to acquire an mean square displacement, and is the time for a single elementary hop then:
This is true because when squaring i, the off-diagonal terms will sum to zero for large n Recall, We have then Average distance a particle has moved is given by: Elementary hop distance x square root of the number of hops If t is the time taken to acquire an mean square displacement, and is the time for a single elementary hop then: Total diffusion time
This is true because when squaring i, the off-diagonal terms will sum to zero for large n Recall, We have then Average distance a particle has moved is given by: Elementary hop distance x square root of the number of hops If t is the time taken to acquire an mean square displacement, and is the time for a single elementary hop then: Total diffusion time Individual hop time
Substituting for n Equate the average distance from analysis of macroscopic diffusion profile for thin source with microscopic ‘rms’ distance
Substituting for n Equate the average distance from analysis of macroscopic diffusion profile for thin source with microscopic ‘rms’ distance ‘Einstein relationship’ : relates microscopic dynamics to macroscopically measured diffusion through a fundamental hopping distance and a fundamental hopping time.
Substituting for n Equate the average distance from analysis of macroscopic diffusion profile for thin source with microscopic ‘rms’ distance ‘Einstein relationship’ : relates microscopic dynamics to macroscopically measured diffusion through a fundamental hopping distance and a fundamental hopping time. The factor 2 represents the probability of hops (left or right) on a 1D lattice
Dimensionality of diffusion 1D 2D 3D
Dimensionality of diffusion 1D 2D 3D Although we deal with real 3D materials, the dimensionality of the diffusive process may well be lower.
NB 1/t is often replaced by a frequency of hopping, n to give: Dimensionality of diffusion 1D 2D 3D Although we deal with real 3D materials, the dimensionality of the diffusive process may well be lower.
NB 1/t is often replaced by a frequency of hopping, n to give: Dimensionality of diffusion 1D 2, 4 or 6 2D 3D Although we deal with real 3D materials, the dimensionality of the diffusive process may well be lower.
Missing anion or cation in a lattice Occur in pairs to maintain electrical neutrality not necessarily together.
Missing anion or cation in a lattice Occur in pairs to maintain electrical neutrality not necessarily together. Vacant lattice site created by an atom moving into an interstititial position
Missing anion or cation in a lattice Occur in pairs to maintain electrical neutrality not necessarily together. Vacant lattice site created by an atom moving into an interstititial position These are intrinsic vacancies Other vacancies may be created by trace amounts of impurities or variable oxidation states of some constituent ions e.g. in NaCl a 2+ impurity (Ca2+, say) will require a missing anion (Cl-) as charge balance.
Missing anion or cation in a lattice Occur in pairs to maintain electrical neutrality not necessarily together. Vacant lattice site created by an atom moving into an interstititial position These are intrinsic vacancies Other vacancies may be created by trace amounts of impurities or variable oxidation states of some constituent ions e.g. in NaCl a 2+ impurity (Ca2+, say) will require a missing anion (Cl-) as charge balance. These are extrinsic vacancies
Small concentrations (up to a few percent) dissolved in lattice, taking up interstitial positions - diffusion often rapid as number of vacant interstitial sites is high.
Small concentrations (up to a few percent) dissolved in lattice, taking up interstitial positions - diffusion often rapid as number of vacant interstitial sites is high. } 2,3,4 all mechanisms proposed to move one atom onto the site of another. Elastic energy required by (3) was considered too great so a cooperative mechanism (4) was postulated.
Small concentrations (up to a few percent) dissolved in lattice, taking up interstitial positions - diffusion often rapid as number of vacant interstitial sites is high. } 2,3,4 all mechanisms proposed to move one atom onto the site of another. Elastic energy required by (3) was considered too great so a cooperative mechanism (4) was postulated. (2) allows direct hopping onto adjacent vacant site, explicitly requires vacancies, (3) + (4) do not.
Small concentrations (up to a few percent) dissolved in lattice, taking up interstitial positions - diffusion often rapid as number of vacant interstitial sites is high. } 2,3,4 all mechanisms proposed to move one atom onto the site of another. Elastic energy required by (3) was considered too great so a cooperative mechanism (4) was postulated. (2) allows direct hopping onto adjacent vacant site, explicitly requires vacancies, (3) + (4) do not. (5) Mechanism actually observed in some fast ion conductors (see later) combination of vacancy (2) and interstitial (1) mechanisms.
Brass - alloy of Cu/Zn Concentration gradient exists between Cu and Cu/Zn - Zn will diffuse out, ‘down’ the gradient’ as sample is heated for long periods of time.
Brass - alloy of Cu/Zn Concentration gradient exists between Cu and Cu/Zn - Zn will diffuse out, ‘down’ the gradient’ as sample is heated for long periods of time. What happens to the inert markers?
Brass - alloy of Cu/Zn Concentration gradient exists between Cu and Cu/Zn - Zn will diffuse out, ‘down’ the gradient’ as sample is heated for long periods of time. What happens to the inert markers? They move closer together the longer time goes on. Conclusion: Zn diffuses faster than Cu.
Brass - alloy of Cu/Zn Concentration gradient exists between Cu and Cu/Zn - Zn will diffuse out, ‘down’ the gradient’ as sample is heated for long periods of time. What happens to the inert markers? They move closer together the longer time goes on. Conclusion: Zn diffuses faster than Cu. Must be vacancy mechanism, because if direct (3) or cooperative (4) then Dcu = DZn
Brass - alloy of Cu/Zn Concentration gradient exists between Cu and Cu/Zn - Zn will diffuse out, ‘down’ the gradient’ as sample is heated for long periods of time. What happens to the inert markers? They move closer together the longer time goes on. Conclusion: Zn diffuses faster than Cu. Must be vacancy mechanism, because if direct (3) or cooperative (4) then Dcu = DZn If markers move in then new sites are being created beyond markers with vacancy flow inwards
Brass - alloy of Cu/Zn Concentration gradient exists between Cu and Cu/Zn - Zn will diffuse out, ‘down’ the gradient’ as sample is heated for long periods of time. What happens to the inert markers? They move closer together the longer time goes on. Conclusion: Zn diffuses faster than Cu. Must be vacancy mechanism, because if direct (3) or cooperative (4) then Dcu = DZn If markers move in then new sites are being created beyond markers with vacancy flow inwards Direct vacancy mechanism is the predominant mechanism in solid state diffusion.
