Langmuir 2009, 25, 1175-1180
1175
Strongly Damped Dynamics of Nematically Ordered Colloidal Clay Platelets in a Magnetic Field Eduardo N. de Azevedo† and M. Engelsberg*,‡ Programa de Po´s-Graduac¸a˜o em Cieˆncia de Materiais, UniVersidade Federal de Pernambuco, Cidade UniVersita´ria, 50670-901, Recife, Pernambuco, Brazil, and Departamento de Fı´sica, UniVersidade Federal de Pernambuco, Cidade UniVersita´ria, 506701-901, Recife, Pernambuco, Brazil ReceiVed September 22, 2008. ReVised Manuscript ReceiVed October 29, 2008 The anisotropy of the diffusivity of water molecules, probed via 1H nuclear magnetic resonance imaging techniques, is used to study the extremely slow dynamics in the nematic phase of synthetic Na-fluorhectorite platelets in aqueous suspension. The anisotropy of the diamagnetic susceptibility of the platelets ∆χ, and the torque experienced in an applied magnetic field, permit one to monitor the time evolution starting from two different initial conditions. The dynamics of the ordered platelets can be modeled by a one-dimensional Fokker-Planck equation, which permits a satisfactory description of the experimental results. From the torque-free evolution, one concludes that the process is diffusive with an extremely slow rotational diffusivity Dφ ) 9.9 × 10- 3 rad2/h. The forced evolution requires a numerical solution of the full Fokker-Planck equation and yields an effective, per platelet, diamagnetic susceptibility anisotropy ∆χ ) - 1.63 × 10- 20 J/T2.
I. Introduction Although the transition from an isotropic to a nematic phase in aqueous clay suspensions can be hindered, in some cases, by the formation of gels, the possibility of nematic order in the gel state, contrary to a previously accepted “house of cards” model,1,2 has been clearly demonstrated in recent years.3 The distinction between a “weak gel”, with some “liquid-like” properties, and a “solid-like” gel, as revealed, for example, by yield stress or frequency-storage modulus measurements,4 requires some comments. The existence of a finite yield stress, below which there is no flow, a characteristic of a “solid-like” gel, depends upon the time scale adopted (typically less than 103 s) in a shear flow curve. A frequency-independent storage modulus is considered to be a more reliable signature of “solid-like” behavior,4 but also, in this case, a lower frequency of ω ) 0.001 Hz is conventionally adopted. In this paper we will consider a nematic clay, which in zero magnetic field displays homeotropic alignment caused by faceto-face anchoring to the walls of a cylindrical container. Under the action of a magnetic field-induced torque, the system slowly acquires nematic order but does not display any appreciable motion of clay platelets in a time scale of 103 s. However, in the much longer time scale of 105 s, it exhibits strongly damped dynamics. Furthermore, even without any external torque, the system displays even slower dynamics, in a time scale of 5 × 105 s, following the removal of the magnetic field. It should be pointed out that the strength of the gel, and the platelet dynamics at a fixed concentration, are controlled by the salt concentration of the medium. In the gel considered here, containing 3 × 10-4 M of NaCl, an increase of the salt concentration to 10-3 M was found * To whom correspondence should be addressed. E-mail: mario@ df.ufpe.br. Phone: 55-81-2126-7626. Fax: 55-81-3271-0359. † Programa de Po´s-Graduac¸a˜o em Cieˆncia de Materiais. ‡ Departamento de Fı´sica. (1) Van Olphen H. An Introduction to Clay Colloid Chemistry; Wiley: New York, 1977. (2) Lockhart, N. C. J. Colloid Interface Sci. 1980, 74, 509–519. (3) Michot, L. J.; Bihannic, I.; Maddi, S.; Funari, S. S.; Baravian, C.; Levitz, P.; Davidson, P. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 16101–16104. (4) Wierenga, A.; Philipse, A. P.; Lekkerkerker, H. N. W. Langmuir 1998, 14, 55–65.
to slow down the dynamics to a point where it is no longer detectable in a time scale of 106 s. An increase in gel strength with increasing salt concentration may be a signature of a regime where, for a considerable fraction of platelets, the double layer repulsion is largely shielded, and van der Waals attractive forces play a dominant role.4 The torque acting on each platelet in a magnetic field is caused by the anisotropy of the diamagnetic susceptibility, which in our clay is sufficiently large to permit a detailed study of the strongly damped dynamics of the clay platelets in a conveniently accessible magnetic field strength. The system under study, displaying the properties described above, consists of a 3% w/w synthetic Na-fluorhectorite (NaFht) aqueous suspension at an ionic strength of ∼3.0 × 10- 4 M, which, under the action of gravity, separates into different strata. The existence of a nematic phase in Na-Fht suspensions has been demonstrated by various techniques.5-7 Diffusion-weighted magnetic resonance imaging (MRI)5 has been particularly useful, permitting in situ elucidation of the roles of anchoring by the walls of the container and of applied magnetic fields in determining the type of ordering present in each stratum. Unlike synthetic laponite, which is frequently employed as model hectorite clay, Na-Fht is markedly polydisperse with a much larger average platelet size. Although gravity-induced separation partially reduces this polydispersity in the nematic phase, a relatively wide range of sizes is still present, somewhat complicating quantitative comparisons between theory and experiment. The presence of a diffuse ionic layer, arising from the relatively large negative surface charge on the Na-Fht platelets and the counterions in the ionic aqueous solution, may further cause departures8 from theoretical predictions based upon hardcore repulsive forces alone. In spite of these nonidealities, the large size of the diamagnetic anisotropy effect and the interest (5) de Azevedo, E. N.; Engelsberg, M.; Fossum, J. O.; de Souza, R. E. Langmuir 2007, 23, 5100–5105. (6) Di Massi, E.; Fossum, J. O.; Gog, T.; Venkataraman, C. Phys. ReV. E 2001, 64, 061704. (7) Fossum, J. O.; Gudding, D.; Fonseca, D. d. M,; Meheust, Y.; Di Masi, E.; Gog, T.; Venkataraman, C Energy 2005, 30, 873–883. (8) Porion, P.; Al-Mukhtar, M.; Fauge´re, A. M.; Delville, A. J. Phys. Chem. B 2004, 108, 10825–10831.
10.1021/la803110f CCC: $40.75 2009 American Chemical Society Published on Web 12/15/2008
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in the strongly damped dynamics in the nematic phase make even a semiquantitative description of Na-Fht quite attractive. We use diffusion-weighted MRI of water protons to study the dynamics of a colloidal suspension of Na-Fht platelets under magnetic-field induced torques as well as in free evolution. Although attractive forces may play an important role in arresting the platelets in a quasi-rigid gel skeleton, we employ an overdamped Langevin torque equation, with a large friction term and a periodic angular potential to describe the dynamics of a collection of effectively Brownian platelets. The data can be interpreted if one assumes a heavily damped rotational motion affected by an external torque. After solving the associated Fokker-Planck (FP) equation, time-dependent order parameters are calculated from the FP probability densities and correlated with the time evolution of the water diffusivity observed experimentally. The results are quite satisfactory and give a very consistent description of the dynamics.
II. Experimental Details Na ion-exchanged7 synthetic fluorhectorite (Na-Fht) (from Corning, Inc., New York) in powder form was employed. An aqueous suspension containing 3% (w/w) clay and a NaCl concentration of 3 × 10- 4 M was prepared and sealed in 10 mm diameter common glass tubes with 0.2-0.3 mm walls. After vigorous shaking and subsequent settling for periods of at least two weeks, the samples were ready for MRI measurements. The gravity-separated suspension exhibits three main strata along the vertical direction y: an isotropic region on top, a nematic phase, and a region containing sediment at the bottom. This last region consists of flocculated material, and its relative size becomes larger if the amount of added salt is increased.9 The volume fraction of platelets in the nematic gel phase of a 3% w/w suspension of Na Fht has been measured9 and found to be Vf ∼ 0.025 for the salt concentration used here. An MRI system (Varian Inova), including a 2 T magnet with 30 cm bore, was employed along with magnetic field gradients b , with amplitude of up to 0.2 T/m, which could be applied in G any desired spatial direction. The pulse sequence for the diffusionweighted images was a conventional spin-echo imaging sequence,10 which included two additional identical magnetic field gradient pulses,11 applied before and after the 180° pulse of the spin-echo sequence. The gradient pulses were applied either in the phase encoding channel (x axis of Figure 1B), or in the slice selection gradient channel (z axis of Figure 1B). The width of the gradient pulses was δ ) 7 ms and the separation between the gradient pulses was ∆ ) 40 ms. Moreover, the fixed amplitude of the pulses was G ) 0.16 T/m for all measurements. Five millimeter sagittal slices perpendicular to the z axis (Figure 1B) were selected and imaged. The parameters used in a typical image, such as shown in Figure 1A, were as follows: field of view of 50 mm × 50 mm, matrix size of 64 × 64, and voxel size 0.8 mm(x) × 0.8 mm(z) × 5 mm(y). From these images, the measured signal intensity was obtained by averaging over all voxels in a rectangular region of interest in the (x, z) plane around the center of the nematic phase (Figure 1A). Although diffusion constants can also be determined from the images,5 the quantity of interest here is measured at a fixed value of the gradient amplitude. For the present purpose we are only interested in the ratio between the signal intensity for a magnetic (9) de Miranda Fonseca, D. Doctoral Thesis. Norwegian University of Science and Tecnology, Trondheim, Norway, 2008. (10) Callaghan P. T. Principles of Nuclear Magnetic Resonance Microscopy; Clarendon Press: Oxford, England, 1991. (11) Stejskal, E. O; Tanner, J. E. J. Chem. Phys. 1965, 42, 288–292.
de AzeVedo and Engelsberg
Figure 1. (A) Sagittal diffusion-weighted MRI of the sample tube, with a magnetic field gradient applied in the (x, z) plane. The image was obtained after the sample had been brought to equilibrium in zero magnetic field. The arrow shows the gravity-separated, homeotropically aligned region in the sample tube schematically depicted in Figure 1B. Gravity is along the y axis. (B) Homeotropic alignment in zero field in Na-Fht caused by strong face-to-face anchoring to the glass walls of a cylindrical container. The application of a magnetic field along z causes a very slow drift of the orientation of the platelets, which adopt a nematic order with the normals b n aligning along the x direction.
field gradient applied along z and the signal intensity for the same gradient applied along x (Figure 1). In the absence of a magnetic field, a homeotropic alignment, where the clay platelets are arranged as shown schematically in Figure 1, prevails. Face-to-wall anchoring of platelets less than a few diameters away from the walls of the glass tube favors a face-to-face nematic order with b n perpendicular to the walls.12 This locally nematic order still persists at a radial distance of approximately 1 mm from the wall. However, for longer distances, the frustration caused by face-to-wall and face-to-face ordering in the cylindrical symmetry leads to randomness in the orientation of vectors b n, which, nevertheless, still remain, on average, perpendicular to the y axis. Images such as Figure 1A with the magnetic field gradient along the y axis confirm this hypothesis.5 When a 2 T magnetic field is applied along the z axis, the platelet normals tend to align with the x axis, which becomes the director of nematic order, and the layer of face-to-face radial order becomes much thinner. Sagittal diffusion-weighted images of the cylindrical tube, in the region of nematic order, were obtained with magnetic field gradients applied along the z axis and along the x axis.
III. Water Diffusivity and Overdamped Dynamics of Clay Platelets Consider an arrangement of clay platelets, such that the directors are perpendicular to the axis of a cylindrical tube, parallel to the y axis, as shown in Figure 1. Such an arrangement is caused by a strong face-to-wall anchoring of the clay platelets. To keep the problem one-dimensional, we further assume the order to be strong, with an order parameter S2y ) (1/2)〈3(cos ϑy)2 - 1〉 ≈ -0.5. In practice, the actual value of S2y is close to -0.4. The normals are therefore considered to be approximately parallel to the (z , x) plane, as shown in Figure 2. The platelets are further assumed to be square with side a and thickness b with a very large aspect ratio a/b . 1 and to possess a negative anisotropy ∆χ ) χ|| - χ⊥ of the diamagnetic susceptibility. The magnetic orientation energy of each platelet is then given by (12) Engelsberg, M.; de Azevedo, E. N. J. Phys. Chem. B 2008, 112, 7045– 7050.
Dynamics of Nematic Clays in a Magnetic Field
Um )
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|∆χ|B2(cos φ)2 2
(1)
This favors an equilibrium alignment where b n tends to be perpendicular to the magnetic field b B, applied along the z axis. Starting from a uniform distribution of angles φ and following the application of a magnetic field b B, there will be a very slow drift of vectors b n toward angles φ ) π/2 and 3π/2 where the magnetic orientation energy is minimum. As the platelet normals slowly drift toward the minimum energy directions, the diffusivity Dz of water molecules along the z direction, as well as Dx along x, will evolve as a consequence of the changing tortuosity of the diffusive paths. The ratio Dx/Dz, which is initially unity, will decrease until it reaches a limiting value. Conversely, if the magnetic field is removed after the limiting value has been attained, the ratio Dx/Dz will increase, at a much slower rate, toward unity in a torque-free regime. Using diffusion-weighted MRI, the ratio of the signal amplitudes ∆zx ) Sz/Sx, for magnetic field gradients applied along x and z, can be measured and directly correlated with the diffusivity ratio Dx/Dz. Although the anisotropy of the diffusivity of water is a very sensitive probe of the type of order present in a colloidal suspension of platelets with a large aspect ratio a/b . 1, a quantitative relationship between diffusivity and order parameter appears to be lacking. For noninteracting platelets, we have recently proposed12 an extension of Nielsen’s formula, which, taking into account the Nagy-Duxbury second-order term,13 becomes
DR )
D0 2 1⁄2
1 + 0.44F〈(cos ϑR) 〉 + 0.05F2〈(cos ϑR)2 〉 R ) x, y, z
,
predictions of eq 2 and experimental NMR results8 in the nematic phase of approximately monodisperse laponite. We found that, in spite of non-negligible electrostatic interactions among laponite platelets, eq 2 reasonably predicts the measured diffusivity anisotropy as a function of volume fraction and order parameter, not too close to the onset of nematic order. Here the order parameter is determined from Na23 residual quadrupolar splitting measurements.8 We will employ eq 2 to predict the variation of the anisotropy in the diffusivity as a function of order parameter at a fixed volume fraction. However, given the polydispersity of our NaFht suspension, the value of F, which for n square platelets per unit volume coincides with the reduced density F ) na3, will be considered as an adjustable parameter. However, its value must ultimately be consistent with other measurements in the same sample. The ratio ∆zx between signal amplitude with magnetic field gradients along z relative to signal amplitude with magnetic field gradient along x is given by11
[
(
∆zx ) exp -γ2δ2G2 ∆ -
δ (D - Dx) 3 z
)
]
(3)
Here Dx and Dz are given by eq 2, with ϑz ) φ, ϑx ) π/2 - φ, and ϑy ≈ π/2, as shown in Figure 1. Furthermore, γ is the gyromagnetic ratio of protons, G denotes the strength of the two magnetic field gradient pulses, δ denotes their duration, and ∆ is the time interval between the gradient pulses. Since we are interested in the time evolution of the signal amplitude ratio ∆zx(t) we need to determine a probability distribution P(φ,t) and calculate a time-dependent 〈(cos(φ)2〉. To that end we write an overdamped Langevin torque equation where the inertia term has been neglected14
(2)
Here, F ) Vf(a/b), where Vf is the volume fraction of platelets. D0 denotes the diffusivity DR of the fluid when 〈cos(ϑR)2〉 ) 0, n and the axis R ) x, y, z, and the angular ϑR is the angle between b brackets denote averages over a probability distribution of angles. Hence, 〈(cos ϑR)2〉 ) (1/3)(2S2R + 1), where S2R is the corresponding order parameter. Equation 2 is expected to be valid for a system where the only interactions come from excluded volume effects between rigid platelets and where the aspect ratio a/b . 1 is the same for all particles. It is interesting to make a comparison between the
η
dφ dUm + ) ξ(t) dt dφ
(4)
Here ξ(t) represents a random torque with a white noise spectrum such that its time correlation function is given by 〈ξ(t)ξ(t - s)〉 ) 2 ηkBTδ(s), where kB is Boltzmann’s constant and T is the absolute temperature. The friction constant is denoted by η, and the magnetic energy Um is given by eq 1. In general, one should also include in eq 4, in addition to the magnetic energy Um, a distortional Frank elasticity15 term, which, in the present case, is expected to be small, compared with the magnetic energy term and the friction. The FP equation14 for the nonequilibrium probability distribution P(φ,t) associated with eq 4 is given by
[
]
kBT ∂2P(φ, t) ∂P(φ, t) ∂ Um ) P(φ, t) + ∂t ∂φ η η ∂φ2
(5)
′ where Um ) dUm/dφ ) (-|∆χ|B2/2) sin(2φ). Equation 5 predicts that the equilibrium probability distribution, corresponding to ∂P/∂t ) 0, is a Boltzmann’s distribution Peq(φ) ) exp(-Um/kBT)/Z. This ignores any dependence upon F, which has been shown to be important in gibbsite for Γ ) |∆χ|B2/2kBT , 1 in the paranematic phase.12,16 In the present case, it should not cause a serious error.
Figure 2. Angles between a platelet normal b n and the Cartesian axes. The magnetic field is applied along the z axis, and the axis of the cylindrical container is along the y axis.
(13) Nagy, T. F.; Duxbury, P. M Phys. ReV. E 2002, 66, 020802(R). (14) Reimann, P. Phys. Rep. 2002, 361, 57–265. (15) de Gennes P. G., Prost J. The Physics of Liquid Crystals; Clarendon Press: Oxford, England, 1993. (16) van der Beek, D.; Petukhov, A. V.; Davidson, P.; Ferre´, J.; Jamet, J. P.; Wensink, H. H.; Vroege, G. J.; Bras, W.; Lekkerkerker, H. N. W. Phys ReV. E 2006, 73, 041402.
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de AzeVedo and Engelsberg
Figure 3. Probability distribution P(φ,t) for the torque-free evolution of eq 6 with η/kBT ) 101 h, for t- ) 1, t0 ) 4.2, t1 ) 10, t2 ) 40, and t3 ) 100 h.
The right-hand side of eq 5 contains a diffusive term and a drift term driven by the magnetic torque. It is possible to test separately the role of each term from diffusion-weighted MRI experiments. To test the effect of the diffusive term, we first allow the system to achieve equilibrium with an applied 2 T magnetic field. Subsequently, the sample is removed from the magnetic field for a given time interval, and the amplitude signal ratio ∆zx of eq 3 is monitored. This requires reintroducing the sample in the magnetic field for a short time, but the measurement does not last enough to appreciably rebuild the lost order. Then the sample is removed again from the magnetic field for a new time interval, and the procedure is repeated several times. Alternatively, one can monitor the evolution under both terms on the right-hand side of eq 5. To that end, the system is first brought to equilibrium in zero field and then placed inside the magnet where the ratio ∆zx is measured as a function of time. a. Torque-free evolution. The solution of eq 5, with the drift term absent, can be written as a Fourier series in the angle interval [0, 2π]. The solution can be simplified if one assumes that the magnetic field is sufficiently strong to permit one to write the initial condition as P(φ,0) ) (1/2)[δ(φ - π/2) + δ(φ - 3π/2]). In that case one obtains
Figure 4. Probability distribution P(φ,t) for forced evolution obtained from a numerical solution of the FP with η/kBT ) 101 h and |∆χ|B2/2kBT ) 8.0 for t0 ) 0, t1 ) 234.4, t2 ) 703.4, and t3 ) 1641 min.
t0 ) 4.2 h and η/kBT ) 101 h is shown in Figure 3. Moreover, from eq 7, applied to τ ) 0, the constant A can also be written in terms of the initial value as A ) 1 - 2〈cos(φ0)2〉. b. Evolution under an Applied Magnetic Torque. In this case, the full solution of the FP equation with a uniform initial distribution P(φ,0) ) 1/2π is needed. To that end, we used the Crank-Nicholson17 implicit finite difference method to represent the second derivative in eq 5. The first derivative ′ P(φ,t))]i,j at grid point (i,j) was approximated by an [∂/∂φ(Um ′ ′ ′ average of ((Um )i+1Pi+1,j - (Um )iPi,j)/δφ and ((Um )i+1Pi+1,j+1 ′ )iPi,j+1)/δφ, where i defines the angular grid, and j defines (Um the temporal grid. Figure 4 shows calculated values of P(φ,t), which were subsequently employed to compute 〈cos(φ)2〉, needed in eqs 2 and 3. The initial condition P(φ,0), corresponding to the equilibrium configuration in zero field of Figure 1b, was approximated by a uniform distribution.
IV. Results and Discussion
(7)
Figure 3 shows diffusion-weighted MRI measured values of the amplitude ratio ∆zx as a function of time in the nematic phase of a 3% w/w synthetic Na-Fht suspension at a ionic strength of ∼3 × 10- 4 M. The system was first brought to equilibrium in a 2 T magnetic field and subsequently removed from the field for various time intervals. The experimental parameters of eq 3 used in the measurements were G ) 0.16 T /m, ∆ ) 40 × 10-3 s, and δ ) 7 × 10-3 s, with γ ) 2.67 × 108 T-1 s-1. The value of D0 is expected, from diffusion measurements,5 to be somewhat smaller than the diffusivity of water DW ) 2.16 × 10- 9 m2/s at 20 °C. We estimate the difference between DW and D0 to be ∼14% of DW. The solid line represents a fit to the data obtained from eqs 2, 3, and 7. The parameter A ) 1 - 2〈cos(φ0)2〉 ) 0.84 of eq 7 was obtained from the initial condition ∆zx(t ) 0) of Figure 5, and the reduced platelet density F of eq 2 was adjusted to best fit the shape of the curve. The same value F ) 1.364 obtained from this fit to the torque-free evolution was employed to fit the forced time evolution. It should be pointed out that this value
Here τ ) (kBT/η)t and the constant A is given by A ) exp(-(4kBT/ η)t0). The probability distribution P(φ,t0) obtained from eq6 for
(17) Crank J. The Mathematics of Diffusion; Oxford University Press: London, 1975.
P(φ, t) )
[
(1 ⁄ 2π) 1 + 2
∑
∞ n)1
]
cos(nπ ⁄ 2) cos(nφ) exp(-kBTn2t ⁄ η) (6)
Figure 3 shows calculated values of P(φ,t) as a function of time obtained from eq 6. We have chosen the value η/kBT ) 101 h in accordance with the results of section IV. Since the alignment produced by the magnetic field will be shown to be substantial, the initial distribution should be relatively sharp. However, the δ- function assumption in P(φ,0) of eq 6 is not applicable. It is still possible, in this case, to use eq 6 to simulate an approximate P(φ,0) by simply shifting the time origin t f t + t0 in eq 6. This will introduce some width at the shifted t ) 0 value, which can be adjusted to accommodate the initial condition. Moreover, it is not the full distribution P(φ,t) that is 2 needed in eq 2 and eq 3 but only 〈cos (φ)2〉 ) ∫2π 0 P(φ,t) cos(φ) dφ, which, by orthogonality, reduces simply to
1 〈cos(φ)2 〉 ) (1 - A exp(-4τ)) 2
Dynamics of Nematic Clays in a Magnetic Field
Figure 5. (2) Experimental values of the signal amplitude ratio ∆zx. The system was brought to equilibrium in 2 T magnetic field, and the torquefree evolution was measured as a function of time elapsed after the removal of the magnetic field. The solid line is a fit based on the diffusion part of eq 5 with an angular diffusivity Dφ ) η/kBT ) 9.9 × 10- 3 rad2/h.
of F is considerably smaller than F ) 3.9 for the onset of a nematic order in a system of thin circular platelets with only excluded volume forces present.18,19 Polydispersity and the relatively large surface charge of Na-Fht may be partly responsible for this discrepancy.8 According to eq 6, the time evolution of ∆zx should be a function of τ ) (kBT/η)t; hence, by adequately scaling the horizontal axis of Figure 5, the friction coefficient η/kBT was obtained. From the satisfactory agreement of Figure 3, it appears that the evolution under torque-free conditions is, to a fair approximation, diffusive with an extremely slow rotational diffusivity Dφ ) kBT/η ) 9.9 × 10- 3 rad2/h. We next consider the forced time evolution in the presence of an applied magnetic field. Figure 6 shows measured values of the amplitude ratio ∆zx as a function of time after the system, initially in equilibrium in zero field, was placed in a 2 T magnetic field. The solid line represents a fit to the data obtained from a numerical solution of the complete FP equation (eq 5) and subsequent computation of 〈cos(φ)2〉. This was used in eqs 2 and 3 to calculate ∆zx. It should be pointed out that the same values of the parameter F and of the friction constant η were employed in the fits of Figure 5 and Figure 6. The fit of Figure 5 permitted a determination of the magnetic energy strength parameter of eq 1 |∆χ|B2/2kBT ) 8.0. Some consistency checks are possible. Although the value ε ) |∆χ|B2/2kBT ) 8.0 was obtained from the forced dynamics in the presence of a magnetic field, the order parameter S2z(ε) ) 〈3 cos(φ)2 - 1〉/2 calculated under equilibrium conditions using Boltzmann’s probability distribution with ε ) 8.0 should be close to the equilibrium value 〈3 cos(φ0)2 - 1〉/2 ) - 0.38 determined experimentally from the value A ) 1 - 2〈cos(φ0)2〉 ) 0.84. The equilibrium value, calculated with Boltzmann‘s probability distribution Peq(φ) in the plane x, z, is S2z(ε ) 8.0) ) - 0.398, in satisfactory agreement with the experimental value. (18) Frenkel, D.; Eppenga, R. Phys. ReV. Lett. 1982, 49, 1089–1092. (19) Eppenga, R.; Frenkel, D. Mol. Phys. 1984, 52, 1303–1334.
Langmuir, Vol. 25, No. 2, 2009 1179
Figure 6. (2) Experimental values of the signal amplitude ratio ∆zx. The system was brought to equilibrium in the absence of a magnetic field and the forced evolution was measured as a function of time elapsed after the application of a B ) 2 T magnetic field. The solid line is a fit based on a numerical solution of the complete FP equation with |∆χ|B2/ 2kBT ) 8.0 and kBT/η ) 101 h.
Another consistency test is related to the value of F ) 1.364 employed in the fits of Figures 3 and 4. In the absence of a magnetic field, a value of the ratio Dy/Dx,z ≈ 1.29 has been measured.5 For the homeotropic alignment prevailing for b B) 0, one could use eq 2, with S2y + 2S2x ) 0, to determine the ratio Dy/Dx,z as a function of F and S2y. For F ) 1.364, the value of the order parameter needed to obtain a ratio Dy/Dx,z ≈ 1.29 is S2y ≈ -0.39, which does not seem to be unreasonable.
V. Conclusions We have used a novel diffusion-weighted MRI experiment to investigate the overdamped motion of Na-Fht platelets in a nematic phase, which in zero field exhibits homeotropic arrangement. In spite of its appearance, which resembles a selfstanding gel, the system exhibits extremely slow but measurable dynamics. The system also shares another characteristic with classical gels. Although the platelets are, to a large degree, arrested by strong electrostatic forces,20 water forms a continuum that flows almost freely through the skeleton formed by the arrested platelets. This is evidenced by the very small change of the diffusivity D0 with respect to the diffusivity DW of pure water. The arrested platelets experience very slow dynamics. If they are allowed to evolve toward equilibrium in a torque-free regime, it appears to be possible to model the dynamics by a Brownian rotational diffusion with a rotational diffusivity Dφ ) kBT/η ) 9.9 × 10- 3 rad2/h. Using the friction constant η/kBT ) 101 h, determined from the torque-free evolution, the forced dynamics in an applied magnetic field was modeled by numerically solving the complete FP equation. Using the calculated time-dependent probability distribution, a value of |∆χ|B2/2kBT ) 8.0 was obtained from a fit to the experimental decay for B ) 2 T and T ) 298 K. This value, which is consistent with that obtained from thermal equilibrium measurements, leads to an effective diamagnetic susceptibility anisotropy per platelet ∆χ ) -1.63 × 10-20 J/T2 (20) Ruzicka, B.; Zulian, L.; Angelini, R.; Sztucki, M.; Moussaı¨d, A.; Ruocco, G Phys. ReV. E 2008, 77, 0204402(R).
1180 Langmuir, Vol. 25, No. 2, 2009
and to a saturation field, defined as BS ) (15kBT/|∆χ|)1/2 , of BS ) 1.94 T. This value of BS is compatible with values obtained in other layered silicates.21-23 Although the uncertainty caused by the polydispersity in NaFht and the effects of the diffuse ionic layer forced us to use an adjustable parameter to characterize an effective reduced platelet density F, the value adopted was not found to be unreasonable. (21) Takahashi, T.; Okhubo, T.; Ikeda, Y. J. Colloid Interface Sci. 2006, 299, 198–203. (22) Uyeda, C; Takeuchi, T.; Yamagishi, A.; Date, M. Physica B 1992, 177, 519–522. (23) Uyeda, C; Takeuchi, T.; Yamagishi, A.; Date, M. J. Phys. Soc. Jpn. 1991, 60, 3234–3237.
de AzeVedo and Engelsberg
Small discrepancies in Figures 5 and 6 can be attributed to the simplifying assumptions that permitted us to describe the system by a one-dimensional FP equation, but the overall agreement appears to be satisfactory. Acknowledgment. We wish to thank Prof. Jon Otto Fossum (Norwegian University of Science and Technology) for useful comments. This work has been supported by Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico CNPq (Brazilian agency). LA803110F
