# -*- coding: utf-8 -*-
# written by Ralf Biehl at the Forschungszentrum Jülich ,
# Jülich Center for Neutron Science (JCNS-1)
# Jscatter is a program to read, analyse and plot data
# Copyright (C) 2015-2026 Ralf Biehl
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
#
import inspect
import math
import os
import sys
import numbers
import numpy as np
from scipy import linalg as la
import scipy
import scipy.integrate
import scipy.interpolate
import scipy.constants
import scipy.special as special
try:
from scipy.special import sph_harm_y as Ylm
except ImportError:
def Ylm(n, m, theta, phi):
return special.sph_harm(m, n, theta, phi)
from .. import dataArray as dA
from .. import dataList as dL
from .. import formel
try:
from ..libs import fscatter
useFortran = True
except ImportError:
fscatter = None
useFortran = False
__all__ = ['simpleDiffusion','cumulant','cumulantDLS','stretchedExp','jumpDiffusion','transDiff',
'methylRotation','methylRotation','diffusionHarmonicPotential',
'diffusionHarmonicPotential','finiteZimm','fixedFiniteZimm','finiteRouse','fixedFiniteRouse',
'diffusionPeriodicPotential','zilmanGranekBicontinious','zilmanGranekLamellar',
'integralZimm','transRotDiffusion','resolution', 'doubleDiffusion']
pi = np.pi
_path_ = os.path.realpath(os.path.dirname(__file__))
#: Planck constant in µeV*ns
h = scipy.constants.Planck / scipy.constants.e * 1E15 # µeV*ns
#: h/2π reduced Planck constant in µeV*ns
hbar = h/2/pi # µeV*ns
kb = 1.3806503e-23 # J/K in SI units
try:
# change in scipy 18
spjn = special.spherical_jn
except AttributeError:
spjn = lambda n, z: special.jv(n + 1 / 2, z) * np.sqrt(pi / 2) / (np.sqrt(z))
def _diff(t, gamma):
return dA(np.c_[t, np.exp(- gamma * t)].T)
[docs]
def transDiff(t, q, D):
r"""
Translational diffusion intermediate scattering function in t domain.
.. math:: I(t,q) = e^{-q^2Dt}
Parameters
----------
t : array
Frequencies in 1/ns
q : float
Wavevector in nm**-1
D : float
Diffusion constant in nm**2/ns
Returns
-------
dataArray
"""
gamma = q * q * D
result = _diff(t, gamma)
result.setColumnIndex(iey=None)
result.columnname = 't;Iqt'
result.modelname = inspect.currentframe().f_code.co_name
result.wavevector = q
result.D = D
return result
[docs]
def simpleDiffusion(t, gamma=None, q=None, D=None, s=0, beta=1, type='lognorm'):
r"""
Intermediate scattering function [g1(t)] for diffusing particles from distribution of relaxation rates.
.. math:: I(q,t,D, \sigma ) = \beta \int g(\Gamma, \Gamma_0, \sigma ) e^{-\Gamma t}
and relaxation rate :math:`\Gamma_0=q^2D`.
Parameters
----------
t : float, array
Times
gamma : float
Mean relaxation rate in inverse t units.
Overrides q and D if given. If q and D given gamma=q*q*D
q : float, array
Wavevector
beta : float
Intercept :math:`\beta` in DLS. The amplitude prefactor.
D : float
Mean diffusion coefficient.
s : float
Relative standard deviation of the diffusion coefficient distribution from e.g. polydispersity.
In absolute units :math:`\sigma=s*q^2*D=w\Gamma`.
For a typical DLS experiment `s≈0.25` just from instrument noise
(e.g. noise ≈ 1e-3 Zetasizer Nano, Malvern)
For s=0 a single exponential is used.
type : 'truncnorm', default 'lognorm'
Distribution shape.
- 'lognorm' lognorm istribution as normal distribution on a log scale.
.. math:: g(x, \m, \sigma) = \frac{ 1 }{\ x\sigma\sqrt{2\pi}} exp(-\frac{\ln(x- \mu)^2}{ 2 \sigma^2 })
This is approximatly what you get from CONTIN or NNLS algorithm in DLS.
- 'truncnorm' normaldistribution cut at zero.
.. math:: g(x, \mu, \sigma)= e^{-0.5(\frac{x-\mu}{\sigma})^2} / (\sigma\sqrt{2\pi}) \ \text{ for x>0}
Returns
-------
out : dataArray
intermediate scattering function or :math:`g_1`
- .D, .wavevector, .beta respective input parameters
- pdf : Probability of the distribution in the interval around pdf[0,i]
(half distance to neighboring points) that sum(pdf[2])==1 similar to CONTIN results.
Notes
-----
Units of q, t and D result in unit-less [q*q*D*t] like q in 1/cm, t in s -> D in cm*cm/s .
Remember that :math:`g_2(t)` (intensity correlation for DLS) is :math:`g_1^2(t)=g_2(t)-1`.
Examples
--------
::
import jscatter as js
import numpy as np
t = js.loglist(1e-6,10,1000) # in seconds
# ≈ a protein with a MW of 140kDa
D = 3e-7 # unit cm^2/s
q = 4*np.pi/633e-7*np.sin(np.pi/4) # 90º HeNe laser in DLS
p=js.grace(1.8,1)
p.multi(1,2)
p[0].xaxis(label=r't / s ',scale='log')
p[0].yaxis(label=r'g\s1\N(t) ')
p[1].xaxis(label=r'\xG\f{} / 1/s ',scale='log')
p[1].yaxis(label=[r'P(\xG\f{})',1,'opposite'],)
for c,s in enumerate([0.25,0.5,1,2],1):
dls = js.dynamic.simpleDiffusion(t=t,q=q,D=D,s=s)
p[0].plot(dls,sy=0,li=[1,3,c],le=f's={s}')
p[1].plot(dls.pdf[0], 20 * dls.pdf[1],sy=0,li=[1,3,c])
p[0].legend(x=0.01,y=0.8)
p[0].title('DLS correlation')
p[1].title('rate distribution')
# p.save(js.examples.imagepath+'/simpleDiffusion.jpg',size=(1.8,1))
.. image:: ../../examples/images/simpleDiffusion.jpg
:align: center
:width: 50 %
:alt: Zimm
"""
if isinstance(D, numbers.Number) and isinstance(q, numbers.Number):
gamma = q*q*D
if s == 0:
# no width or distribution
result = _diff(t, gamma)
else:
if type[0] == 't':
a = (0 - gamma) / (s * gamma)
b = np.inf
result = formel.pDA(_diff, s * gamma, parname='gamma', type='truncnorm', gamma=gamma, t=t, nGauss=30, a=a, b=b)
else:
result = formel.pDA(_diff, s * gamma, parname='gamma', type='lognorm', gamma=gamma, t=t, nGauss=30)
# use interval probability
dif = result.pdf[0] * 0
dif[1:] = np.diff(result.pdf[0])
dif[:-1] += dif[1:]
dif /= 2
result.pdf[1] = result.pdf[1] * dif
result.Y = beta * result.Y
result.beta = beta
result.Diffusioncoefficient = D
result.s = s
result.wavevector = q
result.Gamma = gamma
result.columnname = 't;Iqt'
result.setColumnIndex(iey=None)
result.modelname = inspect.currentframe().f_code.co_name
return result
[docs]
def doubleDiffusion(t, q=None, gamma1=None, D1=None, s1=0,
gamma2=None, D2=None, s2=0, frac=0.5, beta=1, type='lognorm'):
r"""
Intermediate scattering function [g1(t)] for diffusing particles from bimodal distribution of relaxation rates.
.. math:: I_i(q,t,D_i,\sigma_i ) = \int g(\Gamma_i, \Gamma_{i,0} \sigma_i ) e^{-\Gamma_i t} d\Gamma_i
.. math:: I(q,t) = beta [f\ I_1(q,t,D_1,\sigma_1 ) + (1-f)\ I_2(q,t,D_2,\sigma_2 )]
and relaxation rates :math:`\Gamma_{i,0}=q^2D_i`.
Parameters
----------
t : float, array
Times
gamma1,gamma2 : float
Mean relaxation rates in inverse t units.
Overrides q and D if given. If q and D given gamma=q*q*D
q : float, array
Wavevector
beta : float
Intercept :math:`\beta` in DLS. The amplitude prefactor.
D1,D2 : float
Mean diffusion coefficients.
s1,s2 : float
Relative standard deviations of the diffusion coefficient distributions.
In absolute units :math:`\sigma=s*q^2*D`.
For a typical DLS experiment `s≈0.25` just from instrument noise (e.g. noise ≈ 1e-3 Zetasizer Nano, Malvern)
frac : float
Fraction of contribution 1. Contribution 2 is (1-frac)
type : 'truncnorm', default 'lognorm'
Distribution shape.
- 'lognorm' lognorm distribution as normal distribution on a log scale.
.. math:: g(x, \mu, \sigma) = \frac{ 1 }{\ x\sigma\sqrt{2\pi}} exp(-\frac{\ln(x- \mu)^2}{ 2 \sigma^2 })
- 'truncnorm' normal distribution cut at zero.
.. math:: g(x, \mu, \sigma)= e^{-0.5(\frac{x-\mu}{\sigma})^2} / (\sigma\sqrt{2\pi}) \ \text{ for x>0}
Returns
-------
out : dataArray
Intermediate scattering function or :math:`g_1`
- .D1, .D2, .wavevector, .beta respective input parameters
- pdf1, pdf2 : Probability of the distribution in the interval around pdf[0,i]
(half distance to neighboring points) that sum(pdf[2])==1 similar to CONTIN results.
Notes
-----
Units of q, t and D result in unit-less [q*q*D*t] like q in 1/cm, t in s -> D in cm*cm/s .
Examples
--------
::
import jscatter as js
import numpy as np
t = js.loglist(1e-6,10,1000) # in seconds
# ≈ a protein with a MW of 140kDa
D1 = 3e-8 # unit cm^2/s
D2 = 3e-5 # unit cm^2/s
q = 4*np.pi/633e-7*np.sin(np.pi/4) # 90º HeNe laser in DLS
p=js.grace(1.8,1)
p.multi(1,2)
p[0].xaxis(label=r't / s ',scale='log')
p[0].yaxis(label=r'g\s1\N(t) ')
p[1].xaxis(label=r'\xG\f{} / 1/s ',scale='log')
p[1].yaxis(label=[r'P(\xG\f{})',1,'opposite'],)
for c,s in enumerate([0.25,0.5,1,2],1):
dls = js.dynamic.doubleDiffusion(t=t,q=q,D1=D1,s1=s,D2=D2,s2=s,frac=0.4)
p[0].plot(dls,sy=0,li=[1,3,c],le=f's={s}')
p[1].plot(dls.pdf1[0], 20 * dls.pdf1[1],sy=0,li=[1,3,c])
p[1].plot(dls.pdf2[0], 20 * dls.pdf2[1],sy=0,li=[1,3,c])
p[0].legend(x=0.01,y=0.8)
p[0].title('DLS correlation')
p[1].title('rate distribution')
# p.save(js.examples.imagepath+'/doubleDiffusion.jpg',size=(1.8,1))
.. image:: ../../examples/images/doubleDiffusion.jpg
:align: center
:width: 50 %
:alt: Zimm
"""
result1 = simpleDiffusion(t, gamma=gamma1, q=q, D=D1, s=s1, beta=1, type=type)
result2 = simpleDiffusion(t, gamma=gamma2, q=q, D=D2, s=s2, beta=1, type=type)
result = result1.copy()
result.Y = beta * (frac * result1.Y + (1-frac) * result2.Y)
result.beta = beta
result.wavevector = q
result.frac = frac
del result.Diffusioncoefficient
del result.Gamma
del result.pdf
result.Diffusioncoefficient1 = D1
result.s1 = s1
result.Gamma1 = gamma1
result.Diffusioncoefficient2 = D2
result.s2 = s2
result.Gamma2 = gamma2
result.pdf1 = result1.pdf
result.pdf2 = result2.pdf
result.columnname = 't;Iqt'
result.setColumnIndex(iey=None)
result.modelname = inspect.currentframe().f_code.co_name
return result
[docs]
def cumulant(x, k0=1, k1=0, k2=0, k3=0, k4=0, k5=0):
r"""
Cumulant of order ki.
.. math:: I(x) = k_0 exp(-k_1x+1/2k_2x^2-1/6 k_3x^3+1/24k_4x^4-1/120k_5x^5)
Cumulants can only be used in initial slope analysis and not for a full fit of DLS data to long times.
It is necessary to cut large x. See :py:func:`cumulantDLS` for DLS.
Parameters
----------
x : float
Wavevector
k0,k1, k2,k3,k4,k5 : float
Cumulant coefficients; units 1/x
- k0 amplitude
- k1 expected value
- k2 variance with :math:`\sqrt(k2/k1) =` relative standard deviation
- higher order see Wikipedia
Returns
-------
dataArray
Examples
--------
::
import jscatter as js
import numpy as np
t = js.loglist(1e-6,10,1000) # in seconds
p=js.grace()
p.xaxis(scale='log')
k1=1/0.001
for f in [0,0.1,10.5]:
cum=js.dynamic.cumulant(t,k1=k1,k2=f*2*k1)
p.plot(cum,sy=0,li=-1)
"""
x = np.atleast_1d(x)
G = - k1 * x
if k2 > 0:
G += + 1 / 2. * k2 * x ** 2
if k3 > 0: G += - 1 / 6. * k3 * x ** 3
if k4 > 0: G *= + 1 / 24 * k4 * x ** 4
if k5 > 0: G *= - 1 / 120 * k5 * x ** 5
res = np.zeros_like(G)
res[G < 1] = k0 * np.exp(G[G < 1])
result = dA(np.c_[x, res].T)
result.k0tok5 = [k0, k1, k2, k3, k4, k5]
result.modelname = inspect.currentframe().f_code.co_name
result.columnname = 't;Iqt'
result.setColumnIndex(iey=None)
return result
[docs]
def cumulantDLS(t, A, G, sigma, skewness=0, bgr=0., g2=True):
r"""
Cumulant analysis for dynamic light scattering (DLS) or NSE assuming Gaussian size distribution.
See Frisken et al [1]_ :
.. math:: g_1(t) = A exp(-t/G) \big( 1+(sigma/G t)^2/2. - (skewness/G t)^3/6. \big) + bgr
Returns :math:`g_1^2=g_2-1` for DLS or :math:`g_1(t)` for NSE.
Parameters
----------
t : array
Time
A : float
Amplitude at t=0; Intercept
G : float
Mean relaxation time as 1/decay rate in units of t.
sigma : float
- relative standard deviation if a gaussian distribution is assumed
- should be smaller 1 or the Taylor expansion is not valid
- k2=variance=sigma**2/G**2
skewness : float,default 0
Relative skewness k3=skewness**3/G**3
bgr : float; default 0
Constant background
g2 : bool default = True
Determines correlation type as field correlation (actually :math:`g_1^2=g_2-1`) or
intensity correlation :math:`g_1`.
- True is intensity correlations :math:`g_1^2=g_2-1`.
Actually fitting the measured data is prefered, use this for DLS.
This should be used for DLS as :math:`g_1^2(t \rightarrow \infty)` fluctuates around 0
allowing negative values and prevents a bias during fitting.
- False is field correlations :math:`g_1`.
Use this e.g. for NSE as the field correlation is measured directly.
Returns
-------
dataArray
Notes
-----
To fit diffusion constant D e.g. use
::
def cDLS(t, A, D, sigma, q, bgr=0.0, g2=1):
# g2=1 or 0 switches between g1 or g2minus1
# diffusion coefficient in nm*nm/ns if q in 1/nm and t in microseconds
G=1/(D*q**2*1000)
return js.dynamic.cumulantDLS(t=t, A=A, G=G, sigma=sigma, skewness=0, bgr=bgr, g2=True if g2!=0 else False)
Examples
--------
::
import jscatter as js
import numpy as np
# simulate data
t=js.loglist(0.125,10000,1000) #times in microseconds
q=4*np.pi*1.333/632*np.sin(np.deg2rad(90)/2) # 90 degrees for 632 nm , unit is 1/nm**2
D=0.01 # nm**2/ns * 1000 = units nm**2/microseconds
noise=0.001 # typical < 1e-3
G = 1/q**2*D
g1 = 0.9*np.exp(-t/G) # with 20x larger aggragates
data=js.dA(np.c_[t,g1**2 + noise*np.random.randn(len(t))].T) # intensity correlation with noise
data.makeErrPlot(xscale='log')
data.fit(js.dynamic.cumulantDLS,{'A':0.9, 'G':20, 'sigma':0,'bgr':0},{},{'t':'X'},condition=lambda a:a.Y>0.1)
References
----------
.. [1] Revisiting the method of cumulants for the analysis of dynamic light-scattering data
Barbara J. Frisken APPLIED OPTICS 40, 4087 (2001)
"""
t = np.atleast_1d(t)
A = abs(A)
if skewness == 0:
g1 = A * np.exp(-t / G) * (1 + (sigma / G * t) ** 2 / 2.)
else:
g1 = A * np.exp(-t / G) * (1 + (sigma / G * t) ** 2 / 2. - (skewness / G * t) ** 3 / 6.)
if g2:
result = dA(np.c_[t, g1**2 + bgr].T)
result.type = 'g2minus1'
else:
result = dA(np.c_[t, g1 + bgr].T)
result.type = 'g1'
result.columnname = 't;Iqt'
result.A = A
result.relaxationtime = G
result.sigma = sigma
result.skewness = skewness
result.elastic = bgr
result.modelname = inspect.currentframe().f_code.co_name
result.setColumnIndex(iey=None)
return result
[docs]
def stretchedExp(t, gamma, beta, amp=1):
r"""
Stretched exponential function.
.. math:: I(t) = amp\, e^{-(t\gamma)^\beta}
Parameters
----------
t : array
Times
gamma : float
Relaxation rate in units 1/[unit t]
beta : float
Stretched exponent
amp : float default 1
Amplitude
Returns
-------
dataArray
"""
t = np.atleast_1d(t)
res = amp * np.exp(-(t * gamma) ** beta)
result = dA(np.c_[t, res].T)
result.amp = amp
result.gamma = gamma
result.beta = beta
result.modelname = inspect.currentframe().f_code.co_name
result.setColumnIndex(iey=None)
result.columnname = 't;Iqt'
return result
[docs]
def jumpDiffusion(t, Q, t0, l0):
r"""
Incoherent intermediate scattering function of translational jump diffusion in the time domain.
Parameters
----------
t : array
Times, units ns
Q : float
Wavevector, units nm
t0 : float
Residence time, units ns
l0 : float
Mean square jump length, units nm
Returns
-------
dataArray
Notes
-----
We use equ. 3-5 from [1]_ for random jump diffusion as
.. math:: T(Q,t) = exp(-\Gamma(Q)t)
with residence time :math:`\tau_0` and mean jump length :math:`<l^2>^{1/2}_{av}`
and diffusion constant :math:`D` in
.. math:: \Gamma(Q) = \frac{DQ^2}{1+DQ^2\tau_0}
.. math:: D=\frac{ <l^2>_{av}}{6\tau_0}
References
----------
.. [1] Experimental determination of the nature of diffusive motions of water molecules at low temperatures
J. Teixeira, M.-C. Bellissent-Funel, S. H. Chen, and A. J. Dianoux
Phys. Rev. A 31, 1913 – Published 1 March 1985
"""
t = np.atleast_1d(t)
D = l0 ** 2 / 6. / t0
gamma = D * Q * Q / (1 + D * Q * Q * t0)
tdif = np.exp(-gamma * t)
result = dA(np.c_[t, tdif].T)
result.residencetime = t0
result.jumplength = l0
result.diffusioncoefficient = D
result.modelname = inspect.currentframe().f_code.co_name
result.setColumnIndex(iey=None)
result.columnname = 't;Iqt'
return result
[docs]
def methylRotation(t, q, t0=0.001, fraction=1, rhh=0.12, beta=0.8):
r"""
Incoherent intermediate scattering function of CH3 methyl rotation in the time domain.
Parameters
----------
t : array
List of times, units ns
q : float
Wavevector, units nm
t0 : float, default 0.001
Residence time, units ns
fraction : float, default 1
Fraction of protons contributing.
rhh : float, default=0.12
Mean square jump length, units nm
beta : float, default 0.8
exponent
Returns
-------
dataArray
Notes
-----
According to [1]_:
.. math:: I(q,t) = (EISF + (1-EISF) e^{-(\frac{t}{t_0})^{\beta}} )
.. math:: EISF=\frac{1}{3}+\frac{2}{3}\frac{sin(qr_{HH})}{qr_{HH}}
with
:math:`t_0` residence time,
:math:`r_{HH}` proton jump distance.
Examples
--------
::
import jscatter as js
import numpy as np
# make a plot of the spectrum
w=np.r_[-100:100]
ql=np.r_[1:15:1]
iqwCH3=js.dL([js.dynamic.time2frequencyFF(js.dynamic.methylRotation,'elastic',w=np.r_[-100:100:0.1],q=q )
for q in ql])
p=js.grace()
p.plot(iqwCH3,le='CH3')
p.yaxis(min=1e-5,max=10,scale='l')
References
----------
.. [1] M. Bée, Quasielastic Neutron Scattering (Adam Hilger, 1988).
.. [2] Monkenbusch et al. J. Chem. Phys. 143, 075101 (2015)
"""
t = np.atleast_1d(t)
EISF = (1 + 2 * np.sinc(q * rhh / np.pi)) / 3.
Iqt = (1 - fraction) + fraction * (EISF + (1 - EISF) * np.exp(-(t / t0) ** beta))
result = dA(np.c_[t, Iqt].T)
result.wavevector = q
result.residencetime = t0
result.rhh = rhh
result.beta = beta
result.EISF = EISF
result.methylfraction = fraction
result.modelname = inspect.currentframe().f_code.co_name
result.setColumnIndex(iey=None)
result.columnname = 't;Iqt'
return result
[docs]
def diffusionHarmonicPotential(t, q, rmsd, tau, beta=0, ndim=3):
r"""
ISF corresponding to the standard OU process for diffusion in harmonic potential for dimension 1,2,3.
The intermediate scattering function corresponding to the standard OU process
for diffusion in an harmonic potential [1]_. It is used for localized translational motion in
incoherent neutron scattering [2]_ as improvement for the diffusion in a sphere model.
Atomic motion may be restricted to ndim=1,2,3 dimensions and are isotropic averaged.
The correlation is assumed to be exponential decaying.
Parameters
----------
t : array
Time values in units ns
q : float
Wavevector in unit 1/nm
rmsd : float
Root mean square displacement :math:`rmsd=\langle u_x^2 \rangle ^{1/2}` in potential in units nm.
:math:`\langle u_x^2 \rangle ^{1/2}` is the width of the potential
According to [2]_
5*u**2=R**2:math:`5\langle u_x^2 \rangle =R^2` compared to the diffusion in a sphere of radius R.
tau : float
Correlation time :math:`\tau_0` in units ns.
Diffusion constant in sphere Ds=u**2/tau
beta : float, default 0
Exponent in correlation function :math:`\rho(t)`.
- beta=0 : :math:`\rho(t) = exp(-t/\tau_0)`
normal liquids where memory effects are presumably weak or negligible [2]_.
- 0<beta,inf : :math:`\rho(t,beta) = (1+\frac{t}{\beta\tau_0})^{-\beta}`. See [2]_ equ. 21a.
supercooled liquids or polymers, where memory effects may be important correlation functions
with slower decay rates should be introduced [2]_. See [2]_ equ. 21b.
ndim : 1,2,3, default=3
Dimensionality of the diffusion potential.
Returns
-------
dataArray
Notes
-----
We use equ. 18-20 from [2]_ and correlation time :math:`\tau_0`
with equal amplitudes :math:`rmsd=\langle u_x^2 \rangle ^{1/2}` in the dimensions as
3 dim case:
.. math:: I_s(Q,t) = e^{-Q^2\langle u^2_x \rangle (1-\rho(t))}
2 dim case:
.. math:: I_s(Q,t) = \frac{\pi^{0.5}}{2} e^{-g^2(t)} \frac{erfi(g(t))}{g(t)} \ with \
g(t) = \sqrt{Q^2\langle u^2_x \rangle (1-\rho(t))}
1 dim case:
.. math:: I_s(Q,t) = \frac{\pi^{0.5}}{2} \frac{erf(g(t))}{g(t)} \ with \
g(t) = \sqrt{Q^2\langle u^2_x \rangle (1-\rho(t))}
with *erf* as the error function and *erfi* is the imaginary error function *erf(iz)/i*
Examples
--------
::
import numpy as np
import jscatter as js
t=np.r_[0.1:6:0.1]
p=js.grace(1,1)
p.plot(js.dynamic.diffusionHarmonicPotential(t,1,2,1,1),le='1D ')
p.plot(js.dynamic.diffusionHarmonicPotential(t,1,2,1,2),le='2D ')
p.plot(js.dynamic.diffusionHarmonicPotential(t,1,2,1,3),le='3D ')
p.legend()
p.yaxis(label='I(Q,t)')
p.xaxis(label='Q / ns')
p.subtitle('Figure 2 of ref Volino J. Phys. Chem. B 110, 11217')
# p.save(js.examples.imagepath+'/diffusionHarmonicPotential_t.jpg', size=(2,2))
.. image:: ../../examples/images/diffusionHarmonicPotential_t.jpg
:align: center
:width: 50 %
:alt: dynamic_t2f_examples.
References
----------
.. [1] Quasielastic neutron scattering and relaxation processes in proteins: analytical and simulation-based models
G. R. Kneller Phys. ChemChemPhys. ,2005, 7,2641–2655
.. [2] Gaussian model for localized translational motion: Application to incoherent neutron scattering
F. Volino, J.-C. Perrin and S. Lyonnard, J. Phys. Chem. B 110, 11217–11223 (2006)
"""
erf = special.erf
erfi = special.erfi
q2u2 = q ** 2 * rmsd ** 2
if beta <=0:
ft = (1 - np.exp(-t / tau))
else:
ft = (1 - (1+t/tau/beta)**(-beta))
ft[t == 0] = 1e-8 # avoid zero to prevent zero division and overwrite later with EISF
if ndim == 3:
Iqt = np.exp(-q2u2 * ft)
EISF = np.exp(-q2u2)
Iqt[t == 0] = EISF
elif ndim == 2:
q2u2exp = q2u2 * ft
Iqt = 0.5 * pi ** 0.5 * np.exp(-q2u2exp) * erfi(q2u2exp ** 0.5) / q2u2exp ** 0.5
EISF = 0.5 * pi ** 0.5 * np.exp(-q2u2) * erfi(q2u2 ** 0.5) / q2u2 ** 0.5
Iqt[t == 0] = EISF
elif ndim == 1:
q2u2exp = q2u2 * ft
Iqt = 0.5 * pi ** 0.5 * erf(q2u2exp ** 0.5) / q2u2exp ** 0.5
EISF = 0.5 * pi ** 0.5 * erf(q2u2 ** 0.5) / q2u2 ** 0.5
Iqt[t == 0] = EISF
else:
raise Exception('ndim should be one of 1,2,3 ')
result = dA(np.c_[t, Iqt].T)
result.tau = tau
result.Ds = rmsd ** 2 / tau
result.rmsd = rmsd
result.EISF = EISF
result.wavevector = q
result.dimension = ndim
result.modelname = inspect.currentframe().f_code.co_name
result.setColumnIndex(iey=None)
result.columnname = 't;Iqt'
return result
@formel.memoize(maxsize=4)
def _bnmtzimm(t, NN, l, mu, modeamplist, tzp, fixedends):
return fscatter.dynamic.bnmt(t, NN, l, mu, modeamplist, tzp, fixedends)
[docs]
def finiteZimm(t, q, NN=None, pmax=None, l=None, Dcm=None, Dcmfkt=None, tintern=0., mu=0.5, viscosity=1.,
ftype=None, rk=None, Temp=293):
r"""
Zimm dynamics of a finite chain with N beads with internal friction and hydrodynamic interactions.
The Zimm model describes the conformational dynamics of an ideal chain with hydrodynamic interaction between beads.
The single chain diffusion is represented by Brownian motion of beads connected by harmonic springs.
Coherent + incoherent scattering.
Parameters
----------
t : array
Time in units nanoseconds.
q: float, array
Scattering vector in units nm^-1.
If q is list a dataList is returned otherwise a dataArray is returned.
NN : integer
Number of chain beads.
If not given calculated from Dcm,l, mu.
l : float, default 1
Bond length between beads; units nm.
If not given calculated from Dcm, NN, mu .
pmax : integer, list of float, default is NN
- integer => maximum mode number taken into account.
- list => list of amplitudes :math:`a_p > 0` for individual modes
to allow weighing. Not given modes have weight zero.
Dcm : float
Center of mass diffusion in nm²/ns if explicitly is given.
If not given Dcm is calculated
- :math:`=0.196 k_bT/(R_e visc)` for theta solvent with :math:`\nu=0.5`
- :math:`=0.203 k_bT/(R_e visc)` for good solvent with :math:`\nu=0.6`
with :math:`R_e=lN^{\nu}` .
Dcmfkt : array 2xN, function
Function f(q) or array with [qi, f(qi)] as correction for Dcm like Diff = Dcm*f(q).
e.g. for inclusion of structure factor and hydrodynamic function with f(q)=H(Q)/S(q).
Missing values are interpolated.
tintern : float>0
Additional relaxation time due to internal friction between neighboring beads in units ns.
mu : float in range [0.01,0.99]
:math:`\nu` describes solvent quality.
- <0.5 collapsed chain
- =0.5 theta solvent 0.5 (gaussian chain)
- =0.6 good solvent
- >0.6 swollen chain
viscosity : float
:math:`\eta` in units cPoise=mPa*s
e.g. water :math:`visc(T=293 K) =1 mPas`
Temp : float, default 273+20
Temperature in Kelvin.
ftype : 'czif', default = 'zif'
Type of internal friction and interaction modification.
- Default Zimm is used with :math:`t_{intern}=0`
- 'zif' Internal friction between neighboring beads in chain [3]_.
:math:`t_{zp}=t_z p^{-3\nu}+t_{intern}`
- 'czif' Bead confining harmonic potential with internal friction, only for :math:`\nu=0.5` [6]_ .
The beads are confined in an additional harmonic potential with :math:`\frac{1}{2}k_c(r_n-0)^2` leading to
a more compact configuration. :math:`rk= k_c/k` describes the relative strength
compared to the force between beads :math:`k`.
rk : None , float
:math:`rk= k_c/k` describes the relative force constant for *ftype* 'czif'.
Returns
-------
S(q,t)/S(q,0) : dataArray : for single q, dataList : for multiple q
- [time; Sqt; Sqt_inf; Sqtinc; Sqtz]
- time units ns
- Sqt as S(q,t)/S(q,0) coherent scattering with diffusion and mode contributions
- Sqt_inf is coherent scattering with ONLY diffusion (no internal modes)
- Sqtinc is incoherent scattering with diffusion and mode contributions (no separate diffusion)
- Sqtz is coherent scattering with diffusion and mode contributions, but no Dcmfkt => f(q)=1
- .q wavevector
- .modecontribution :math:`a_p` of coherent modes i in sequence as in PRL 71, 4158 equ (3)
- .Re
- .tzimm => Zimm time or rotational correlation time
- .t_p characteristic times
- .... use .attr for all attributes
Notes
-----
The Zimm model describes beads connected by harmonic springs with hydrodynamic interaction and free ends.
The :math:`\nu` parameter scales between theta solvent :math:`\nu=0.5` and good solvent :math:`\nu=0.6`
(excluded volume or swollen chain). The coherent intermediate scattering function :math:`S(q,t)` is
.. math:: S(q,t) = \frac{1}{N} e^{-q^2D_{cm}t}\sum_{n,m}^N e^{-\frac{1}{6}q^2B(n,m,t)}
.. math:: B(n,m,t)=|n-m|^{2\nu}l^2 + \sum_{p=1}^{N-1} A_p cos(\pi pn/N)cos(\pi pm/N) (1-e^{-t/t_{zp}})
and for incoherent intermediate scattering function the same with :math:`n=m` in the first sum.
with
- :math:`A_p = a_p\frac{4R_e^2}{\pi^2}\frac{1}{p^{2\nu+1}}` mode amplitude (usual :math:`a_p=1`)
- :math:`t_{zp} = t_z p^{-3\nu}` mode relaxation time
- :math:`t_z = \eta R_e^3/(\sqrt(3\pi) k_bT)` Zimm mode relaxation time
- :math:`R_e=l N^{\nu}` end to end distance
- :math:`k=3kT/l^2` force constant between beads
- :math:`\xi=6\pi\eta l` single bead friction in solvent with viscosity :math:`\eta`
- :math:`a_p` additional amplitude for suppression of specific modes e.g. by topological constraints (see [5]_).
- :math:`D_{cm} = \frac{8}{3(6\pi^3)^{1/2}} \frac{k_bT}{\eta R_e} = 0.196 \frac{k_bT}{\eta R_e}`
Modifications (*ftype*) for internal friction and additional interaction:
- ZIF : Zimm with internal friction between neighboring beads in chain [3]_ [4]_.
- :math:`t_{zp}=t_z p^{{-3\nu}}+t_{intern}`
- :math:`\xi_i=t_{intern}k=t_{intern}3k_bT/l^2` internal friction per bead
- CZIF : Compacted Zimm with internal friction [6]_.
Restricted to :math:`\nu=0.5` , a combination with excluded volume is not valid.
In [9]_ the beads are confined in an additional harmonic potential around the origin with
:math:`\frac{1}{2}k_c(r_n-0)^2` leading to a more compact configuration.
:math:`rk= k_c/k` describes the relative strength compared to the force between beads :math:`k`.
Typically :math:`rk << 1` .
- The mode amplitude prefactor changes from Zimm type to modified confined amplitudes
.. math:: A_p =\frac{4Nl^2}{\pi^2}\frac{1}{p^2}\Rightarrow
A_p^c = \frac{4Nl^2}{\pi^2}\frac{1}{\frac{N^2k_c}{\pi^2k}+p^2}
- The mode relaxation time changes from Zimm type to modified confined
with :math:`t_{z} = \frac{\eta N^{3/2} l^3}{\sqrt(3\pi) k_bT}`
.. math:: t_{zp} = t_z \frac{1}{p^{3/2}} \Rightarrow
t_{zp}^c = t_z \frac{p^{1/2}}{\frac{N^2k_c}{\pi^2k} + p^2}
- :math:`R_e^c` allows to determine :math:`k_c/k` from small angle scattering data
.. math:: (R_e^c)^2 = \frac{2l^2}{\sqrt{k_c/k}}tanh(\frac{N}{2}\sqrt{k_c/k})
- For a free diffusing chain we assume here (not given in [9]_ ) that the additional potential
is :math:`\frac{1}{2}k_c(r_n-r_0)^2` with :math:`r_0` as the polymer center of mass.
As the Langevin equation only depends on position distances the internal motions are not affected.
The center of mass diffusion :math:`D_{cm}` can be calculated similar to the Zimm :math:`D_{cm}` in [1]_
assuming a Gaussian configuration with width :math:`R_e`. We find
.. math:: D_{cm} = \frac{kT}{\xi_{p=0}} = \frac{8}{3(6\pi^3)^{1/2}} \frac{kT}{\eta R_e}
- With :math:`rk=k_c/k \rightarrow 0` the original Zimm is recovered for amplitudes,
relaxation and :math:`R_e` .
From above the triple Dcm,l,N are fixed.
- If 2 are given 3rd is calculated.
- If all 3 are given the given values are used.
For an example see `example_Zimm` and
:ref:`collectivezimmdynamics` .
Examples
--------
Coherent and incoherent contributions to Zimm dynamics.
To mix the individual q dependent contributions of coherent and incoherent these have to be weighted by
the according formfactor respectively incoherent scattering length and instrument specific measurement technique.
Typically, diffusion and mode contributions cannot be separated.
At larger Q the diffusion contributes marginally while at low Q diffusion dominates.
::
import jscatter as js
import numpy as np
t = js.loglist(0.02, 100, 40)
q=np.r_[0.1:2:0.2]
l=0.38 # nm , bond length amino acids
zz = js.dynamic.finiteZimm(t, q, 124, 7, l=0.38, Dcm=0.37, tintern=0., Temp=273 + 60)
p=js.grace(2,2)
p.multi(2,1)
p[0].xaxis(scale='log')
p[0].yaxis(label='I(q,t)\scoherent')
p[1].xaxis(label=r't / ns',scale='log')
p[1].yaxis(label=r'I(q,t)\sincoherent')
p[0].title('Zimm dynamics in a solvent')
for i, z in enumerate(zz, 1):
p[0].plot(z.X, z.Y, line=[1, 1, i], symbol=0, legend='')
p[0].plot(z.X, z._Sqt_inf, line=[3, 2, i], symbol=0, legend='')
p[1].plot(z.X, z._Sqtinc, line=[1, 2, i], symbol=0, legend=fr'q={z.q:.1f} nm\S-1')
p[1].legend(x=0.02,y=0.8,charsize=0.5)
p[0].text('only diffusion', x=0.02,y=0.55)
p[0].text('diffusion + modes', x=15,y=0.65,rot=305)
# p.save(js.examples.imagepath+'/Zimmcohinc.jpg',size=(1,1))
.. image:: ../../examples/images/Zimmcohinc.jpg
:align: center
:width: 50 %
:alt: Zimm
References
----------
.. [1] Doi Edwards Theory of Polymer dynamics
in appendix the equation is found
.. [2] Nonflexible Coils in Solution: A Neutron Spin-Echo Investigation of
Alkyl-Substituted Polynorbonenes in Tetrahydrofuran
Michael Monkenbusch et al.Macromolecules 2006, 39, 9473-9479
The exponential is missing a "t"
http://dx.doi.org/10.1021/ma0618979
about internal friction
.. [3] Exploring the role of internal friction in the dynamics of unfolded proteins using simple polymer models
Cheng et al.JOURNAL OF CHEMICAL PHYSICS 138, 074112 (2013) http://dx.doi.org/10.1063/1.4792206
.. [4] Rouse Model with Internal Friction: A Coarse Grained Framework for Single Biopolymer Dynamics
Khatri, McLeish| Macromolecules 2007, 40, 6770-6777 http://dx.doi.org/10.1021/ma071175x
mode contribution factors from
.. [5] Onset of Topological Constraints in Polymer Melts: A Mode Analysis by Neutron Spin Echo Spectroscopy
D. Richter et al.PRL 71,4158-4161 (1993)
.. [6] Looping dynamics of a flexible chain with internal friction at different degrees of compactness.
Samanta, N., & Chakrabarti, R. (2015).
Physica A: Statistical Mechanics and Its Applications, 436, 377–386.
https://doi.org/10.1016/j.physa.2015.05.042
"""
# convert to Pa*s
viscosity *= 1e-3
q = np.atleast_1d(q)
# check mu between 0.1 and 0.9
mu = max(mu, 0.01)
mu = min(mu, 0.99)
# avoid l=0 from stupid users
if l == 0: l = None
# and linear interpolate prefactor
ffact = 8 / (3 * 6 ** 0.5 * np.pi ** (3 / 2))
fact = ffact + (mu - 0.5) / (0.6 - 0.5) * (0.203 - 0.196)
NN = int(NN)
if pmax is None: pmax = NN
# if a list pmax of modes is given these are amplitudes for the modes
# pmax is length of list
if isinstance(pmax, numbers.Number):
pmax = min(int(pmax), NN)
modeamplist = np.ones(pmax)
elif isinstance(pmax, list):
modeamplist = np.abs(pmax)
else:
raise TypeError('pmax should be integer or list of amplitudes')
# create correction for diffusion
if Dcmfkt is not None:
if formel._getFuncCode(Dcmfkt):
# is already an interpolation function
Dcmfunktion = Dcmfkt
elif isinstance(Dcmfkt,dA):
Dcmfunktion = lambda qq: Dcmfkt.interp(qq)
elif np.shape(Dcmfkt)[0] == 2:
Dcmfunktion = lambda qq: dA(Dcmfkt).interp(qq)
else:
raise TypeError('Shape of Dcmfkt is not 2xN!')
else:
# by default no correction
Dcmfunktion = lambda qq: 1.
if ftype == 'czif':
# compacted zimm with internal friction
if mu != 0.5:
raise ValueError('For ftype "czif" only mu=0.5 is allowed. ')
if Dcm is None and l is not None and NN is not None:
# Re = end to end distance
Re = (2 * l ** 2 / rk ** 0.5 * np.tanh(NN / 2 * rk ** 0.5))**0.5
# center of mass diffusion constant in nm^2/ns
Dcm = fact * kb * Temp / (Re * 1e-9 * viscosity) * 1e9
elif Dcm is not None and l is None and NN is not None:
Re = fact * kb * Temp / (Dcm * 1e-9 * viscosity) * 1e9
l = Re * (rk ** 0.5 / 2 / np.tanh(NN / 2 * rk ** 0.5)) ** 0.5
elif Dcm is not None and l is not None and NN is None:
Re = fact * kb * Temp / (Dcm * 1e-9 * viscosity) * 1e9
NN = 2 / rk ** 0.5 * np.arctanh(rk * Re ** 2 / 2 / l ** 2 )
elif Dcm is not None and l is not None and NN is not None:
Re = 2 * l ** 2 / rk ** 0.5 * np.tanh(NN / 2 * rk ** 0.5)
else:
raise TypeError('finiteZimm takes at least 2 arguments from Dcm, NN, l')
# determine mode relaxation times
# slowest zimm time
tz1 = viscosity * NN ** (3 / 2) * (l * 1e-9) ** 3 / (np.sqrt(3 * pi) * kb * Temp) * 1e9
# mode amplitudes
p = np.r_[1:len(modeamplist) + 1]
modeamplist = 4 * NN * l ** 2 / pi ** 2 * modeamplist
tzp = tz1 * p ** 0.5 / (NN ** 2 / np.pi ** 2 * rk + p ** 2) + abs(tintern)
modeamplist = modeamplist / (NN ** 2 / np.pi ** 2 * rk + p ** (2 * mu + 1))
else:
# ZIF with constant internal friction time added as default
if Dcm is None and l is not None and NN is not None:
Re = l * NN ** mu # end to end distance
Dcm = fact * kb * Temp / (Re * 1e-9 * viscosity) * 1e9 # diffusion constant in nm^2/ns
elif Dcm is not None and l is None and NN is not None:
Re = fact * kb * Temp / (Dcm * 1e-9 * viscosity) * 1e9 # end to end distance
l = Re / NN ** mu # bond length
elif Dcm is not None and l is not None and NN is None:
Re = fact * kb * Temp / (Dcm * 1e-9 * viscosity) * 1e9 # end to end distance
NN = int((Re / l) ** (1. / mu))
elif Dcm is not None and l is not None and NN is not None:
Re = l * NN ** mu
else:
raise TypeError('finiteZimm takes at least 2 arguments from Dcm,NN,l')
# determine mode relaxation times
# slowest zimm time
tz1 = viscosity * (Re * 1e-9) ** 3 / (np.sqrt(3 * pi) * kb * Temp) * 1e9
# mode amplitudes
p = np.r_[1:len(modeamplist) + 1]
modeamplist = 4 * Re ** 2 / pi ** 2 * modeamplist
# characteristic Zimm time of mode p adding internal friction ti
tzp = tz1 * p ** (-3 * mu) + abs(tintern)
modeamplist = modeamplist / (p ** (2 * mu + 1))
ftype = 'zif'
# prepend 0 and append infinite time
t = np.r_[0, np.atleast_1d(t)]
# calc array of mode contributions including first constant element as list
# do the calculation as an array of bnm=[n*m , len(t)] elements
# sum up contributions for modes: all, diff+ mode1, only diffusion, t=0 amplitude for normalisation
if useFortran:
BNM = _bnmtzimm(t=t, NN=NN, l=l, mu=mu, modeamplist=modeamplist, tzp=tzp, fixedends=0)
BNMmodes = BNM[:, -len(modeamplist):]
BNMi = BNM[:, len(t):2*len(t)]
BNMinf = BNM[:, 2 * len(t) + 1] # coherent t = inf
BNM = BNM[:, :len(t)]
else:
raise ImportError('finiteZimm only with working Fortran.')
result = dL()
for qq in q:
# diffusion for all t
Sqt = np.exp(-qq ** 2 * Dcm * Dcmfunktion(qq) * t[1:]) # only diffusion contribution
Sqt0 = np.exp(-qq ** 2 * Dcm * t[1:]) # only diffusion contribution
# amplitude at t=0
expB0 = np.sum(np.exp(-qq ** 2 / 6. * BNM[:, 0])) # is S(qq,t=0) coherent
expB0i = np.sum(np.exp(-qq ** 2 / 6. * BNMi[:, 0])) # is S(qq,t=0) incoherent
# diffusion for infinite times in modes
expBinf = np.sum(np.exp(-qq ** 2 / 6. * BNMinf)) # is S(qq,t=inf)
# contribution all modes
expB = np.sum(np.exp(-qq ** 2 / 6. * BNM[:, 1:]), axis=0) # coherent
expBi = np.sum(np.exp(-qq ** 2 / 6. * BNMi[:, 1:]), axis=0) # incoherent
# contribution only first modes
result.append(np.c_[t[1:],
Sqt * expB / expB0, # Zimm with H/S
Sqt * expBinf / expB0, # no internal modes
Sqt * expBi / expB0i, # incoherent contributions
Sqt0 * expB / expB0].T) # pure Zimm , no H/S modification
result[-1].modecontribution = (np.sum(np.exp(-qq ** 2 / 6. * BNMmodes), axis=0) / expB0).flatten()
result[-1].q = qq
result[-1].Re = Re
result[-1].ll = l
result[-1].pmax = pmax
result[-1].Dcm = Dcm
result[-1].effectiveDCM = Dcm * Dcmfunktion(qq)
DZimm = fact * kb * Temp / (Re * 1e-9 * viscosity) * 1e9
result[-1].DZimm = DZimm
result[-1].mu = mu
result[-1].viscosity = viscosity
result[-1].Temperature = Temp
result[-1].tzimm = tz1
result[-1].moderelaxationtimes = tzp
result[-1].tintern = tintern
result[-1].modeAmplist = modeamplist
result[-1].Drot = 1. / 6. / tz1
result[-1].N = NN
result[-1].columnname = ' time; Sqt; Sqt_inf; Sqtinc; Sqtz'
result[-1].ftype = ftype
result[-1].rk = rk
if len(result) == 1:
return result[0]
result.setColumnIndex(iey=None)
result.modelname = inspect.currentframe().f_code.co_name
return result
[docs]
def fixedFiniteZimm(t, q, NN=None, pmax=None, l=None, Dcm=None, Dcmfkt=None, tintern=0., mu=0.5, viscosity=1.,
ftype=None, rk=None, Temp=293, fixedends=1):
r"""
Zimm dynamics of a chain **with fixed ends** with internal friction and hydrodynamic interactions.
Opposite to the :py:func:`finiteZimm` here one or both ends are fixed.
This might be a chain tethered to a particle that defines the diffusion. Chains are non interacting.
Parameters
----------
fixedends : 0,1,2, default = 1
Number of fixed ends. 0 is only for comparison and corresponds to :py:func:`finiteZimm` .
t : array
Time in units nanoseconds.
q: float, array
Scattering vector in units nm^-1.
If q is list a dataList is returned otherwise a dataArray is returned.
NN : integer
Number of chain beads.
l : float, default 1
Bond length between beads; units nm.
pmax : integer, list of float, default is NN
- integer => maximum mode number taken into account.
- list => list of amplitudes :math:`a_p > 0` for individual modes
to allow weighing. Not given modes have weight zero.
Dcm : float
Center of mass diffusion in nm²/ns.
Dcmfkt : array 2xN, function
Function f(q) or array with [qi, f(qi)] as correction for Dcm like Diff = Dcm*f(q).
e.g. for inclusion of structure factor and hydrodynamic function with f(q)=H(Q)/S(q).
Missing values are interpolated.
tintern : float>0
Additional relaxation time due to internal friction between neighboring beads in units ns.
mu : float in range [0.01,0.99]
:math:`\nu` describes solvent quality.
- <0.5 collapsed chain
- =0.5 theta solvent 0.5 (gaussian chain)
- =0.6 good solvent
- >0.6 swollen chain
viscosity : float
:math:`\eta` in units cPoise=mPa*s e.g. water :math:`visc(T=293 K) =1 mPas`
Temp : float, default 273+20
Temperature in Kelvin.
ftype : 'czif', default = 'zif'
Type of internal friction and interaction modification.
- Default Zimm is used with :math:`t_{intern}=0`
- 'zif' Internal friction between neighboring beads in chain [3]_.
:math:`t_{zp}=t_z p^{-3\nu}+t_{intern}`
- 'czif' Bead confining harmonic potential with internal friction, only for :math:`\nu=0.5` [6]_ .
The beads are confined in an additional harmonic potential with :math:`\frac{1}{2}k_c(r_n-0)^2` leading to
a more compact configuration. :math:`rk= k_c/k` describes the relative strength
compared to the force between beads :math:`k`.
rk : None , float
:math:`rk= k_c/k` describes the relative force constant for *ftype* 'czif'.
Returns
-------
S(q,t)/S(q,0) : dataArray : for single q, dataList : for multiple q
- [time; Sqt; Sqt_inf; Sqtinc; Sqtz]
- time units ns
- Sqt is S(q,t)/S(q,0) coherent scattering with diffusion and mode contributions
- Sqt_inf is coherent scattering with ONLY diffusion (no internal modes)
- Sqtinc is incoherent scattering with diffusion and mode contributions (no separate diffusion)
- Sqt0 is coherent scattering with diffusion and mode contributions, but no Dcmfkt => f(q)=1
- .q wavevector
- .modecontribution :math:`a_p` of coherent modes i in sequence as in PRL 71, 4158 equ (3)
- .Re is :math:`R_e=lN^{\nu}`
- .tzimm => Zimm time or rotational correlation time
- .t_p characteristic times
- .... use .attr for all attributes
Notes
-----
The Zimm model describes beads connected by harmonic springs with hydrodynamic interaction (see 4.2 in [1]_).
We find
.. math:: S(q,t) = \frac{1}{N} e^{-q^2D_{cm}t}\sum_{n,m}^N e^{-\frac{1}{6}q^2B(n,m,t)}
:math:`B(n,m,t)` describes the internal motions characterised by eigenmodes of the equation 4.II.6 in [1]_
:math:`\frac{\zeta_p}{\zeta} k \frac{\partial^2 \Phi_{pn}}{\partial^2 n}=-k_p\Phi_{pn}`
where :math:`\Phi_{pn}` describes the delocalisation of bead n in mode p.
The boundary conditions select the eigenmodes from the general form :math:`\Phi_{pn} = A sin(kn) + Bcos(kn)`
- Two free ends :math:`\partial\Phi_{pn}/\partial n=0 \text{ for n=0 and n=N}`
select A=0 and k=pπ/N (equ. 4.II.7+9 in [1]_):
.. math:: B(n,m,t)=|n-m|^{2\nu}l^2 + \sum_{p=1}^{N-1} A_p cos(\pi pn/N)cos(\pi pm/N) (1-e^{-t/t_{zp}})
- One fixed and one free end :math:`\partial\Phi_{pn}/\partial n=0 \text{ at n=0 and } \Phi_{pn}=0` at N
select B=0 and k=(p-1/2)π/N (see [2]_):
.. math:: B(n,m,t)=|n-m|^{2\nu}l^2 + \sum_{p=1-1/2}^{N-1-1/2} A_p sin(\pi pn/N) sin(\pi pm/N) (1-e^{-t/t_{zp}})
- Two fixed ends :math:`\Phi_{pn}=0` at n=0 and n=N select B=0 and k=pπ/N:
.. math:: B(n,m,t)=|n-m|^{2\nu}l^2 + \sum_{p=1}^{N-1} A_p sin(\pi pn/N)sin(\pi pm/N) (1-e^{-t/t_{zp}})
For fixed ends the center of mass diffusion Dcm is that of the object where the chain is fixed.
The chain dimension is defined by :math:`R_e=lN^{\nu}`.
Unfortunately there are some papers that give wrong equations for fixed end Zimm or Rouse dynamics.
The correct equations above can be retrieved from [1]_ appendix 4.II as solution of the
differential equation 4.II.6 above which describes standing waves in a string or in a open tube.
The classical Zimm/Rouse describe two open ends.
For detailed description of parameters see :py:func:`finiteZimm`.
Examples
--------
Let us assume we have a core shell micelle of diblock copolymers with a hydrophobic part that assemble in the core
and the hydrophilic part extended into the solvent. The core is solvent matched and invisible.
For low aggregation number the hydrophobic tails extending into the solvent dont interact and the
motions are that of a Zimm chain with one fixed end. The center of mass diffusion is that of the micelle
and much slower than :math:`D_{Zimm}` but could be determined by DLS dilution series or PFG-NMR.
(See e.g. Mark et al. https://doi.org/10.1103/PhysRevLett.119.047801
for silica nanoparticles with grafted chains)
For comparison we think of a triblock with hydrophobic ends that will make a loop that both ends
are fixed (but not at the same position). The hydrophilic is of same size.
We neglect any influence of the core onto the chain configuration.
We allow a Dcm ≈50 times slower than DZimm of the hydrophilic tail.
We observe two relaxations, a faster of the internal dynamics and a slower because of diffusion.
First compare one fixed end (full lines) with the free Zimm (broken lines).
For long times the diffusion gets equal visible at low q for long times.
Obviously the amplitude of mode relaxations is much stronger than for open ends due to the different eigenmodes.
::
import jscatter as js
import numpy as np
t = js.loglist(0.1, 1000, 40)
q=np.r_[0.1:2:0.25]
l=0.38 # nm , bond length amino acids
p=js.grace(1.5,1.5)
p.xaxis(label='q / ns',scale='log')
p.yaxis(label='I(q,t)/I(q,0)')
p.title('Compare 1 fixed to free ends')
p.subtitle('solid line = one fixed end; broken lines = open ends')
# free ends just for comapring
fFZ0 = js.dynamic.fixedFiniteZimm(t, q, 124, 40, l=l, mu=0.5, Dcm=0.004,fixedends=0)
# one fixed end
fFZ1 = js.dynamic.fixedFiniteZimm(t, q, 124, 40, l=l, mu=0.5, Dcm=0.004,fixedends=1)
for i, z in enumerate(fFZ1, 1):
p.plot(z.X, z.Y, line=[1, 3, i], symbol=0, legend=fr'q={z.q:.1f} nm\S-1')
for i, z in enumerate(fFZ0, 1):
p.plot(z.X, z.Y, line=[3, 3, i], symbol=0, legend='')
p.legend(x=0.2,y=0.4,charsize=0.6)
# p.save(js.examples.imagepath+'/fixedZimm_vs_freeZimm.jpg',size=(1.,1.))
.. image:: ../../examples/images/fixedZimm_vs_freeZimm.jpg
:align: center
:width: 50 %
:alt: Zimm open vs. fixed ends
Now we compare one and two open ends.
The differences are marginally and will be difficult to discriminate in real measurements.
::
import jscatter as js
import numpy as np
t = js.loglist(0.1, 1000, 40)
q=np.r_[0.1:2:0.25]
l=0.38 # nm , bond length amino acids
p=js.grace(1.5,1.5)
p.xaxis(label='q / ns',scale='log')
p.yaxis(label='I(q,t)/I(q,0)')
p.title('Compare 1 fixed to 2 fixed ends')
p.subtitle('solid line = one fixed end; broken lines = 2 fixed ends')
# two fixed ends
fFZ2 = js.dynamic.fixedFiniteZimm(t, q, 124, 40, l=l, mu=0.5, Dcm=0.004, fixedends=2)
# one fixed end
fFZ1 = js.dynamic.fixedFiniteZimm(t, q, 124, 40, l=l, mu=0.5, Dcm=0.004, fixedends=1)
for i, z in enumerate(fFZ1, 1):
p.plot(z.X, z.Y, line=[1, 3, i], symbol=0, legend=fr'q={z.q:.1f} nm\S-1')
for i, z in enumerate(fFZ2, 1):
p.plot(z.X, z.Y, line=[3, 3, i], symbol=0, legend='')
p.legend(x=0.2,y=0.4,charsize=0.6)
# p.save(js.examples.imagepath+'/fixedZimm_1vs2_fixed.jpg',size=(1,1))
.. image:: ../../examples/images/fixedZimm_1vs2_fixed.jpg
:align: center
:width: 50 %
:alt: Zimm 1 vs. 2 fixed ends
References
----------
.. [1] The Theory of Polymer dynamics
Doi, M., & Edwards, S. F. (1988). Clarendon Press.
.. [2] Normal Modes of Stretched Polymer Chains
Y. Marciano and F. Brochard-Wyart
Macromolecules 1995, 28, 985-990 https://doi.org/10.1021/ma00108a028
"""
assert fixedends in [0, 1, 2]
# convert to Pa*s
viscosity *= 1e-3
q = np.atleast_1d(q)
# check mu between 0.1 and 0.9
mu = max(mu, 0.01)
mu = min(mu, 0.99)
# avoid l=0 from stupid users
if l == 0: l = None
# and linear interpolate prefactor
ffact = 8 / (3 * 6 ** 0.5 * np.pi ** (3 / 2))
fact = ffact + (mu - 0.5) / (0.6 - 0.5) * (0.203 - 0.196)
NN = int(NN)
if pmax is None: pmax = NN
# if a list pmax of modes is given these are amplitudes for the modes
# pmax is length of list
if isinstance(pmax, numbers.Number):
pmax = min(int(pmax), NN)
modeamplist = np.ones(pmax)
elif isinstance(pmax, list):
modeamplist = np.abs(pmax)
else:
raise TypeError('pmax should be integer or list of amplitudes')
# create correction for diffusion
if Dcmfkt is not None:
if formel._getFuncCode(Dcmfkt):
# is already an interpolation function
Dcmfunktion = Dcmfkt
elif np.shape(Dcmfkt)[0] == 2:
Dcmfunktion = lambda qq: dA(Dcmfkt).interp(qq)
else:
raise TypeError('Shape of Dcmfkt is not 2xN!')
else:
# by default no correction
Dcmfunktion = lambda qq: 1.
if ftype == 'czif':
# compacted zimm with internal friction
if mu != 0.5:
raise ValueError('For ftype "czif" only mu=0.5 is allowed. ')
if Dcm is not None and l is not None and NN is not None:
Re = 2 * l ** 2 / rk ** 0.5 * np.tanh(NN / 2 * rk ** 0.5)
else:
raise TypeError('fixedFiniteZimm needs NN, l, rk')
# determine mode relaxation times
# slowest zimm time
tz1 = viscosity * NN ** (3 / 2) * (l * 1e-9) ** 3 / (np.sqrt(3 * pi) * kb * Temp) * 1e9
# mode amplitudes
p = np.r_[1:len(modeamplist) + 1]
modeamplist = 4 * NN * l ** 2 / pi ** 2 * modeamplist
tzp = tz1 * p ** 0.5 / (NN ** 2 / np.pi ** 2 * rk + p ** 2) + abs(tintern)
modeamplist = modeamplist / (NN ** 2 / np.pi ** 2 * rk + p ** (2 * mu + 1))
else:
# ZIF with constant internal friction time added as default
if Dcm is not None and l is not None and NN is not None:
Re = l * NN ** mu
else:
raise TypeError('finiteZimm takes at least 2 arguments from Dcm,NN,l')
# determine mode relaxation times
# slowest zimm time
tz1 = viscosity * (Re * 1e-9) ** 3 / (np.sqrt(3 * pi) * kb * Temp) * 1e9
# mode amplitudes
p = np.r_[1:len(modeamplist) + 1]
modeamplist = 4 * Re ** 2 / pi ** 2 * modeamplist
# characteristic Zimm time of mode p adding internal friction ti
tzp = tz1 * p ** (-3 * mu) + abs(tintern)
modeamplist = modeamplist / (p ** (2 * mu + 1))
ftype = 'zif'
# prepend 0 and append infinite time
t = np.r_[0, np.atleast_1d(t)]
# calc array of mode contributions including first constant element as list
# do the calculation as an array of bnm=[n*m , len(t)] elements
# sum up contributions for modes: all, diff+ mode1, only diffusion, t=0 amplitude for normalisation
if useFortran:
BNM = _bnmtzimm(t=t, NN=NN, l=l, mu=mu, modeamplist=modeamplist, tzp=tzp, fixedends=int(fixedends))
BNMmodes = BNM[:, -len(modeamplist):] # coh after infinite time for each mode
BNMi = BNM[:, len(t):2*len(t)] # incoherent with mode contributions
BNMinf = BNM[:, 2 * len(t) + 1] # coherent t = inf
BNM = BNM[:, :len(t)] # coh with mode contributions
else:
raise ImportError('finiteZimm only with working Fortran.')
result = dL()
for qq in q:
# diffusion for all t
Sqt = np.exp(-qq ** 2 * Dcm * Dcmfunktion(qq) * t[1:]) # only diffusion contribution
Sqt0 = np.exp(-qq ** 2 * Dcm * t[1:]) # only diffusion contribution
# amplitude at t=0
expB0 = np.sum(np.exp(-qq ** 2 / 6. * BNM[:, 0])) # is S(qq,t=0) coherent
expB0i = np.sum(np.exp(-qq ** 2 / 6. * BNMi[:, 0])) # is S(qq,t=0) incoherent
# diffusion for infinite times in modes
expBinf = np.sum(np.exp(-qq ** 2 / 6. * BNMinf)) # is S(qq,t=inf)
# contribution all modes
expB = np.sum(np.exp(-qq ** 2 / 6. * BNM[:, 1:]), axis=0) # coherent
expBi = np.sum(np.exp(-qq ** 2 / 6. * BNMi[:, 1:]), axis=0) # incoherent
# contribution only first modes
result.append(np.c_[t[1:],
Sqt * expB / expB0, # Zimm with H/S
Sqt * expBinf / expB0, # no internal modes
Sqt * expBi / expB0i, # incoherent contributions
Sqt0 * expB / expB0].T) # pure Zimm , no H/S modification
result[-1].modecontribution = (np.sum(np.exp(-qq ** 2 / 6. * BNMmodes), axis=0) / expB0).flatten()
result[-1].q = qq
result[-1].Re = Re
result[-1].ll = l
result[-1].pmax = pmax
result[-1].Dcm = Dcm
result[-1].effectiveDCM = Dcm * Dcmfunktion(qq)
DZimm = fact * kb * Temp / (Re * 1e-9 * viscosity) * 1e9
result[-1].DZimm = DZimm
result[-1].mu = mu
result[-1].viscosity = viscosity
result[-1].Temperature = Temp
result[-1].tzimm = tz1
result[-1].moderelaxationtimes = tzp
result[-1].tintern = tintern
result[-1].modeAmplist = modeamplist
result[-1].Drot = 1. / 6. / tz1
result[-1].N = NN
result[-1].columnname = ' time; Sqt; Sqt_inf; Sqtinc; Sqtz'
result[-1].ftype = ftype
result[-1].rk = rk
result[-1].fixedends = fixedends
if len(result) == 1:
return result[0]
result.setColumnIndex(iey=None)
result.modelname = inspect.currentframe().f_code.co_name
return result
@formel.memoize(maxsize=4)
def _bnmtrouse(t, NN, l, modeamplist, trp, fixedends=0):
return fscatter.dynamic.bnmt(t, NN, l, 0.5, modeamplist, trp, fixedends)
[docs]
def finiteRouse(t, q, NN=None, pmax=None, l=None, frict=None, Dcm=None, Wl4=None, Dcmfkt=None, tintern=0.,
Temp=293, ftype=None, specm=None, specb=None, rk=None):
r"""
Rouse dynamics of a finite chain with N beads of bonds length l and internal friction.
The Rouse model describes the conformational dynamics of an ideal chain.
The single chain diffusion is represented by Brownian motion of beads connected by harmonic springs.
No excluded volume, random thermal force, drag force with solvent,
no hydrodynamic interaction and optional internal friction.
Coherent + incoherent scattering.
Parameters
----------
t : array
Time in units nanoseconds
q : float, list
Scattering vector, units nm^-1
For a list a dataList is returned otherwise a dataArray is returned
NN : integer
Number of chain beads.
l : float, default 1
Bond length between beads; unit nm.
pmax : integer, list of floats
- integer => maximum mode number (:math:`a_p=1`)
- list => :math:`a_p` list of amplitudes>0 for individual modes
to allow weighing; not given modes have weight zero
frict : float
Friction of a single bead/monomer in units `Pas*m=kg/s=1e-6 g/ns`
:math:`\xi = 6\pi\eta l`, .
A monomer bead with `l=R=0.1nm = 0.1e-9m` in H2O(20°C) (1 mPas) => 1.89e-12 Pas*m.
Rouse dynamics in a melt needs the bead friction with effective viscosity of the melt.
Wl4 : float
:math:`W_l^4` Characteristic value to calc friction and Dcm.
:math:`D_{cm}=\frac{W_l^4}{3R_e^2}` and characteristic Rouse variable
:math:`\Omega_Rt=(q^2/6)^2 W_l^4 t`
Dcm : float
Center of mass diffusion in nm^2/ns.
- :math:`D_{cm}=k_bT/(N\xi)` with :math:`\xi` = friction of single bead in solvent
- :math:`D_{cm}=W_l^4/(3Nl^2)=W_l^4/(3Re^2)`
Dcmfkt : array 2xN, function
Function f(q) or array with [qi, f(qi) ] as correction for Dcm like Diff = Dcm*f(q).
e.g. for inclusion of structure factor or hydrodynamic function with f(q)=H(Q)/S(q).
Missing values are interpolated.
tintern : float>0
Relaxation time due to internal friction between neighboring beads in ns.
ftype : 'rni', 'rap','nonspec', 'specrif', 'crif', default = 'rif'
Type of internal friction. See [7]_ for a description and respective references.
- *'rif'*: Internal friction between neighboring beads in chain. :math:`t_{rp}=t_r p^{-2}+t_{intern}`
- *'rni'*: Rouse model with non-local interactions (RNI).
Additional friction between random close approaching beads. :math:`t_{rp}=t_r p^{-2}+N/p t_{intern}`
- *'rap'*: Rouse model with anharmonic potentials due to stiffness of the chain
:math:`t_{rp}=t_r p^{-2}+t_{intern}ln(N/p\pi)`
- *'specrif'*: Specific interactions of strength :math:`b` between beads separated by *m* bonds.
See [7]_. :math:`t_{rp}=t_r p^{-2} (1+bm^2)^{-1} + (1+m^2/(1+bm^2))t_{intern}`
- *'crif'*: Bead confining potential with internal friction. The beads are confined in an additional
harmonic potential with :math:`\frac{1}{2}k_c(r_n-0)^2` leading to a more compact configuration.
:math:`rk= k_c/k` describes the relative strength compared to the force between beads :math:`k`.
Temp : float
Temperature Kelvin = 273+T[°C]
specm,specb: float
Parameters *m, b* used in internal friction models 'spec' and 'specrif'.
rk : None , float
:math:`rk= k_c/k` describes the relative force constant for *ftype* 'crif'.
Returns
-------
S(q,t)/S(q,0) : dataArray : for single q, dataList : multiple q
- [time; Sqt; Sqt_inf; Sqtinc]
- time units ns
- Sqt is S(q,t)/S(q,0) coherent scattering with diffusion and mode contributions
- Sqt_inf is coherent scattering with ONLY diffusion
- Sqtinc is incoherent scattering with diffusion and mode contributions (no separate diffusion)
- .q wavevector
- .Iq normalized form factor
- .Wl4
- .Re end to end distance :math:`R_e^2=l^2N`
- .trouse rotational correlation time or rouse time
:math:`tr_1 = \xi N^2 l^2/(3 \pi^2 k_bT)= <R_e^2>/(3\pi D_{cm}) = N^2\xi/(pi^2k)`
- .tintern relaxation time due to internal friction
- .tr_p characteristic times :math:`tr_p=tr_1 p^{-2}+t_{intern}`
- .beadfriction
- .ftype type of internal friction
- .... use .attr to see all attributes
Notes
-----
The Rouse model describes beads connected by harmonic springs without hydrodynamic interactions and open ends.
The coherent intermediate scattering function :math:`S(q,t)` is [1]_ [2]_ :
.. math:: S(q,t) = \frac{1}{N} e^{-q^2D_{cm}t} \sum_{n,m}^N e^{-\frac{1}{6}q^2B(n,m,t)}
.. math:: B(n,m,t)=|n-m|^{2\nu}l^2 +
\sum_{p=1}^{N-1} A_p cos(\pi pn/N)cos(\pi pm/N) (1-e^{-t/t_{rp}})
and for incoherent intermediate scattering function the same with :math:`n=m` in the first sum.
with
- :math:`A_p = a_p\frac{4R_e^2}{\pi^2}\frac{1}{p^2}` mode amplitude (usual :math:`a_p=1`)
- :math:`t_{rp} = \frac{t_r}{p^2}` mode relaxation time with Rouse time
:math:`t_r =\frac{\xi N R_e^2 }{3\pi^2 k_bT} = \frac{R_e^2}{3\pi^2 D_{cm}} = \frac{N^2 \xi}{\pi^2 k}`
- :math:`D_{cm}=kT/{N\xi}` center of mass diffusion
- :math:`k=3k_bT/l^2` force constant k between beads.
- :math:`\xi=6\pi visc R` single bead friction :math:`\xi` in solvent (e.g. surrounding melt)
- :math:`t_{intern}=\xi_i/k` additional relaxation time due to internal friction :math:`\xi_i`
Modifications (*ftype*) for internal friction and additional interaction (see [7]_ and [9]_):
- RIF : Rouse with internal friction between neighboring beads in chain [3]_ [4]_.
- :math:`t_{rp}=t_r p^{-2}+t_{intern}`
- :math:`\xi_i=t_{intern}k=t_{intern}3k_bT/l^2` internal friction per bead
- RNI : Rouse model with non-local interactions as additional friction between spatial close beads [5]_ .
- :math:`t_{rp}=t_r p^{-2}+Nt_{intern}/p`
- RAP : Rouse model with anharmonic potentials in bonds describing the stiffness of the chain [6]_.
- :math:`t_{rp}=t_r p^{-2}+t_{intern}ln(N/p\pi)`
- SPECRIF : Specific interactions of relative strength :math:`b` between beads separated by *m* bonds.
Internal friction between neighboring beads as in RIF is added.
- :math:`t_{rp}=t_r p^{-2} (1+bm^2)^{-1} + (1+\frac{m^2}{1+bm^2})t_{intern}`
- :math:`b=k_{specific}/k_{neighbor}` relative strength of both interactions.
- The interaction is between **all** pairs separated by m.
- CRIF : Compacted Rouse with internal friction [9]_.
The beads are confined in an additional harmonic potential with
:math:`\frac{1}{2}k_c(r_n-0)^2` leading to a more compact configuration.
:math:`rk= k_c/k` describes the relative strength compared to the force between beads :math:`k`.
Typically :math:`rk << 1` .
- The mode amplitude prefactor changes from Rouse type to modified confined amplitudes
.. math:: A_p =\frac{4R_e^2}{\pi^2}\frac{1}{p^2}\Rightarrow
A_p^c = \frac{4R_e^2}{\pi^2}\frac{1}{\frac{N^2k_c}{\pi^2k}+p^2}
- The mode relaxation time changes from Rouse type to modified confined
.. math:: t_{rp} = \frac{t_r}{p^2} \Rightarrow
t_{rp}^c = \frac{t_r}{\frac{N^2k_c}{\pi^2k} + p^2}
- :math:`R_e` allows to determine :math:`k_c/k` from small angle scattering data
.. math:: R_e^2 = \frac{2l^2}{\sqrt{k_c/k}}tanh(\frac{N}{2}\sqrt{k_c/k})
- We assume here that the additional potential is :math:`\frac{1}{2}k_c(r_n-r_0)^2` with :math:`r_0`
as the polymer center of mass. As the Langevin equation only depends on relative distances the
internal motions are not affected. The center of mass diffusion :math:`D_{cm}=f(R_e)` is not affected
as the mode dependent friction coefficients don't change [9]_.
- With :math:`rk=k_c/k \rightarrow 0` the original Rouse is recovered for amplitudes,
relaxation and :math:`R_e` .
A combination of different effects is possible [7]_ (but not implemented).
The amplitude :math:`A_p` allows for additional suppression of specific modes
e.g. by topological constraints (see [8]_).
From above the triple Dcm,l,NN are fixed.
- If 2 are given 3rd is calculated
- If all 3 are given the given values are used
For an example see `example_Zimm` and
:ref:`collectivezimmdynamics` how to include collective effects.
Examples
--------
Coherent and incoherent contributions to Rouse dynamics.
To mix the individual q dependent contributions have to be weighted with the according formfactor respectivly
incoherent scattering length and instrument specific measurement technique.
::
import jscatter as js
import numpy as np
t = js.loglist(0.02, 100, 40)
q=np.r_[0.1:2:0.2]
l=0.38 # nm , bond length amino acids
rr = js.dynamic.finiteRouse(t, q, 124, 7, l=0.38, Dcm=0.37, tintern=0., Temp=273 + 60)
p=js.grace()
p.multi(2,1)
p[0].xaxis(scale='log')
p[0].yaxis(label='I(q,t)\scoherent')
p[1].xaxis(label=r't / ns',scale='log')
p[1].yaxis(label=r'I(q,t)\sincoherent')
p[0].title('Rouse dynamics in a solvent')
for i, z in enumerate(rr, 1):
p[0].plot(z.X, z.Y, line=[1, 1, i], symbol=0, legend='q=%g' % z.q)
p[0].plot(z.X, z._Sqt_inf, line=[3, 2, i], symbol=0, legend='q=%g diff' % z.q)
p[1].plot(z.X, z._Sqtinc, line=[1, 2, i], symbol=0, legend='q=%g diff' % z.q)
#p.save(js.examples.imagepath+'/Rousecohinc.jpg')
.. image:: ../../examples/images/Rousecohinc.jpg
:align: center
:width: 50 %
:alt: Rouse
References
----------
.. [1] Doi Edwards Theory of Polymer dynamics
in the appendix the equation is found
.. [2] Nonflexible Coils in Solution: A Neutron Spin-Echo Investigation of
Alkyl-Substituted Polynorbonenes in Tetrahydrofuran
Michael Monkenbusch et al.Macromolecules 2006, 39, 9473-9479
The exponential is missing a "t"
http://dx.doi.org/10.1021/ma0618979
about internal friction
.. [3] Exploring the role of internal friction in the dynamics of unfolded proteins using simple polymer models
Cheng et al.JOURNAL OF CHEMICAL PHYSICS 138, 074112 (2013) http://dx.doi.org/10.1063/1.4792206
.. [4] Rouse Model with Internal Friction: A Coarse Grained Framework for Single Biopolymer Dynamics
Khatri, McLeish| Macromolecules 2007, 40, 6770-6777 http://dx.doi.org/10.1021/ma071175x
.. [5] Origin of internal viscosities in dilute polymer solutions
P. G. de Gennes
J. Chem. Phys. 66, 5825 (1977); https://doi.org/10.1063/1.433861
.. [6] Microscopic theory of polymer internal viscosity: Mode coupling approximation for the Rouse model.
Adelman, S. A., & Freed, K. F. (1977).
The Journal of Chemical Physics, 67(4), 1380–1393. https://doi.org/10.1063/1.435011
.. [7] Internal friction in an intrinsically disordered protein - Comparing Rouse-like models with experiments
A. Soranno, F. Zosel, H. Hofmann
J. Chem. Phys. 148, 123326 (2018) http://aip.scitation.org/doi/10.1063/1.5009286
.. [8] Onset of topological constraints in polymer melts: A mode analysis by neutron spin echo spectroscopy
D. Richter, L. Willner, A. Zirkel, B. Farago, L. J. Fetters, and J. S. Huang
Phys. Rev. Lett. 71, 4158 https://doi.org/10.1103/PhysRevLett.71.4158
.. [9] Looping dynamics of a flexible chain with internal friction at different degrees of compactness.
Samanta, N., & Chakrabarti, R. (2015).
Physica A: Statistical Mechanics and Its Applications, 436, 377–386.
https://doi.org/10.1016/j.physa.2015.05.042
"""
# assure flatt arrays
t = np.atleast_1d(t)
q = np.atleast_1d(q)
# avoid l=0
if l == 0: l = None
NN = int(NN)
if pmax is None: pmax = NN
# if a list pmax of modes is given these are amplitudes for the modes
# pmax is length of list
if isinstance(pmax, numbers.Number):
pmax = min(int(pmax), NN)
modeamplist = np.ones(pmax)
elif isinstance(pmax, list):
modeamplist = np.abs(pmax)
else:
raise TypeError('pmax should be integer or list of amplitudes')
# create correction for diffusion
if Dcmfkt is not None:
if formel._getFuncCode(Dcmfkt):
# is already an interpolation function
Dcmfunktion = Dcmfkt
elif np.shape(Dcmfkt)[0] == 2:
Dcmfunktion = lambda qq: dA(Dcmfkt).interp(qq)
else:
raise TypeError('Shape of Dcmfkt is not 2xN!')
else:
# by default no correction
Dcmfunktion = lambda qq: 1.
# calc the cases of not given parameters for Dcm,NN,l
# kB*Temp is in SI so convert all to SI then back to ns
if rk is not None:
# [9]_ equ 17 for rk->0 this goes to l*NN**0.5
Re = 2 * l ** 2 / rk ** 0.5 * np.tanh(NN / 2 * rk ** 0.5)
else:
# end to end distance
Re = l * np.sqrt(NN)
# friction or Dcm must be given
# Dcm is independent of rk as no HI in Rouse
if Dcm is not None and frict is not None:
pass
elif Dcm is not None and frict is None:
frict = kb * Temp / NN / (Dcm * 1e-9) # diffusion constant in nm^2/ns
elif Dcm is None and frict is not None:
Dcm = kb * Temp / NN / frict * 1e9 # diffusion constant in nm^2/ns
elif Dcm is None and frict is None and Wl4 is not None:
Dcm = Wl4 / (3 * Re ** 2)
frict = kb * Temp / NN / (Dcm * 1e-9)
else:
raise TypeError('fqtfiniteRouse takes at least 1 arguments from Dcm, frict, Wl4')
# slowest relaxation time is rouse time
tr1 = frict * NN ** 2 * l ** 2 / (3 * pi ** 2 * kb * Temp) * 1e-9
# different models for internal friction
p = np.r_[1:len(modeamplist) + 1]
modeamplist = 4 * Re ** 2 / pi ** 2 * modeamplist
if ftype == 'rni':
# rouse with non-local interactions
# frict = f_s + p *f_i
trp = tr1 / p ** 2 + NN * abs(tintern) / p
modeamplist = modeamplist / p ** 2
elif ftype == 'rap':
# rouse model with anharmonic potentials
trp = tr1 / p ** 2 + abs(tintern) * np.log(NN / p * np.pi)
modeamplist = modeamplist / p ** 2
elif ftype == 'specrif':
# rouse model with specific interactions between bead separated by specm of relative strength specb
# + rif
trp = tr1 / p ** 2 / (1 + specb * specm ** 2) + (1 + specm ** 2 / (1 + specb * specm ** 2)) * abs(tintern)
modeamplist = modeamplist / p ** 2
elif ftype == 'crif':
# compacted rouse with internal friction
trp = tr1 / (NN ** 2 / np.pi ** 2 * rk + p ** 2) + abs(tintern)
modeamplist = modeamplist / (NN ** 2 / np.pi ** 2 * rk + p ** 2)
else:
# RIF with constant internal friction time added as default
trp = tr1 / p ** 2 + abs(tintern)
modeamplist = modeamplist / p ** 2
ftype = 'rif'
# prepend 0
t = np.r_[0, np.atleast_1d(t)]
# do the calculation as an 2D array of bnm=[n*m , len(t) +len(t)+len(modeamplist)] elements
if useFortran:
RNM = _bnmtrouse(t=t, NN=NN, l=l, modeamplist=modeamplist, trp=trp)
RNMmodes = RNM[:, -len(modeamplist):]
RNMi = RNM[:, len(t):(2*len(t))] # incoherent
RNMinf = RNM[:, 2*len(t)+1] # coherent t = inf
RNM = RNM[:, :len(t)] # coherent
else:
raise ImportError('finiteRouse only with working Fortran.')
result = dL()
for qq in q:
# diffusion for all t
exp_q2Dt = np.exp(-qq ** 2 * Dcm * Dcmfunktion(qq) * t[1:]) # only diffusion contribution
# amplitude at t=0
expB0 = np.sum(np.exp(-qq ** 2 / 6. * RNM[:, 0])) # is S(qq,t=0) # coherent t=0
expB0i = np.sum(np.exp(-qq ** 2 / 6. * RNMi[:, 0])) # is S(qq,t=0) # incoherent t=0
# diffusion for infinite times in modes
expBinf = np.sum(np.exp(-qq ** 2 / 6. * RNMinf)) # is S(qq,t=inf)
# contribution all modes
expB = np.sum(np.exp(-qq ** 2 / 6. * RNM[:, 1:]), axis=0) # coherent
expBi = np.sum(np.exp(-qq ** 2 / 6. * RNMi[:, 1:]), axis=0) # incoherent
# contribution only first modes
result.append(dA(np.c_[t[1:], # times
exp_q2Dt * expB / expB0, # Sqt
exp_q2Dt * expBinf / expB0, # Sqt_inf
exp_q2Dt * expBi / expB0i].T)) # inc
result[-1].setColumnIndex(iey=None)
result[-1].columnname = 'time; Sqt; Sqt_inf; Sqtinc'
result[-1].modecontribution = (np.sum(np.exp(-qq ** 2 / 6. * RNMmodes), axis=0) / expB0).flatten()
result[-1].Iq = expB0 / NN**2 # normalized form factor
result[-1].q = qq
result[-1].Re = Re
result[-1].ll = l
result[-1].pmax = pmax
result[-1].Dcm = Dcm
result[-1].effectiveDCM = Dcm * Dcmfunktion(qq)
result[-1].Dcmrouse = kb * Temp / NN / frict * 1e9
result[-1].Temperature = Temp
result[-1].trouse = tr1
result[-1].tintern = tintern
result[-1].moderelaxationtimes = trp
result[-1].modeamplitudes = modeamplist
result[-1].beadfriction = frict
result[-1].Drot = 1. / 6. / tr1
result[-1].N = NN
result[-1].internalfriction_g_ns = (tintern * 1e-9) * 3 * kb * Temp / (l * 1e-9) ** 2 * 1e-6
result[-1].ftype = ftype
result[-1].rk = rk
if specm is not None:
result[-1].specm = specm
result[-1].specb = specb
if len(result) == 1:
return result[0]
result.setColumnIndex(iey=None)
result.modelname = inspect.currentframe().f_code.co_name
return result
[docs]
def fixedFiniteRouse(t, q, NN=None, pmax=None, l=None, frict=None, Dcm=None, Wl4=None, Dcmfkt=None, tintern=0.,
Temp=293, ftype=None, specm=None, specb=None, rk=None, fixedends=1):
r"""
Rouse dynamics of a chain **with fixed ends** with N beads of bonds length l and internal friction.
Opposite to the :py:func:`finiteRouse` here one or both ends are fixed.
This might be a chain tethered to a particle that defines the diffusion. Chains are non interacting.
Parameters
----------
fixedends : 0,1,2, default = 1
Number of fixed ends. 0 is only for comparison and corresponds to :py:func:`finiteZimm` .
t : array
Time in units nanoseconds
q : float, list
Scattering vector, units nm^-1
For a list a dataList is returned otherwise a dataArray is returned
NN : integer
Number of chain beads.
l : float, default 1
Bond length between beads; unit nm.
pmax : integer, list of floats
- integer => maximum mode number (:math:`a_p=1`)
- list => :math:`a_p` list of amplitudes>0 for individual modes
to allow weighing; not given modes have weight zero
frict : float
Friction of a single bead/monomer in units `Pas*m=kg/s=1e-6 g/ns`:math:`\xi = 6\pi\eta l`, .
A monomer bead with `l=R=0.1nm = 0.1e-9m` in H2O(20°C) (1 mPas) => 1.89e-12 Pas*m.
Rouse dynamics in a melt needs the bead friction with effective viscosity of the melt
which should be much higher than water.
Polymer melts are typically examined above the glass temperature of the polymer.
Wl4 : float
:math:`W_l^4` Characteristic value to calc friction and Dcm.
:math:`\Omega_Rt=(q^2/6)^2 W_l^4 t`
Dcm : float
Center of mass diffusion in nm^2/ns.
Dcmfkt : array 2xN, function
Function f(q) or array with [qi, f(qi) ] as correction for Dcm like Diff = Dcm*f(q).
e.g. for inclusion of structure factor or hydrodynamic function with f(q)=H(Q)/S(q).
Missing values are interpolated.
tintern : float>0
Relaxation time due to internal friction between neighboring beads in ns.
ftype : 'rni', 'rap','nonspec', 'specrif', 'crif', default = 'rif'
Type of internal friction. See [7]_ for a description and respective references.
- *'rif'*: Internal friction between neighboring beads in chain. :math:`t_{rp}=t_r p^{-2}+t_{intern}`
- *'rni'*: Rouse model with non-local interactions (RNI).
Additional friction between random close approaching beads. :math:`t_{rp}=t_r p^{-2}+N/p t_{intern}`
- *'rap'*: Rouse model with anharmonic potentials due to stiffness of the chain
:math:`t_{rp}=t_r p^{-2}+t_{intern}ln(N/p\pi)`
- *'specrif'*: Specific interactions of strength :math:`b` between beads separated by *m* bonds.
See [7]_. :math:`t_{rp}=t_r p^{-2} (1+bm^2)^{-1} + (1+m^2/(1+bm^2))t_{intern}`
- *'crif'*: Bead confining potential with internal friction. The beads are confined in an additional
harmonic potential with :math:`\frac{1}{2}k_c(r_n-0)^2` leading to a more compact configuration.
:math:`rk= k_c/k` describes the relative strength compared to the force between beads :math:`k`.
Temp : float
Temperature Kelvin = 273+T[°C]
specm,specb: float
Parameters *m, b* used in internal friction models 'spec' and 'specrif'.
rk : None , float
:math:`rk= k_c/k` describes the relative force constant for *ftype* 'crif'.
Returns
-------
S(q,t)/S(q,0) : dataArray for single q, dataList for multiple q
- [time; Sqt; Sqt_inf; Sqtinc]
- time units ns
- Sqt is S(q,t)/S(q,0) coherent scattering with diffusion and mode contributions
- Sqt_inf is coherent scattering with ONLY diffusion
- Sqtinc is incoherent scattering with diffusion and mode contributions (no separate diffusion)
- .q wavevector
- .Wl4
- .Re end to end distance :math:`R_e^2=l^2N`
- .trouse rotational correlation time or rouse time
:math:`tr_1 = \xi N^2 l^2/(3 \pi^2 k_bT)= <R_e^2>/(3\pi D_{cm}) = N^2\xi/(pi^2k)`
- .tintern relaxation time due to internal friction
- .tr_p characteristic times :math:`tr_p=tr_1 p^{-2}+t_{intern}`
- .beadfriction
- .ftype type of internal friction
- .... use .attr to see all attributes
Notes
-----
The Rouse model describes beads connected by harmonic springs without hydrodynamic interactions and open ends.
The coherent intermediate scattering function :math:`S(q,t)` is [1]_ [2]_ :
.. math:: S(q,t) = \frac{1}{N} e^{-q^2D_{cm}t} \sum_{n,m}^N e^{-\frac{1}{6}q^2B(n,m,t)}
:math:`B(n,m,t)` describes the internal motions characterised by eigenmodes of the equation 4.II.6 in [1]_
:math:`\frac{\zeta_p}{\zeta} k \frac{\partial^2 \Phi_{pn}}{\partial^2 n}=-k_p\Phi_{pn}`
where :math:`\Phi_{pn}` describes the delocalisation of bead n in mode p.
The boundary conditions select the eigenmodes from the general form :math:`\Phi_{pn} = A sin(kn) + Bcos(kn)`
- Two free ends :math:`\partial\Phi_{pn}/\partial n=0 \text{ for n=0 and n=N}`
select A=0 and k=pπ/N (equ. 4.II.7+9 in [1]_):
.. math:: B(n,m,t)=|n-m|^{2\nu}l^2 + \sum_{p=1}^{N-1} A_p cos(\pi pn/N)cos(\pi pm/N) (1-e^{-t/t_{zp}})
- One fixed and one free end :math:`\partial\Phi_{pn}/\partial n=0 \text{ at n=0 and } \Phi_{pn}=0` at N
select B=0 and k=(p-1/2)π/N (see [3]_):
.. math:: B(n,m,t)=|n-m|^{2\nu}l^2 + \sum_{p=1-1/2}^{N-1-1/2} A_p sin(\pi pn/N) sin(\pi pm/N) (1-e^{-t/t_{zp}})
With the ends interchanged this can be written like [4]_ (same result)
.. math:: B(n,m,t)=|n-m|^{2\nu}l^2 + \sum_{p=1-1/2}^{N-1-1/2} A_p cos(\pi pn/N) cos(\pi pm/N) (1-e^{-t/t_{zp}})
- Two fixed ends :math:`\Phi_{pn}=0` at n=0 and n=N select B=0 and k=pπ/N:
.. math:: B(n,m,t)=|n-m|^{2\nu}l^2 + \sum_{p=1}^{N-1} A_p sin(\pi pn/N)sin(\pi pm/N) (1-e^{-t/t_{zp}})
For fixed ends the center of mass diffusion Dcm is that of the object where the chain is fixed.
The chain dimension is defined by :math:`R_e=lN^{0.5}`.
Unfortunately there are some papers that give wrong equations for fixed ends Zimmm or Rouse dynamics.
The correct equations above can be retrieved from [1]_ appendix 4.II as solution of the
differential equation 4.II.6 above which describes standing waves in a string or in a open tube.
The classical Zimm/Rouse describe two open ends.
For detailed description of parameters see :py:func:`finiteRouse`.
Examples
--------
Let us assume we have a core-shell particle with grafted chains on a core
(See e.g. Mark et al. [4]_ for silica nanoparticles with grafted chains)
For low aggregation number the chains dont interact and the
motions are that of a Rouse chain with one fixed end. The center of mass diffusion is that of the core-shell
particle and much slower than :math:`D_{Rouse}` but could be determined by PFG-NMR.
For comparison we think of both ends grafted that will make a loop that both ends
are fixed (but not at the same position).
We neglect any influence of the core onto the chain configuration (at least its only a half space).
We allow a Dcm ≈50 times slower than DRouse.
We observe two relaxations, a faster of the internal dynamics and a slower because of diffusion.
First compare one fixed end (full lines) with the free Rouse (broken lines).
For long times the diffusion gets equal visible at low q for long times.
The internal relaxation times are similar as the single chains relax in the same way which we see at larger Q.
The amplitude of mode relaxations is weaker for open ends (the diffusion plateau is higher)
due to the different eigenmodes. The effect is already present in the first mode (pmax=1).
It seems to result from the decorrelation of fixed and open end
compared to the correlated motion with both ends open.
::
import jscatter as js
import numpy as np
t = js.loglist(0.1, 1000, 40)
q=np.r_[0.1:2:0.25]
l=0.38 # nm , bond length amino acids
p=js.grace(1.5,1.5)
p.xaxis(label='q / ns',scale='log')
p.yaxis(label='I(q,t)/I(q,0)')
p.title('Compare 1 fixed to free ends')
p.subtitle('solid line = one fixed end; broken lines = open ends')
# free ends just for comapring
fFR0 = js.dynamic.fixedFiniteRouse(t, q, 124, 40, l=l, Dcm=0.004,frict=9e-14,fixedends=0)
# one fixed end
fFR1 = js.dynamic.fixedFiniteRouse(t, q, 124, 40, l=l, Dcm=0.004,frict=9e-14,fixedends=1)
for i, z in enumerate(fFR1, 1):
p.plot(z.X, z.Y, line=[1, 3, i], symbol=0, legend=fr'q={z.q:.1f} nm\S-1')
for i, z in enumerate(fFR0, 1):
p.plot(z.X, z.Y, line=[3, 3, i], symbol=0, legend='')
p.legend(x=0.2,y=0.4,charsize=0.6)
# p.save(js.examples.imagepath+'/fixed_vs_freeRouse.jpg',size=(1.1,1.1))
.. image:: ../../examples/images/fixed_vs_freeRouse.jpg
:align: center
:width: 50 %
:alt: Zimm open vs. fixed ends
Now we compare one and two open ends.
The differences are marginally and only significant at larger Q for long times as the plateau is different.
The changes will be difficult to discriminate in real measurements.
::
import jscatter as js
import numpy as np
t = js.loglist(0.01, 1000, 50)
q=np.r_[0.1:4:0.4]
l=0.38 # nm , bond length amino acids
p=js.grace(1.5,1.5)
p.xaxis(label='q / ns',scale='log')
p.yaxis(label='I(q,t)/I(q,0)')
p.title('Compare 1 fixed to 2 fixed ends')
p.subtitle('solid line = one fixed end; broken lines = 2 fixed ends')
# two fixed ends
fFR2 = js.dynamic.fixedFiniteRouse(t, q, 124, l=l, Dcm=0.004, frict=9e-14, fixedends=2)
# one fixed end
fFR1 = js.dynamic.fixedFiniteRouse(t, q, 124, l=l, Dcm=0.004, frict=9e-14, fixedends=1)
for i, z in enumerate(fFR1, 1):
p.plot(z.X, z.Y, line=[1, 3, i], symbol=0, legend=fr'q={z.q:.1f} nm\S-1')
for i, z in enumerate(fFR2, 1):
p.plot(z.X, z.Y, line=[3, 3, i], symbol=0, legend='')
p.legend(x=100,y=0.95,charsize=0.6)
# p.save(js.examples.imagepath+'/fixedRouse_1vs2.jpg',size=(2,2))
.. image:: ../../examples/images/fixedRouse_1vs2.jpg
:align: center
:width: 50 %
:alt: Zimm 1 vs. 2 fixed ends
References
----------
.. [1] Doi Edwards Theory of Polymer dynamics
in the appendix the equation is found
.. [2] Nonflexible Coils in Solution: A Neutron Spin-Echo Investigation of
Alkyl-Substituted Polynorbonenes in Tetrahydrofuran
Michael Monkenbusch et al.Macromolecules 2006, 39, 9473-9479
The exponential is missing a "t"
http://dx.doi.org/10.1021/ma0618979
.. [3] Normal Modes of Stretched Polymer Chains
Y. Marciano and F. Brochard-Wyart
Macromolecules 1995, 28, 985-990 https://doi.org/10.1021/ma00108a028
.. [4] Polymer Chain Conformation and Dynamical Confinement in a Model One-Component Nanocomposite
C. Mark, O. Holderer, J. Allgaier, E. Hübner, W. Pyckhout-Hintzen, M. Zamponi, A. Radulescu,
A. Feoktystov, M. Monkenbusch, N. Jalarvo, and D. Richter
Phys. Rev. Lett. 119, 047801 (2017), https://doi.org/10.1103/PhysRevLett.119.047801
about internal friction
.. [7] Internal friction in an intrinsically disordered protein - Comparing Rouse-like models with experiments
A. Soranno, F. Zosel, H. Hofmann
J. Chem. Phys. 148, 123326 (2018) http://aip.scitation.org/doi/10.1063/1.5009286
"""
assert fixedends in [0, 1, 2]
# assure flatt arrays
t = np.atleast_1d(t)
q = np.atleast_1d(q)
# avoid l=0
if l == 0: l = None
NN = int(NN)
if pmax is None: pmax = NN
# if a list pmax of modes is given these are amplitudes for the modes
# pmax is length of list
if isinstance(pmax, numbers.Number):
pmax = min(int(pmax), NN)
modeamplist = np.ones(pmax)
elif isinstance(pmax, list):
modeamplist = np.abs(pmax)
else:
raise TypeError('pmax should be integer or list of amplitudes')
# create correction for diffusion
if Dcmfkt is not None:
if formel._getFuncCode(Dcmfkt):
# is already an interpolation function
Dcmfunktion = Dcmfkt
elif np.shape(Dcmfkt)[0] == 2:
Dcmfunktion = lambda qq: dA(Dcmfkt).interp(qq)
else:
raise TypeError('Shape of Dcmfkt is not 2xN!')
else:
# by default no correction
Dcmfunktion = lambda qq: 1.
# calc the cases of not given parameters for Dcm,NN,l
# kB*Temp is in SI so convert all to SI then back to ns
if rk is not None:
# [9]_ equ 17 for rk->0 this goes to l*NN**0.5
Re = 2 * l ** 2 / rk ** 0.5 * np.tanh(NN / 2 * rk ** 0.5)
else:
# end to end distance
Re = l * np.sqrt(NN)
# friction or Dcm must be given
# Dcm is independent of rk as no HI in Rouse
if Dcm is None and frict is None:
raise TypeError('fqtfiniteRouse needs Dcm, NN, l and frict')
# slowest relaxation time is rouse time
tr1 = frict * NN ** 2 * l ** 2 / (3 * pi ** 2 * kb * Temp) * 1e-9
# different models for internal friction
p = np.r_[1:len(modeamplist) + 1]
modeamplist = 4 * Re ** 2 / pi ** 2 * modeamplist
if ftype == 'rni':
# rouse with non-local interactions
# frict = f_s + p *f_i
trp = tr1 / p ** 2 + NN * abs(tintern) / p
modeamplist = modeamplist / p ** 2
elif ftype == 'rap':
# rouse model with anharmonic potentials
trp = tr1 / p ** 2 + abs(tintern) * np.log(NN / p * np.pi)
modeamplist = modeamplist / p ** 2
elif ftype == 'specrif':
# rouse model with specific interactions between bead separated by specm of relative strength specb
# + rif
trp = tr1 / p ** 2 / (1 + specb * specm ** 2) + (1 + specm ** 2 / (1 + specb * specm ** 2)) * abs(tintern)
modeamplist = modeamplist / p ** 2
elif ftype == 'crif':
# compacted rouse with internal friction
trp = tr1 / (NN ** 2 / np.pi ** 2 * rk + p ** 2) + abs(tintern)
modeamplist = modeamplist / (NN ** 2 / np.pi ** 2 * rk + p ** 2)
else:
# RIF with constant internal friction time added as default
trp = tr1 / p ** 2 + abs(tintern)
modeamplist = modeamplist / p ** 2
ftype = 'rif'
# prepend 0
t = np.r_[0, np.atleast_1d(t)]
# do the calculation as an array of bnm=[n*m , len(t)] elements
# sum up contributions for modes: all, diff+ mode1, only diffusion, t=0 amplitude for normalisation
if useFortran:
RNM = _bnmtrouse(t=t, NN=NN, l=l, modeamplist=modeamplist, trp=trp, fixedends=fixedends)
RNMmodes = RNM[:, -len(modeamplist):]
RNMi = RNM[:, len(t):(2*len(t))] # incoherent
RNMinf = RNM[:, 2 * len(t) + 1] # coherent t = inf
RNM = RNM[:, :len(t)] # coherent
else:
raise ImportError('fixedFiniteRouse only with working Fortran.')
result = dL()
for qq in q:
# diffusion for all t
Sqt = np.exp(-qq ** 2 * Dcm * Dcmfunktion(qq) * t[1:]) # only diffusion contribution
# amplitude at t=0
expB0 = np.sum(np.exp(-qq ** 2 / 6. * RNM[:, 0])) # is S(qq,t=0)/Sqt # coherent
expB0i = np.sum(np.exp(-qq ** 2 / 6. * RNMi[:, 0])) # is S(qq,t=0)/Sqt incoherent
# diffusion for infinite times in modes
expBinf = np.sum(np.exp(-qq ** 2 / 6. * RNMinf)) # is S(qq,t=inf)/Sqt
# contribution all modes
expB = np.sum(np.exp(-qq ** 2 / 6. * RNM[:, 1:]), axis=0) # coherent
expBi = np.sum(np.exp(-qq ** 2 / 6. * RNMi[:, 1:]), axis=0) # incoherent
# contribution only first modes
result.append(dA(np.c_[t[1:],
Sqt * expB / expB0,
Sqt * expBinf / expB0,
Sqt * expBi / expB0i].T))
result[-1].setColumnIndex(iey=None)
result[-1].modecontribution = (np.sum(np.exp(-qq ** 2 / 6. * RNMmodes), axis=0) / expB0).flatten()
result[-1].q = qq
result[-1].Re = Re
result[-1].ll = l
result[-1].pmax = pmax
result[-1].Dcm = Dcm
result[-1].effectiveDCM = Dcm * Dcmfunktion(qq)
result[-1].Dcmrouse = kb * Temp / NN / frict * 1e9
result[-1].Temperature = Temp
result[-1].trouse = tr1
result[-1].tintern = tintern
result[-1].moderelaxationtimes = trp
result[-1].modeamplitudes = modeamplist
result[-1].beadfriction = frict
result[-1].Drot = 1. / 6. / tr1
result[-1].N = NN
result[-1].internalfriction_g_ns = (tintern * 1e-9) * 3 * kb * Temp / (l * 1e-9) ** 2 * 1e-6
result[-1].columnname = 'time; Sqt; Sqt_inf; Sqtinc'
result[-1].ftype = ftype
result[-1].rk = rk
result[-1].fixedends = fixedends
if specm is not None:
result[-1].specm = specm
result[-1].specb = specb
if len(result) == 1:
return result[0]
result.setColumnIndex(iey=None)
result.modelname = inspect.currentframe().f_code.co_name
return result
@formel.memoize()
def _msd_trap(t, u, rt, gamma=1):
# defined here to memoize it
# msd in trap ; equ 4 right part
res = np.zeros_like(t) + u ** 2
res[t < rt * 30] = 6 * u ** 2 * (1 - formel.Ea(-(t[t < rt * 30] / rt) ** gamma, gamma))
return res
[docs]
def diffusionPeriodicPotential(t, q, u, rt, Dg, gamma=1):
r"""
Fractional diffusion of a particle in a periodic potential.
The diffusion describes a fast dynamics inside of the potential trap with a mean square displacement
before a jump and a fractional long time diffusion. For fractional coefficient gamma=1 normal diffusion
is recovered.
Parameters
----------
t : array
Time points, units ns.
q : float
Wavevector, units 1/nm
u : float
Root mean square displacement in the trap, units nm.
rt : float
Relaxation time of fast dynamics in the trap; units ns ( = 1/lambda in [1]_ )
gamma : float
Fractional exponent gamma=1 is normal diffusion
Dg : float
Long time fractional diffusion coefficient; units nm**2/ns.
Returns
-------
dataArray :
[t, Iqt , Iqt_diff, Iqt_trap]
Notes
-----
We use equ. 4 of [1]_ for fractional diffusion coefficient :math:`D_{\gamma}` with fraction :math:`\gamma` as
.. math:: I(Q,t) = exp(-\frac{1}{6}Q^2 msd(t))
.. math:: msd(t) = \langle (x(t)-x(0))^2 \rangle =
6\Gamma^{-1}(\gamma+1)D_{\gamma}t^{\gamma} + 6\langle u^2 \rangle (1-E_{\gamma}(-(\lambda t)^{\gamma}))
with the Mittag Leffler function :math:`E_{\gamma}(-at^{\gamma})` and Gamma function :math:`\Gamma`
and :math:`\lambda =1/r_t`.
The first term in *msd* describes the long time fractional diffusion
while the second describes the additional mean-square displacement inside the trap :math:`\langle u^2 \rangle `.
For :math:`\gamma=1 \to E_{\gamma}(-at^{\gamma}) \to exp(-at)` simplifying the equation to normal diffusion
with traps.
Examples
--------
Example similar to protein diffusion in a mesh of high molecular weight PEG as found in [1]_.
::
import jscatter as js
import numpy as np
t=js.loglist(0.1,50,100)
p=js.grace()
for i,q in enumerate(np.r_[0.1:2:0.3],1):
iq=js.dynamic.diffusionPeriodicPotential(t,q,0.5,5,0.036)
p.plot(iq,symbol=[1,0.3,i],legend='q=$wavevector')
p.plot(iq.X,iq._Iqt_diff,sy=0,li=[1,0.5,i])
p.title('Diffusion in periodic potential traps')
p.subtitle('lines show long time diffusion contribution')
p.yaxis(max=1,min=1e-2,scale='log',label='I(Q,t)/I(Q,0)')
p.xaxis(min=0,max=50,label='t / ns')
p.legend(x=110,y=0.8)
# p.save(js.examples.imagepath+'/fractalDiff.jpg')
.. image:: ../../examples/images/fractalDiff.jpg
:align: center
:height: 300px
:alt: fractalDiff
References
----------
.. [1] Gupta, S.; Biehl, R.; Sill, C.; Allgaier, J.; Sharp, M.; Ohl, M.; Richter, D.
Macromolecules 49 (5), 1941 (2016). https://doi.org/10.1021/acs.macromol.5b02281
"""
# q=np.atleast_1d(q)
# mean square displacement for diffusion in periodic potential no trap; equ 4 left part
msd = lambda t, Dg, u, rt, gamma=1: 6 * Dg * t ** gamma / scipy.special.gamma(gamma + 1)
# Trap contribution in _msd_trap. This is memoized as it is independent of the wavevector
# but for fitting with several Q it is needed multiple times. Cache size is 128 entries.
# the above but extrapolation to t=0 without trap as contribution of long time diffusion at short times
msd_0 = lambda t, Dg, u, rt, gamma=1: 6 * Dg * t ** gamma / scipy.special.gamma(gamma + 1) + 6 * u ** 2
# intermediate scattering function of diffusion in periodic...
sqt = lambda q, t, Dg, u, rt, gamma=1: np.exp(-q ** 2 / 6 * (msd(t, Dg, u, rt, gamma)))
sqttrap = lambda q, t, Dg, u, rt, gamma=1: np.exp(-q ** 2 / 6 * (_msd_trap(t, u, rt, gamma)))
sqt_0 = lambda q, t, Dg, u, rt, gamma=1: np.exp(-q ** 2 / 6 * msd_0(t, Dg, u, rt, gamma))
result = dA(np.c_[t, sqt(q, t, Dg, u, rt, gamma) * sqttrap(q, t, Dg, u, rt, gamma),
sqt_0(q, t, Dg, u, rt, gamma),
sqttrap(q, t, Dg,u, rt, gamma)].T)
result.wavevector = q
result.fractionalDiffusionCoefficient = Dg
result.displacement_u = u
result.relaxationtime = rt
result.fractionalCoefficient_gamma = gamma
result.modelname = inspect.currentframe().f_code.co_name
result.setColumnIndex(iey=None)
result.columnname = 't;Iqt;Iqt_diff;Iqt_trap'
return result
[docs]
def zilmanGranekBicontinious(t, q, xi, kappa, eta, mt=1, amp=1, eps=1, nGauss=60):
r"""
Dynamics of bicontinuous micro emulsion phases. Zilman-Granek model as equ B10 in [1]_. Coherent scattering.
On very local scales (however larger than the molecular size) Zilman and Granek represent the amphiphile layer
in the bicontinuous network as consisting of an ensemble of independent patches at random orientation of size
equal to the correlation length xi.
Uses Gauss integration and multiprocessing.
Parameters
----------
t : array
Time values in ns
q : float
Scattering vector in 1/A
xi : float
Correlation length related to the size of patches which are locally planar
and determine the width of the peak in static data. unit A
A result of the teubnerStrey model to e.g. SANS data. Determines kmin=eps*pi/xi .
kappa : float
Apparent single membrane bending modulus, unit kT
eta : float
Solvent viscosity, unit kT*A^3/ns=100/(1.38065*T)*eta[unit Pa*s]
Water about 0.001 Pa*s = 0.000243 kT*A^3/ns
amp : float, default = 1
Amplitude scaling factor
eps : float, default=1
Scaling factor in range [1..1.3] for kmin=eps*pi/xi and rmax=xi/eps. See [1]_.
mt : float, default 0.1
Membrane thickness in unit A as approximated from molecular size of material. Determines kmax=pi/mt.
About 12 Angstrom for tenside C10E4.
nGauss : int, default 60
Number of points in Gauss integration
Returns
-------
dataList
Notes
-----
See equ B10 in [1]_ :
.. math:: S(q,t) = \frac{2\pi\xi^2}{a^4} \int_0^1 d\mu \int_0^{r_{max}} dr rJ_0(qr\sqrt{1-\mu^2})
e^{-kT/(2\pi\kappa)q^2\mu^2 \int_{k_{min}}^{k_{max}} dk[1-J_0(kr)e^{w(k)t}]/k^3}
with :math:`\mu = cos(\sphericalangle(q,surface normal))` , :math:`J_0` as Bessel function of order 0
- For technical reasons, in order to avoid numerical difficulties,
the real space upper (rmax integration) cutoff was realized by multiplying the
integrand with a Gaussian having a width of eps*xi and integrating over [0,3*eps*xi].
Examples
--------
::
import jscatter as js
import numpy as np
t=js.loglist(0.1,30,20)
p=js.grace()
iqt=js.dynamic.zilmanGranekBicontinious(t=t,q=np.r_[0.03:0.2:0.04],xi=110,kappa=1.,eta=0.24e-3,nGauss=60)
p.plot(iqt)
# to use the multiprocessing in a fit of data use memoize
data=iqt # this represent your measured data
tt=list(set(data.X.flatten)) # a list of all time values
tt.sort()
# use correct values from data for q -> interpolation is exact for q and tt
zGBmem=js.formel.memoize(q=data.q,t=tt)(js.dynamic.zilmanGranekBicontinious)
def mfitfunc(t, q, xi, kappa, eta, amp):
# this will calculate in each fit step for for Q (but calc all) and then take from memoized values
res= zGBmem(t=t, q=q, xi=xi, kappa=kappa, eta=eta, amp=amp)
return res.interpolate(q=q,X=t)[0]
# use mfitfunc for fitting with multiprocessing
References
----------
.. [1] Dynamics of bicontinuous microemulsion phases with and without amphiphilic block-copolymers
M. Mihailescu, M. Monkenbusch et al
J. Chem. Phys. 115, 9563 (2001); http://dx.doi.org/10.1063/1.1413509
"""
tt = np.r_[0., t]
qq = np.r_[q]
result = dL()
nres = formel.doForList(_zgbicintegral, looplist=qq, loopover='q', t=tt, xi=xi, kappa=kappa, eta=eta, mt=mt,
eps=eps, nGauss=nGauss)
for qi, res in zip(qq, nres):
S0 = res[0]
result.append(dA(np.c_[t, res[1:]].T))
result[-1].setColumnIndex(iey=None)
result[-1].Y *= amp / S0
result[-1].q = qi
result[-1].xi = xi
result[-1].kappa = kappa
result[-1].eta = eta
result[-1].eps = eps
result[-1].mt = mt
result[-1].amp = amp
result[-1].setColumnIndex(iey=None)
result[-1].columnname = 't;Iqt'
return result
def _zgbicintegral(t, q, xi, kappa, eta, eps, mt, nGauss):
"""integration of gl. B10 in Mihailescu, JCP 2001"""
quad = formel.parQuadratureFixedGauss
aquad = formel.parQuadratureAdaptiveGauss
def _zgintegrand_k(k, r, t, kappa, eta):
"""kmin-kmax integrand of gl. B10 in Mihailescu, JCP 2001"""
tmp = -kappa / 4. / eta * k ** 3 * t
res = (1. - special.j0(k * r) * np.exp(tmp)) / k ** 3
return res
def _zgintegral_k(r, t, xi, kappa, eta):
"""kmin-kmax integration of gl. B10 in Mihailescu, JCP 2001
integration is done in 2 intervals to weight the lower stronger.
"""
kmax = pi / mt
# use higher accuracy at lower k
res0 = aquad(_zgintegrand_k, eps * pi / xi, kmax / 8., 'k', r=r, t=t[None, :], kappa=kappa, eta=eta,
rtol=0.1 / nGauss, maxiter=250)
res1 = aquad(_zgintegrand_k, kmax / 8., kmax, 'k', r=r, t=t[None, :], kappa=kappa, eta=eta, rtol=1. / nGauss,
maxiter=250)
return res0 + res1
def _zgintegrand_mu_r(r, mu, q, t, xi, kappa, eta):
"""Mu-r integration of gl. B10 in Mihailescu, JCP 2001
aus numerischen Gruenden Multiplikation mit Gaussfunktion mit Breite xi"""
tmp = (-1 / (2 * pi * kappa) * q * q * mu * mu * _zgintegral_k(r, t, xi, kappa, eta)[0] - r * r / (
2 * (eps * xi) ** 2))
tmp[tmp < -500] = -500 # otherwise overflow error in np.exp
y = r * special.j0(q * r * np.sqrt(1 - mu ** 2)) * np.exp(tmp - r ** 2 / (2 * (eps * xi) ** 2))
return y
def _gaussBorder(mu, q, t, xi, kappa, eta):
# For technical reasons, in order to avoid numerical difficulties, the real
# space upper cutoff was realized by multiplying the integrand with a
# Gaussian having a width of eps*xi.
y = quad(_zgintegrand_mu_r, 0, eps * 3 * xi, 'r', mu=mu, q=q, t=t, xi=xi, kappa=kappa, eta=eta, n=nGauss)
return y
y = quad(_gaussBorder, 0., 1., 'mu', q=q, t=t, xi=xi, kappa=kappa, eta=eta, n=nGauss)
return y
[docs]
def zilmanGranekLamellar(t, q, df, kappa, eta, mu=0.001, eps=1, amp=1, mt=0.1, nGauss=40):
r"""
Dynamics of lamellar microemulsion phases. Zilman-Granek model as Equ 16 in [1]_. Coherent scattering.
Oriented lamellar phases at the length scale of the inter membrane distance and beyond are performed
using small-angle neutrons scattering and neutron spin-echo spectroscopy.
Parameters
----------
t : array
Time in ns
q : float
Scattering vector
df : float
- film-film distance. unit A
- This represents half the periodicity of the structure,
generally denoted by d=0.5df which determines the peak position and determines kmin=eps*pi/df
kappa : float
Apparent single membrane bending modulus, unit kT
mu : float, default 0.001
Angle between q and surface normal in unit rad.
For lamellar oriented system this is close to zero in NSE.
eta : float
Solvent viscosity, unit kT*A^3/ns = 100/(1.38065*T)*eta[unit Pa*s]
Water about 0.001 Pa*s = 0.000243 kT*A^3/ns
eps : float, default=1
Scaling factor in range [1..1.3] for kmin=eps*pi/xi and rmax=xi/eps
amp : float, default 1
Amplitude scaling factor
mt : float, default 0.1
Membrane thickness in unit A as approximated from molecular size of material. Determines kmax=pi/mt
About 12 Angstrom for tenside C10E4.
nGauss : int, default 40
Number of points in Gauss integration
Returns
-------
dataList
Examples
--------
::
import jscatter as js
import numpy as np
t=js.loglist(0.1,30,20)
ql=np.r_[0.08:0.261:0.03]
p=js.grace()
iqt=js.dynamic.zilmanGranekLamellar(t=t,q=ql,df=100,kappa=1,eta=2*0.24e-3)
p.plot(iqt)
p.yaxis(label=r'I(Q,t)',min=1e-6,max=1)
p.xaxis(label=r'Q / nm\S-1')
# p.save(js.examples.imagepath+'/zilmanGranekLamellar.jpg', size=(2,2))
.. image:: ../../examples/images/zilmanGranekLamellar.jpg
:align: center
:width: 50 %
:alt: dynamic_t2f_examples.
Notes
-----
See equ 16 in [1]_ :
.. math:: S(q,t) \propto \int_0^{r_{max}} dr r J_0(q_{\perp}r)
exp \Big( -\frac{kT}{2\pi\kappa} q^2\mu^2
\int_{k_{min}}^{k_{max}} \frac{dk}{k^3} [1-J_0(kr) e^{w^\infty(k)t}] \Big)
with :math:`w^{\infty(k) = k^3\kappa/4\overline{\eta}}`, :math:`\mu = cos(\sphericalangle(q,surface normal))` ,
:math:`J_0` as Bessel function of order 0. For details see [1]_.
The integrations are done by nGauss point Gauss quadrature, except for the kmax-kmin integration which is done by
adaptive Gauss integration with rtol=0.1/nGauss k< kmax/8 and rtol=1./nGauss k> kmax/8.
References
----------
.. [1] Neutron scattering study on the structure and dynamics of oriented lamellar phase microemulsions
M. Mihailescu, M. Monkenbusch, J. Allgaier, H. Frielinghaus, D. Richter, B. Jakobs, and T. Sottmann
Phys. Rev. E 66, 041504 (2002)
"""
tt = np.r_[0., t]
qq = np.atleast_1d(q)
result = dL()
nres = formel.doForList(_zglamintegral, looplist=qq, loopover='q', t=tt, kappa=kappa, eta=eta, df=df, mu=mu,
mt=mt, eps=eps, nGauss=nGauss)
for qi, res in zip(qq, nres):
S0 = res[0]
result.append(dA(np.c_[t, res[1:]].T))
result[-1].setColumnIndex(iey=None)
result[-1].Y *= amp / S0
result[-1].q = qi
result[-1].df = df
result[-1].kappa = kappa
result[-1].eta = eta
result[-1].eps = eps
result[-1].mt = mt
result[-1].amp = amp
result[-1].setColumnIndex(iey=None)
result[-1].columnname = 't;Iqt'
return result
def _zglamintegral(t, q, df, kappa, eta, eps, mu, mt, nGauss):
"""integration of gl. 16"""
# quad=scipy.integrate.quad
quad = formel.parQuadratureFixedGauss
aquad = formel.parQuadratureAdaptiveGauss
def _zgintegrand_k(k, r, t, kappa, eta):
"""kmin-kmax integrand o"""
tmp = -kappa / 4. / eta * k ** 3 * t
res = (1. - special.j0(k * r) * np.exp(tmp)) / k ** 3
return res
def _zgintegral_k(r, t, df, kappa, eta):
"""
kmin-kmax integration of gl. B10 in Mihailescu, JCP 2001
"""
kmax = pi / mt
# use higher accuracy at lower k
res0 = aquad(_zgintegrand_k, eps * pi / df, kmax / 8., 'k', r=r, t=t[None, :], kappa=kappa, eta=eta,
rtol=0.1 / nGauss, maxiter=250)
res1 = aquad(_zgintegrand_k, kmax / 8., kmax, 'k', r=r, t=t[None, :], kappa=kappa, eta=eta, rtol=1. / nGauss,
maxiter=250)
return res0 + res1
def _zgintegrand_r(r, mu, q, t, df, kappa, eta):
"""Mu-r integration """
smu = np.sin(mu)
tmp = (-1 / (2 * pi * kappa) * q * q * (1 - smu ** 2) * _zgintegral_k(r, t, df, kappa, eta)[0])
tmp[tmp < -500] = -500 # otherwise overflow error in np.exp
y = r * special.j0(q * r * smu) * np.exp(tmp)
return y
y = quad(_zgintegrand_r, 0, df / eps, 'r', mu=mu, q=q, t=t, df=df, kappa=kappa, eta=eta, n=nGauss)
return y
[docs]
def integralZimm(t, q, Temp=293, viscosity=1.0e-3, amp=1, rtol=0.02, tol=0.02, limit=50):
r"""
Conformational dynamics of an ideal chain with hydrodynamic interaction, coherent scattering.
Integral version Zimm dynamics.
Parameters
----------
t : array
Time points in ns
q : float
Wavevector in 1/nm
Temp : float
Temperature in K
viscosity : float
Viscosity in cP=mPa*s
amp : float
Amplitude
rtol,tol : float
Relative and absolute tolerance in scipy.integrate.quad
limit : int
Limit in scipy.integrate.quad.
Returns
-------
dataArray
Notes
-----
The Zimm model describes the conformational dynamics of an ideal chain with hydrodynamic
interaction between beads. We use equ 85 and 86 from [1]_ as
.. math:: S(Q,t) = \frac{12}{Q^2l^2} \int_0^{\infty} e^{-u-(\Omega_Z t)^{2/3} g(u(\Omega_Z t)^{2/3})} du
with
.. math:: g(y) = \frac{2}{\pi} \int_0^{\infty} x^{-2}cos(xy)(1-e^{-2^{-0.5}x^{2/3}}) dx
.. math:: \Omega_z = \frac{kTQ^3}{6\pi\eta_s}
See [1]_ for details.
Examples
--------
::
import jscatter as js
import numpy as np
t=np.r_[0:10:0.2]
p=js.grace()
for q in np.r_[0.26,0.40,0.53,0.79,1.06]:
iqt=js.dynamic.integralZimm(t=t,q=q,viscosity=0.2e-3)
p.plot((iqt.X*iqt.q**3)**(2/3.),iqt.Y)
p.xaxis(label=r'(Q\S3\Nt)\S2/3')
p.yaxis(label=r'I(Q,t)/I(Q,0)')
p.title('integral Zimm rescaled by characteristic time')
# p.save(js.examples.imagepath+'/integralZimm.jpg')
.. image:: ../../examples/images/integralZimm.jpg
:width: 50 %
:align: center
:alt: integralZimm
References
----------
.. [1] Neutron Spin Echo Investigations on the Segmental Dynamics of Polymers in Melts, Networks and Solutions
in Neutron Spin Echo Spectroscopy Viscoelasticity Rheology
Volume 134 of the series Advances in Polymer Science pp 1-129
DOI 10.1007/3-540-68449-2_1
"""
quad = scipy.integrate.quad
tt = np.r_[t] * 1e-9
tt[t == 0] = 1e-20 # avoid zero
# Zimm diffusion coefficient
OmegaZ = (q * 1e9) ** 3 * kb * Temp / (6 * pi * viscosity)
_g_integrand = lambda x, y: math.cos(y * x) / x / x * (1 - math.exp(-x ** (3. / 2.) / math.sqrt(2)))
_g = lambda y: 2. / pi * quad(_g_integrand, 0, np.inf, args=(y,), epsrel=rtol, epsabs=tol, limit=limit)[0]
_z_integrand = lambda u, t: math.exp(-u - (OmegaZ * t) ** (2. / 3.) * _g(u * (OmegaZ * t) ** (2. / 3.)))
y1 = [quad(_z_integrand, 0, np.inf, args=(ttt,), epsrel=rtol, epsabs=tol, limit=limit)[0] for ttt in tt]
result = dA(np.c_[t, amp * np.r_[y1]].T)
result.setColumnIndex(iey=None)
result.columnname = 't;Iqt'
result.q = q
result.OmegaZimm = OmegaZ
result.Temperature = Temp
result.viscosity = viscosity
result.amplitude = amp
return result
[docs]
def transRotDiffusion(t, q, cloud, Dr, Dt=0, lmax='auto'):
r"""
Translational + rotational diffusion of an object (dummy atoms); dynamic structure factor in time domain.
A cloud of dummy atoms can be used for coarse graining of a non-spherical object e.g. for amino acids in proteins.
On the other hand its just a way to integrate over an object e.g. a sphere or ellipsoid (see example).
We use [2]_ for an objekt of arbitrary shape modified for incoherent scattering.
Parameters
----------
t : array
Times in ns.
q : float
Wavevector in units 1/nm
cloud : array Nx3, Nx4 or Nx5 or float
- A cloud of N dummy atoms with positions cloud[:3] in units nm that describe an object .
- If given, cloud[3] is the incoherent scattering length :math:`b_{inc}` otherwise its equal 1.
- If given, cloud[4] is the coherent scattering length :math:`b_{coh}` otherwise its equal 1.
- If cloud is single float the value is used as radius of a sphere with 10x10x10 grid points.
Dr : float
Rotational diffusion constant (scalar) in units 1/ns.
Dt : float, default=0
Translational diffusion constant (scalar) in units nm²/ns.
lmax : int
Maximum order of spherical bessel function.
'auto' -> lmax > 2π r.max()*q/6.
Returns
-------
dataArray :
Columns [t; Iqtinc; Iqtcoh; Iqttrans]
- .radiusOfGyration
- .Iq_coh coherent scattering (formfactor)
- .Iq_inc incoherent scattering
- .wavevector
- .rotDiffusion
- .transDiffusion
- .lmax
Notes
-----
We calculate the field autocorrelation function given in equ 24 in [2]_ for an arbitrary rigid object
without additional internal dynamic as
.. math:: I(q,t) = e^{-q^2D_tt} I_{rot}(q,t) = e^{-q^2D_tt} \sum_l S_{l,i/c}(q)e^{-l(l+1)D_rt}
where :math:`I_{rot}(q,t)` is the rotational diffusion contribution and
.. math:: S_{l,c}(q) &= 4\pi \sum_m |\sum_i b_{i,coh} j_l(qr_i) Y_{l,m}(\Omega_i)|^2 & coherent scattering \\
S_{l,i}(q) &= \sum_m \sum_i (2l+1) b_{i,inc}^2 |j_l(qr_i)|^2 & incoherent scattering\\
and coh/inc scattering length :math:`b_{i,coh/inc}`, position vector :math:`r_i` and orientation of atoms
:math:`\Omega_i`, spherical Bessel function :math:`j_l(x)`, spherical harmonics :math:`Y_{l,m}(\Omega_i)`.
- The incoherent intermediate scattering function is res.Y/res.Iq_inc or res._Iqtinc/res.Iq_inc
- The coherent intermediate scattering function is res._Iqtcoh/res.Iq_coh
- For real scattering data as backscattering or spinecho coherent and incoherent have to be mixed according
to the polarisation conditions of the experiment accounting also for spin flip probability of coherent and
incoherent scattering. For the simple case of non-polarised measurement we get
.. math:: I(q,t)/I(q,0) = \frac{I_{coh}(q,t)+I_{inc}(q,t)}{I_{coh}(q,0)+I_{inc}(q,0)}
Examples
--------
A bit artificial look at only rotational diffusion of a superball build from dummy atoms.
(rotational diffusion should only show if also translational diffusion is seen)
Change p to change from spherical shape (p=1) to cube (p>10) or star like (p<0.5)
(use grid.show() to take a look at the shape)
The coherent contribution is suppressed for low q if the particle is spherical .
::
import jscatter as js
import numpy as np
R=2;NN=10
ql=np.r_[0.4:2.:0.3,2.1:15:2]
t=js.loglist(0.001,50,50)
# get superball
p2=1
grid=js.ff.superball(ql,R,p=p2,nGrid=NN,returngrid=True)
Drot=js.formel.Drot(R)
Dtrans=js.formel.Dtrans(R)
p=js.grace(1.5,1)
p.new_graph(xmin=0.23,xmax=0.43,ymin=0.25,ymax=0.55)
iqt=js.dL([js.dynamic.transRotDiffusion(t,q,grid.XYZ,Drot,lmax=30) for q in ql])
for i,iiqt in enumerate(iqt,1):
p[0].plot(iiqt.X,iiqt.Y/iiqt.Iq_inc,li=[1,3,i],sy=0,le=f'q={iiqt.wavevector:.1f} nm\S-1')
p[0].plot(iiqt.X,iiqt._Iqtcoh/iiqt.Iq_coh,li=[3,3,i],sy=0,le=f'q={iiqt.wavevector:.1f} nm\S-1')
p[1].plot(iqt.wavevector,iqt.Iq_coh.array/grid.numberOfAtoms(),li=1)
p[1].plot(iqt.wavevector,iqt.Iq_inc.array/grid.numberOfAtoms(),li=1)
p[0].xaxis(scale='l',label='t / ns',max=200,min=0.001)
p[0].yaxis(scale='n',label='I(q,t)/I(q,0)')
p[1].xaxis(scale='n',label='q / nm\S-1')
p[1].yaxis(scale='l',label='I(q,t=0)')
p[0].legend(x=60,y=1.1,charsize=0.7)
p[0].title(f'rotational diffusion of superball with p={p2:.2f}')
p[0].subtitle(f'coh relevant only at high q for sphere')
p[1].subtitle('coh + inc scattering')
p[0].text(x=0.0015,y=0.8,string=r'lines inc\ndashed coh',charsize=1.5)
#p.save(js.examples.imagepath+'/rotDiffusion.jpg')
# Second example
# non-polarized experiment
p=js.grace(1.5,1)
grid=js.ff.superball(ql,R,p=1.,nGrid=10,returngrid=True)
iqt=js.dL([js.dynamic.transRotDiffusion(t,q,grid.XYZ,Drot,Dtrans,lmax=30) for q in ql])
for i,iiqt in enumerate(iqt,1):
p.plot(iiqt.X,(iiqt._Iqtinc+iiqt._Iqtcoh)/(iiqt.Iq_inc+iiqt.Iq_coh),li=[1,3,i],sy=0,le=f'q={iiqt.wavevector:.1f} nm\S-1')
p.plot(iiqt.X,iiqt._Iqtcoh/iiqt.Iq_coh,li=[3,3,i],sy=0,le=f'q={iiqt.wavevector:.1f} nm\S-1')
p.xaxis(scale='l',label='t / ns',max=200,min=0.001)
p.yaxis(scale='n',label='I(q,t)/I(q,0)')
p[0].legend(x=60,y=1.1,charsize=0.7)
p[0].title(f'translational/rotational diffusion of superball with p={p2:.2f}')
p[0].text(x=0.0015,y=0.5,string=r'lines coh+inc\ndashed only coh',charsize=1.5)
#p.save(js.examples.imagepath+'/transrotDiffusion.jpg')
.. image:: ../../examples/images/rotDiffusion.jpg
:width: 50 %
:align: center
:alt: rotDiffusion
.. image:: ../../examples/images/transrotDiffusion.jpg
:width: 50 %
:align: center
:alt: transrotDiffusion
References
----------
.. [1] Incoherent scattering law for neutron quasi-elastic scattering in liquid crystals.
Dianoux, A., Volino, F. & Hervet, H. Mol. Phys. 30, 37–41 (1975).
.. [2] Effect of rotational diffusion on quasielastic light scattering from fractal colloid aggregates.
Lindsay, H., Klein, R., Weitz, D., Lin, M. & Meakin, P. Phys. Rev. A 38, 2614–2626 (1988).
"""
#: Lorentzian
expo = lambda t, ll1D: np.exp(-ll1D * t)
if isinstance(cloud, numbers.Number):
R = cloud
NN = 10
grid = np.mgrid[-R:R:1j * NN, -R:R:1j * NN, -R:R:1j * NN].reshape(3, -1).T
inside = lambda xyz, R: la.norm(grid, axis=1) < R
cloud = grid[inside(grid, R)]
if cloud.shape[1] == 5:
# last columns are incoherent and coherent scattering length
blinc = cloud[:, 3]
blcoh = cloud[:, 4]
cloud = cloud[:, :3]
elif cloud.shape[1] == 4:
# last column is scattering length
blinc = cloud[:, 3]
blcoh = np.ones(cloud.shape[0])
cloud = cloud[:, :3]
else:
blinc = np.ones(cloud.shape[0])
blcoh = blinc
t = np.array(t, float)
bi2 = blinc ** 2
r, p, th = formel.xyz2rphitheta(cloud).T
pp = p[:, None]
tt = th[:, None]
qr = q * r
if not isinstance(lmax, numbers.Integral):
# lmax = pi * r.max() * q / 6. # a la Cryson
lmax = min(max(2 * int(pi * qr.max() / 6.), 6), 100)
# We calc here the field autocorrelation function as in equ 24
# incoherent with i=j -> Sum_m(Ylm) leads to (2l+1)/4pi
bjlylminc = [(bi2 * spjn(l, qr) ** 2 * (2 * l + 1)).sum() for l in np.r_[:lmax + 1]]
# add time dependence
Iqtinc = np.c_[[bjlylminc[l].real * expo(t, l * (l + 1) * Dr) for l in np.r_[:lmax + 1]]].sum(axis=0)
Iq_inc = np.sum(bjlylminc).real
# coh is sum over i then (abs)squared and sum over m see Lindsay equ 19 or 20
bjlylmcoh = [4 * np.pi * np.sum(np.abs((blcoh * spjn(l, qr) * Ylm(l, np.r_[-l:l + 1], pp, tt).T).sum(axis=1)) ** 2)
for l in np.r_[:lmax + 1]]
Iqtcoh = np.c_[[bjlylmcoh[l].real * expo(t, l * (l + 1) * Dr) for l in np.r_[:lmax + 1]]].sum(axis=0)
Iq_coh = np.sum(bjlylmcoh).real
Iq_trans = np.exp(-q ** 2 * Dt * t)
result = dA(np.c_[t, Iq_trans * Iqtinc, Iq_trans * Iqtcoh].T)
result.modelname = inspect.currentframe().f_code.co_name
result.setColumnIndex(iey=None)
result.columnname = 't; Iqtinc; Iqtcoh; Iqttrans'
result.radiusOfGyration = np.sum(r ** 2) ** 0.5
result.Iq_coh = Iq_coh
result.Iq_inc = Iq_inc
result.wavevector = q
result.rotDiffusion = Dr
result.transDiffusion = Dt
result.lmax = lmax
return result
def _gauss_norm(x, mean, sigma):
return np.exp(-0.5 * (x - mean) ** 2 / sigma ** 2) / np.sqrt(2 * pi) / sigma
# noinspection PyIncorrectDocstring
[docs]
def resolution(t, s0=1, m0=0, s1=None, m1=None, s2=None, m2=None, s3=None, m3=None,
s4=None, m4=None, s5=None, m5=None, s6=None, m6=None, s7=None, m7=None,
a0=1, a1=1, a2=1, a3=1, a4=1, a5=1, a6=1, a7=1, bgr=0, resolution=None):
r"""
Resolution in time domain as multiple Gaussians for inelastic measurement
as back scattering or time of flight instrument.
Multiple Gaussians define the function to describe a resolution measurement.
Use ```resolution_w``` to fit with the appropriate normalized Gaussians and this function
to implicit Fourier transform a signal. See Notes.
Parameters
----------
t : array
Times
s0,s1,... : float
Width of Gaussian functions representing a resolution measurement.
The number of si not None determines the number of Gaussians.
m0, m1,.... : float, None
Means of the Gaussian functions representing a resolution measurement.
a0, a1,.... : float, None
Amplitudes of the Gaussian functions representing a resolution measurement.
bgr : float, default=0
Background
resolution : dataArray
Resolution with attributes sigmas, amps which are used instead of si, ai.
- If resolution is from `w` domain this represents the Fourier transform from `w` to `t` domain.
`means` are NOT used from `w` domain as these result only in a phase shift, instead m0..m5 are used.
If mi is not give zero is assumed.
- If from `t` domain the resolution is recalculated with same parameters for new t.
Returns
-------
dataArray
Notes
-----
In a typical inelastic experiment the resolution is measured by e.g. a vanadium measurement (elastic scatterer).
In `t` domain (Neutron Spin Echo) this is a carbon black sample for small Q or e.g. Zirconium for higher Q.
In `w` domain the resolution is described by a multi Gaussian function as in resw=resolution_w(w,...) with
amplitudes :math:`ai_w`, width :math:`si_w` and common mean :math:`m_w`.
resolution(t,resolution_w=resw) defines the Fourier transform of resolution_w using the same coefficients
as an implicit Fourier transform.
:math:`mi_t` are set by default to zero as :math:`mi_w` lead only to a phase shift.
It is easiest to shift w values in w domain as it corresponds to a shift of the elastic line.
Beside the fitting of resolution measurements the pair of `resolution_w` and `resolution` allows
a Fourier transform from `w` to `t` domain of any signal. If `resolution_w` is used for fitting
of data in the `w` domain then ```ft = resolution(t=..,resolution=resolution_w_fit)```
represents the Fourier transform of the fitted data.
The used Gaussians are normalized that they are a pair of Fourier transforms:
.. math:: R_t(t,m_i,s_i,a_i)=\sum_i a_i s_i e^{-\frac{1}{2}s_i^2 t^2} \Leftrightarrow
R_w(w,m_i,s_i,a_i)= \sum_i a_i e^{-\frac{1}{2}(\frac{w-m_i}{s_i})^2}
under the Fourier transform defined as
.. math:: F(f(t)) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt
.. math:: F(f(w)) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(\omega) e^{i\omega t} d\omega
Examples
--------
Using the result of a fit in w domain to represent the resolution in time domain :
::
import jscatter as js
import numpy as np
# resw is a resolution in w domain maybe as a result from a fit to vanadium data
# resw contains all parameters
w = np.r_[-100:100:0.5]
resw = js.dynamic.resolution_w(w, s0=12, m0=0, a0=2)
# representing the Fourier transform of resw as a gaussian transforms to time domain
t = np.r_[0:1:0.01]
rest = js.dynamic.resolution(t,resolution=resw)
t2 = np.r_[0:0.5:0.005]
rest2 = js.dynamic.resolution(t2,resolution=rest)
"""
# we keep None to allow change of single Gaussians
if resolution is None:
means = [m0, m1, m2, m3, m4, m5, m6, m7]
sigmas = [s0, s1, s2, s3, s4, s5, s6, s7]
amps = [a0, a1, a2, a3, a4, a5, a6, a7]
else:
if resolution.modelname[-1] == 'w':
means = [0 if m is None else m for m in [m0, m1, m2, m3, m4, m5, m6, m7]]
sigmas = [1. / s if s is not None else s for s in resolution.sigmas]
amps = resolution.amps
else:
means = resolution.means
sigmas = resolution.sigmas
amps = resolution.amps
t = np.atleast_1d(t)
# filter Nones
sma = np.array([[s, m, a] for s, m, a in zip(sigmas, means, amps) if None not in [s, m, a]]).T
Y = np.sum(sma[2][:, None] * _gauss_norm(x=t, mean=sma[1][:, None], sigma=sma[0][:, None]), axis=0)
result = dA(np.c_[t, Y + bgr].T)
result.setColumnIndex(iey=None)
result.modelname = inspect.currentframe().f_code.co_name
result.columnname = 't; Rqt'
result.means = means
result.sigmas = sigmas
result.amps = amps
return result