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Dynamics of interacting Brownian particles
Volume 112, number 3
R~AMICS
CHEhiICAL
OF ~TE~~~~
BRO~IA~
T. TSANG Deparrmenr of Physics and Astronomy,
Hoard
PHYSICS LETTERS
7 December 1984
PARTICLES
University, Washingron, D.C. 20059, USA
and Code 6120. Naval Research Laboratory, Washington, LXC 2037.5, USA Received 5 September
I984
A memory function formulation
has been used for the dynamics of interacting Brow-‘-N.... particles in colloidal systems
and
polymer dispersions. By introducing approx‘unations at the level of the force correlation function. the theoretical scattering functions agree well with the experimental photon correfation spectroscopy data.
Charged macromolecules or particles in aqueous environments can exhibit considerable ordering due to the repulsive Coulomb interactions. Such orderings of polymer dispersions, coiIoids, virus particles and bioIogical systems have been studied by light scattering. More recently, photon correlation spectroscopy (PCS) has been used to study the dynamics of these systems; these investigations have been extensively reviewed [l--3]. In these experiments, the intermediate scattering functions F(K,t) have been measured 141, where K is the scattering vector and t is the correlation decay time. When experiments were performed on aqueous dispersions of polystyrene spheres in the presence of dissolved eiectrolytes(e.g. =0.0X M NaCl), the Coulomb interactions are effectively screened, hence identical exponential decays of F(K,r) versus @r have been observed at various values of K, in agreement with the simple Brownian motion or free-particle results. The experimental free-particle diffusion constant D is also in good agreement with the StokesEinstein value. However, when the dissolved electrolytes were carefulIy removed, the decay of F(K,r) would no longer be exponential and there is also considerable dependence of F(K,r) on ii, The details of the many-body dynamics are rather complicated for these systems of interacting Brownian particles, hence a simplified approach has been developed in the present work to evaluate the scattering functions. Following the hiori projection operator formaIism j5,6], approximations have been introduced at the level of the memo-_ ‘30
ry frtnction. From the static or equilibrium structure of the system and the Stokes-Einstein diffusion constant, the scattering functions may be readily determined without introducing any arbitrary or adjustable parameters. The theoretical results may then be cornpared with some of the typical experimental data [7] on aqueous dispersions of polystyrene particles, and the agreement appears to be satisfactory. The static structure factor S(K) is defined as the initial (t = 0) value of the scattering function; that is, S(K) = F(K-0). The radial distribution function g(r) may be obtained by the Fourier transform of S(K), where r is the distance_ It is convenient to choose the particle mass as the unit of mass and to introduce the normalized scattering function f(K,t) = F(K,t)/F(K,O). Given S(K) and the Stokes-Einstein diffusion constant i), we wiII now proceed to evaIuatef(K,t). It has been shown previously [S] that the K dependence off(K,t) may be removed by introducing the effective mean square displacement 1M: f(K, r) = exp [-$ K%f(K, t)] . Then M(K,t) may be expressed
(0 in the form
M(K, t) = W(8) - i [S(K) - 1] /~(~~~~(~) ,
(21
where W(t) and J(t) are functions of t only. The explicit K dependence of f(K,t) and M(K,I) is through the S(K) term in (2). Physically, W(t) is the second moment or mean square displacement of a particle in the system, thus 0 009-~614/84/$03_00 0 Elsevier Science Publishers born-~oBand Physics Publishing Division)
13-V.
7 December 1984
CHEMICAL PHYSICS LETTERS
Volume 112. number 3
it can be related [9] to the normalized velocity correlation function G(t). For free particles, we have the very simple form d$/dt = -71$ or G(t) = e-r*, where y is the Stokes-Einstein friction constant due to the viscosity of water or collisions with water molecules_ However, there are considerable complications for systems of interacting Brownian particles_ The Mori memory function formalism [5,6] gives the general fomr as
(3)
exponential Substitution
so that no particle is counted of (4) and (8) into (3) gives
d$Jdt = -y$
2a edad
- j
twice [ II].
$(t - 7) dT _
(9)
0
The initial condition is r$ = 1 at t = 0. This equation may be readily solved by the method of Laplace transform (LT). Let the LT of Q be denoted by U: U(p) = J
e-Pt
G(f) dr _
(10)
0
where I’(t) is the memory function due to the interactions between the particles and is proportional to the force correlation function (FCF) d(f): l--(r) = (3k7)-I@),
Then the LT of (9) is pU - 1 = -yU
(4)
where k is the Boltzmann perature_ Ii is convenient
constant and T is the temto write @(t) as:
00) = G(O) R(t) .
(3
where r$(O) is the mean square force (MSF) or the initial value of FCF and R(t) is the normalized memory function (NMF). By definition, we have R(0) = 1. For interacting particles, we have previously considered [8] a harmonic force field with force constant Q, potential V = $ (or? and force -v’ = -dV/dr = --Qr. In general, cz is time-dependent. As an approximation, a(t) may be replaced by its initial value o(0) = cz. For a particle in this system with friction constant y = kT/D and harmonic force constant a, the average distance (r(t)) is related to the initial distance r(0) by the relation [lo] (r(f))/r(O)
= e-nflr,
(6)
when r2 Z+ Q. Since the interacting force is proportional to r, the same form may be used for NMF also: R(t) = e-a*/T _
(7)
The value of MSF Q(O) may be obtained Boltzmann averaging of (Y’)‘:
by the
- 2aU(p + a/y)-’
e-fiv/7 (If’)” dr/ r 0
4nr2 e-flv12 dr
= 6akT, where /3 = (kT)-l.
The factor off
is included
(8) in the
01)
Thus we get U= (yp + a)/[yp2 + (y2 +a)p + 3ay]. This may be written in the form of partial fractions_ For y2 s a, we have U = (p i- y)-l - (2a/y2)( p + 3a/y)-I, and the inverse transform is G(t) =
e-Yr
_
(2a/-$)e-jadY_
(!2)
The second moment [9] is [U(t) = 2kTJk X $(T)dT. Fort a y-l, we get Ii’(t) = (2kTr/y)
- (4kTf9a)
E(3ar/y)
(t - T)
_
(13
where the function Eis defined by E(x)=x - 1 i-e-r_ The lu(t)versus r curve is shown as the solid line in the top diagram of fig. 1. The slope is 2kT/y (same as free particles) for small t and 2kT/3y for large t. In a harmonic force field, there is also 2 general tendency for the particles to move toward the potential minimum. The function J(t) is the reduction in second moment due to these motions and has been given previously [S] as J(t) = (kT/a) [I - exp (-2
Jo(r) 0
dr/y)]
.
(14)
The NMF R(t) in (7) may .be used as an approximate description of the decay of the force constant, o(t) = aR(t) = a eAar/Y_ Substitution into (12) gives: J(r) = (kT/a) [ 1 - exp(-2
Q(O) = 7 4x? 0
.
•f 2 e-“‘/Y)] _
(1%
For the J(t) versus r curve, the slope is 2kTjy for small t and is 0 for large t. We wiIl now compare our theoretical results with experimental data [7] for a system of polystyrene par221
Volume
1 12, number
3
CHEMICAL
PHYSICS
LETTERS
7 December
1984
6-
M
0
KxxxJ
20030 t
Fig. 3. Effective mean square displacemenrM(K,f) versus t. SoIid curve, theoretical for S = 0.18; daslted curve, theoretical for free particle; circles, experimental for case C, K = 0.89.
t Fig. 1. Effective mean square displacement fif(K.t) versus t. Top diagram: solid curve, theoreticat for S = 1; dashed curve, tlieoreticai for ircc particle; crosses, esperiment3l for case -4. K = 3.22; circles, experimental for case A’, K = 2.42. Bottom diagram: solid curve, theoretical for S = 1.8; dashed curve, theoretical for free particle; triangles, experimental for caseB, K = 2.12.
titles of 250 A mean radius in aqueous environment at the concentration of 1.25 X I Om3 g/cm3. As before [S], the particle mass 5.9 X 1O-l7 g is used as the unit of mass, 1O-5 cm = 1000 A is arbitrarily chosen as the unit of length. The unit of time is chosen to be 3.78 X 10m7 s such that X-T= I at the experimental temperature (293 K) and D = I/y in these units. The static structure factor S(K) and its Fourier transform y(r) have been reported previously [7,8]. Fromg(r), we have previously obtained II = 2.8. The free-particle diffusion constant D = 3.23 X 1O-* and friction constant y = l/D = 3096 are determined by PCS from the same environment in the presence of electrolytes; these values are also in very good agreement with the Stokes-Einstein values. From these values of 0 and y, we have calculated W(r) and J(t) from (13) and (15). M(K,r) is then calculated from (2) for: (A) 1”;= 3.22, S = I, M(r) = IV(r), (A’) K = 2.42, S = 1, M = IV, (B) K=2_12,S= 1.8,M=W-0.44J,(C)R=0.89,S= 0.18.M = IV + 4_55f_ These results are shown as the solid curves in figs. 1 and 2. By using (l), the PCS experimental/(K,r) data of ref. [7] are converted into
M(K,t). These data are shown as: (A) crosses in the upper diagram of fig. 1, (A’) circles in upper diagram of fig. 1, (B) triangles in lower diagram of fig. 1, (C) circles in fig. 2. There is general agreement between the experimental and theoretical results. The freeparticle (simple Brownian motion) result M = 2Dt is shown as the dashed line for comparison. They ieviate considerably from the PCS data. Although the initial slopes at t = 0 are dependent on the wave vector A’, the slopes of all the theoretical M(K,r) versus t curves at “long times” (t > T/U) have the common value of 2013, indicating a factor of 3 reduction from the free-particle behavior due to the interactions between the particles. These curves are similar to the phenomenotogical two-exponential model [ 121 proposed for these systems. For more dilute polymer dispersions or for systems where electrolytes may be present, the value of a would be small, thus the “transition” to the long-time behavior would occur at larger values oft. Sometimes, r would be so iong that the resulting PCS signal would be very weak and it may be difficult to observe this “transition”. In conclusion, the dynamics of systems of interacting Brownian particles may be understood by using the memory function formalism and by introducing approximations at the level of force correlation functions_ Starting from the experimental static structure factor S(K) and the Stokes-Einstein diffusion constant D, we are able to calculate explicitly the scattering function F(K,t) without introducing any arbi-
Volume 112, number 3
CHEMICAL PHYSICS LET-PERS
7 December 1984
trary or adjustable parameters. The general agreement With the PCS experimental data appears to be satisfactory_
[3] W. Hessand R. Klein, Advan. Phys. 32 (1983) 173. [4] S.H. Chen, in: Physical chemistry, an advanced treatise,
The partial financial support by National Aeronautical and Space Administration grant NAG-S-1 56 is gratefully acknowledged.
[6] H. Mori, Pro8r. Theor. Phys. (Kyoto) 34 (1965) 399. [7] J-C. Brown,P.N. Pusey, J.W. Goodwin and R.H. Ottewili, J. Phys. A8 (1975) 664. [S] T. Tsang and H-T_ Tazg. J. Chem. Phys. 76 (1982) 3873. 191 D.A. McQuarrie, Statistical mechanics (Harper and Row, New York, 1976). [lo] S. Chandrasckhar, Rev. Mod. Phys. 15 (1943) 1. [ 111 XI. F&man_ J. Chem. Phys. 51 (1969) 3270.
References [ 1 J H.Z. Cummins and E.R. Pike, Photon correlation spectroscopy and velocity (Plenum Press, New York, 1977). [2] V. Degiorgio, hf. Corti and hi. Giglio, Light scattering in liquids and macromolecular New York, 1980).
Vol. 8A, eds. H. Eyring, D_ Henderson and W. Jost (Academic Press, New York, 1971). [5] H. hlori, Progr. Theor. Phys. (Kyoto) 33 (1965) 423.
[ 131 P.N. Pusey, Phil. Trans. Roy. Sot. A293 (1979)
429.
solutions (Plenum Press,
223
R~AMICS
CHEhiICAL
OF ~TE~~~~
BRO~IA~
T. TSANG Deparrmenr of Physics and Astronomy,
Hoard
PHYSICS LETTERS
7 December 1984
PARTICLES
University, Washingron, D.C. 20059, USA
and Code 6120. Naval Research Laboratory, Washington, LXC 2037.5, USA Received 5 September
I984
A memory function formulation
has been used for the dynamics of interacting Brow-‘-N.... particles in colloidal systems
and
polymer dispersions. By introducing approx‘unations at the level of the force correlation function. the theoretical scattering functions agree well with the experimental photon correfation spectroscopy data.
Charged macromolecules or particles in aqueous environments can exhibit considerable ordering due to the repulsive Coulomb interactions. Such orderings of polymer dispersions, coiIoids, virus particles and bioIogical systems have been studied by light scattering. More recently, photon correlation spectroscopy (PCS) has been used to study the dynamics of these systems; these investigations have been extensively reviewed [l--3]. In these experiments, the intermediate scattering functions F(K,t) have been measured 141, where K is the scattering vector and t is the correlation decay time. When experiments were performed on aqueous dispersions of polystyrene spheres in the presence of dissolved eiectrolytes(e.g. =0.0X M NaCl), the Coulomb interactions are effectively screened, hence identical exponential decays of F(K,r) versus @r have been observed at various values of K, in agreement with the simple Brownian motion or free-particle results. The experimental free-particle diffusion constant D is also in good agreement with the StokesEinstein value. However, when the dissolved electrolytes were carefulIy removed, the decay of F(K,r) would no longer be exponential and there is also considerable dependence of F(K,r) on ii, The details of the many-body dynamics are rather complicated for these systems of interacting Brownian particles, hence a simplified approach has been developed in the present work to evaluate the scattering functions. Following the hiori projection operator formaIism j5,6], approximations have been introduced at the level of the memo-_ ‘30
ry frtnction. From the static or equilibrium structure of the system and the Stokes-Einstein diffusion constant, the scattering functions may be readily determined without introducing any arbitrary or adjustable parameters. The theoretical results may then be cornpared with some of the typical experimental data [7] on aqueous dispersions of polystyrene particles, and the agreement appears to be satisfactory. The static structure factor S(K) is defined as the initial (t = 0) value of the scattering function; that is, S(K) = F(K-0). The radial distribution function g(r) may be obtained by the Fourier transform of S(K), where r is the distance_ It is convenient to choose the particle mass as the unit of mass and to introduce the normalized scattering function f(K,t) = F(K,t)/F(K,O). Given S(K) and the Stokes-Einstein diffusion constant i), we wiII now proceed to evaIuatef(K,t). It has been shown previously [S] that the K dependence off(K,t) may be removed by introducing the effective mean square displacement 1M: f(K, r) = exp [-$ K%f(K, t)] . Then M(K,t) may be expressed
(0 in the form
M(K, t) = W(8) - i [S(K) - 1] /~(~~~~(~) ,
(21
where W(t) and J(t) are functions of t only. The explicit K dependence of f(K,t) and M(K,I) is through the S(K) term in (2). Physically, W(t) is the second moment or mean square displacement of a particle in the system, thus 0 009-~614/84/$03_00 0 Elsevier Science Publishers born-~oBand Physics Publishing Division)
13-V.
7 December 1984
CHEMICAL PHYSICS LETTERS
Volume 112. number 3
it can be related [9] to the normalized velocity correlation function G(t). For free particles, we have the very simple form d$/dt = -71$ or G(t) = e-r*, where y is the Stokes-Einstein friction constant due to the viscosity of water or collisions with water molecules_ However, there are considerable complications for systems of interacting Brownian particles_ The Mori memory function formalism [5,6] gives the general fomr as
(3)
exponential Substitution
so that no particle is counted of (4) and (8) into (3) gives
d$Jdt = -y$
2a edad
- j
twice [ II].
$(t - 7) dT _
(9)
0
The initial condition is r$ = 1 at t = 0. This equation may be readily solved by the method of Laplace transform (LT). Let the LT of Q be denoted by U: U(p) = J
e-Pt
G(f) dr _
(10)
0
where I’(t) is the memory function due to the interactions between the particles and is proportional to the force correlation function (FCF) d(f): l--(r) = (3k7)-I@),
Then the LT of (9) is pU - 1 = -yU
(4)
where k is the Boltzmann perature_ Ii is convenient
constant and T is the temto write @(t) as:
00) = G(O) R(t) .
(3
where r$(O) is the mean square force (MSF) or the initial value of FCF and R(t) is the normalized memory function (NMF). By definition, we have R(0) = 1. For interacting particles, we have previously considered [8] a harmonic force field with force constant Q, potential V = $ (or? and force -v’ = -dV/dr = --Qr. In general, cz is time-dependent. As an approximation, a(t) may be replaced by its initial value o(0) = cz. For a particle in this system with friction constant y = kT/D and harmonic force constant a, the average distance (r(t)) is related to the initial distance r(0) by the relation [lo] (r(f))/r(O)
= e-nflr,
(6)
when r2 Z+ Q. Since the interacting force is proportional to r, the same form may be used for NMF also: R(t) = e-a*/T _
(7)
The value of MSF Q(O) may be obtained Boltzmann averaging of (Y’)‘:
by the
- 2aU(p + a/y)-’
e-fiv/7 (If’)” dr/ r 0
4nr2 e-flv12 dr
= 6akT, where /3 = (kT)-l.
The factor off
is included
(8) in the
01)
Thus we get U= (yp + a)/[yp2 + (y2 +a)p + 3ay]. This may be written in the form of partial fractions_ For y2 s a, we have U = (p i- y)-l - (2a/y2)( p + 3a/y)-I, and the inverse transform is G(t) =
e-Yr
_
(2a/-$)e-jadY_
(!2)
The second moment [9] is [U(t) = 2kTJk X $(T)dT. Fort a y-l, we get Ii’(t) = (2kTr/y)
- (4kTf9a)
E(3ar/y)
(t - T)
_
(13
where the function Eis defined by E(x)=x - 1 i-e-r_ The lu(t)versus r curve is shown as the solid line in the top diagram of fig. 1. The slope is 2kT/y (same as free particles) for small t and 2kT/3y for large t. In a harmonic force field, there is also 2 general tendency for the particles to move toward the potential minimum. The function J(t) is the reduction in second moment due to these motions and has been given previously [S] as J(t) = (kT/a) [I - exp (-2
Jo(r) 0
dr/y)]
.
(14)
The NMF R(t) in (7) may .be used as an approximate description of the decay of the force constant, o(t) = aR(t) = a eAar/Y_ Substitution into (12) gives: J(r) = (kT/a) [ 1 - exp(-2
Q(O) = 7 4x? 0
.
•f 2 e-“‘/Y)] _
(1%
For the J(t) versus r curve, the slope is 2kTjy for small t and is 0 for large t. We wiIl now compare our theoretical results with experimental data [7] for a system of polystyrene par221
Volume
1 12, number
3
CHEMICAL
PHYSICS
LETTERS
7 December
1984
6-
M
0
KxxxJ
20030 t
Fig. 3. Effective mean square displacemenrM(K,f) versus t. SoIid curve, theoretical for S = 0.18; daslted curve, theoretical for free particle; circles, experimental for case C, K = 0.89.
t Fig. 1. Effective mean square displacement fif(K.t) versus t. Top diagram: solid curve, theoreticat for S = 1; dashed curve, tlieoreticai for ircc particle; crosses, esperiment3l for case -4. K = 3.22; circles, experimental for case A’, K = 2.42. Bottom diagram: solid curve, theoretical for S = 1.8; dashed curve, theoretical for free particle; triangles, experimental for caseB, K = 2.12.
titles of 250 A mean radius in aqueous environment at the concentration of 1.25 X I Om3 g/cm3. As before [S], the particle mass 5.9 X 1O-l7 g is used as the unit of mass, 1O-5 cm = 1000 A is arbitrarily chosen as the unit of length. The unit of time is chosen to be 3.78 X 10m7 s such that X-T= I at the experimental temperature (293 K) and D = I/y in these units. The static structure factor S(K) and its Fourier transform y(r) have been reported previously [7,8]. Fromg(r), we have previously obtained II = 2.8. The free-particle diffusion constant D = 3.23 X 1O-* and friction constant y = l/D = 3096 are determined by PCS from the same environment in the presence of electrolytes; these values are also in very good agreement with the Stokes-Einstein values. From these values of 0 and y, we have calculated W(r) and J(t) from (13) and (15). M(K,r) is then calculated from (2) for: (A) 1”;= 3.22, S = I, M(r) = IV(r), (A’) K = 2.42, S = 1, M = IV, (B) K=2_12,S= 1.8,M=W-0.44J,(C)R=0.89,S= 0.18.M = IV + 4_55f_ These results are shown as the solid curves in figs. 1 and 2. By using (l), the PCS experimental/(K,r) data of ref. [7] are converted into
M(K,t). These data are shown as: (A) crosses in the upper diagram of fig. 1, (A’) circles in upper diagram of fig. 1, (B) triangles in lower diagram of fig. 1, (C) circles in fig. 2. There is general agreement between the experimental and theoretical results. The freeparticle (simple Brownian motion) result M = 2Dt is shown as the dashed line for comparison. They ieviate considerably from the PCS data. Although the initial slopes at t = 0 are dependent on the wave vector A’, the slopes of all the theoretical M(K,r) versus t curves at “long times” (t > T/U) have the common value of 2013, indicating a factor of 3 reduction from the free-particle behavior due to the interactions between the particles. These curves are similar to the phenomenotogical two-exponential model [ 121 proposed for these systems. For more dilute polymer dispersions or for systems where electrolytes may be present, the value of a would be small, thus the “transition” to the long-time behavior would occur at larger values oft. Sometimes, r would be so iong that the resulting PCS signal would be very weak and it may be difficult to observe this “transition”. In conclusion, the dynamics of systems of interacting Brownian particles may be understood by using the memory function formalism and by introducing approximations at the level of force correlation functions_ Starting from the experimental static structure factor S(K) and the Stokes-Einstein diffusion constant D, we are able to calculate explicitly the scattering function F(K,t) without introducing any arbi-
Volume 112, number 3
CHEMICAL PHYSICS LET-PERS
7 December 1984
trary or adjustable parameters. The general agreement With the PCS experimental data appears to be satisfactory_
[3] W. Hessand R. Klein, Advan. Phys. 32 (1983) 173. [4] S.H. Chen, in: Physical chemistry, an advanced treatise,
The partial financial support by National Aeronautical and Space Administration grant NAG-S-1 56 is gratefully acknowledged.
[6] H. Mori, Pro8r. Theor. Phys. (Kyoto) 34 (1965) 399. [7] J-C. Brown,P.N. Pusey, J.W. Goodwin and R.H. Ottewili, J. Phys. A8 (1975) 664. [S] T. Tsang and H-T_ Tazg. J. Chem. Phys. 76 (1982) 3873. 191 D.A. McQuarrie, Statistical mechanics (Harper and Row, New York, 1976). [lo] S. Chandrasckhar, Rev. Mod. Phys. 15 (1943) 1. [ 111 XI. F&man_ J. Chem. Phys. 51 (1969) 3270.
References [ 1 J H.Z. Cummins and E.R. Pike, Photon correlation spectroscopy and velocity (Plenum Press, New York, 1977). [2] V. Degiorgio, hf. Corti and hi. Giglio, Light scattering in liquids and macromolecular New York, 1980).
Vol. 8A, eds. H. Eyring, D_ Henderson and W. Jost (Academic Press, New York, 1971). [5] H. hlori, Progr. Theor. Phys. (Kyoto) 33 (1965) 423.
[ 131 P.N. Pusey, Phil. Trans. Roy. Sot. A293 (1979)
429.
solutions (Plenum Press,
223