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Nuclear relaxation of liquids in confinements
Magnetic Resonance Imaging, Vol. 12, No. 2, pp. 179-181, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in the USA. All rights reserved 0730-725X/94 $6.00 + .OO
Pergamon
0730-725X(93)EOOOS-9
0 Contributed Paper NUCLEAR RELAXATION OF LIQUIDS IN CONFINEMENTS J.-P. KORB,* A. DELVILLE,? SW Xv, AND J. JONAS~ *Laboratoire de Physique de la Mat&e Conde&e, Ecole Polytechnique, CNRS, 91128 Palaiseau, France, tCentre de Recherche sur la Mat&e Diviske, CNRS, lb rue de la FCrollerie, 45071 OrlCans, France, and SMaterial and Research Laboratory and Department of Chemistry, University of Illinois, Urbana, IL 61801, USA We study theoretically the effects of geometrical confinement on the dipolar relaxation of a non-interacting liquid in porous media. Application to the ‘H relaxation of methylcyclohexane liquid in porous silica glasses is given. The case of an interacting liquid is considered by molecular dynamics simulations. Geometrical confinement and surface interaction lead to similar freauencv behaviour of relaxation rates according to the layering of local density and anisotropy of the molecular mob&. Keywords: Relaxation; Confinement; Silica Glasses; Surface interactions.
porous system has been derived.2 We consider an ensemble of spins I = l/2 belonging to the same species of uniform density diffusing within an infinite layer of finite thickness d, between two flat solid surfaces, in presence of a strong constant magnetic field B,, oriented at the angle /3 from the normal axis n. The possibility of finite size L of the layer has also been considered. Only the intermolecular dipolar relaxation process is considered. We impose a distance of minimal approach 6 <
INTRODUCTION
Nuclear relaxation of a liquid in porous media depends on geometrical confinement and surface interactions. However it is difficult to identify their respective roles, especially when the average pore size approaches molecular dimensions. Experimental attempts have been made, with porous silica glasses, by modifying the surface and varying the polarity of the guest liquids.’ But some results are paradoxical for small pores: First, the spin-lattice relaxation rate T;’ largely depends on the surface modification while the spin-spin relaxation rate TF’ and spin-lattice relaxation rate in the rotating frame Tip’ remain unchanged.’ Second, there is a logarithmic frequency behaviour of T;’ and T,’ for quadrupolar and dipolar nuclei either for interacting or non interacting liquids. Is2All these seemingly paradoxical results indicate that geometrical confinement dominates over surface modification, at low frequency and for small pores. In the absence of a theoretical treatment it is difficult however, to take both effects into account. The aim of this work is precisely to propose such a treatment by considering both the case of non-interacting and interacting liquids. We apply such a theory to the ‘H relaxation of methylcyclohexane liquid in porous silica glasses.2
It-
(JL(u)) = -
-
u
10 d2iY2 ‘l
x[3ln[l+&]+2(1+$) x
1 - i 07~ + 07~ Arctan( wT1)
II (1)
EFFECT OF GEOMETRICAL CONFINEMENT ON DIPOLAR RELAXATION
The frequency dependence of Eq. (1) is displayed in Fig. 1. For small values of d/6, one has a logarithmic dependence when WT~< 1 as observed experimentally. 1,2This logarithmic dependence decreases when
A theory of nuclear dipolar relaxation by translational diffusion of a non-interacting liquid in a model 179
Magnetic Resonance Imaging 0 Volume 12, Number 2, 1994
I. Calculated
variation
o
of ( JL (a)) vs ml for different
(rad s-1)
d/6. Fig. 3. Semilogarithmic plot of the ‘H spin-lattice tion rates of liquid methylcyclohexane as a function
relaxaof w in
porous silica glasses at 300K. (Continuous lines + theory.) d/6 increases and at the limit d/6 130, (JL) tends to a size independent value. These variations are then intermediate between the usual two-dimensional and threedimensional results. An immediate consequence of Eq. (1) is the quadratic pore size dependence (0~l/d’) for T;‘, TF1 and T1,‘,* which is at variance with the linear dependence (0: l/d) observed for T;’ for interacting liquids according to the two-fraction fastexchange model.’ APPLICATION TO WEAK INTERACTING LIQUIDS IN POROUS SILICA GLASSES We apply this theory to the ‘H relaxation of methylcyclohexane liquid in sol-gel porous silica glasses with narrow pore-size distribution.* The observed pore size dependences of the relaxation rates l/T,, l/T2 and l/T,, (Fig. 2) confirm the theoretical predictions (0: 1/R*) for very weak interacting solvents in systems of pore sizes in the range of 18.4-87.2 A and in the
bulk. At the limit of small pores, the logarithmic frequency dependences of 1/TIP and 1/T, observed over several decades of frequency are interpreted in terms of a model of unbounded two-dimensional diffusion in a layered geometry (Fig. 3). The leveling off of the 1T,, at low frequency is interpreted as a bounded twodimensional diffusion due to the finite length L of the pores.2 An estimate of a finite pore size of L = 100 A is in excellent agreement with the experimental results of the transmission electron microscopy study of platinum-carbon replicated xerogels.3 Finally the temperature dependence of T, is shown in Fig. 4, both in the bulk and in a porous silica glass of radius R = 96 A. One notes clearly the phase transition on the bulk diagram for T = 147 K. This temperature is shifted to 124 K in the porous system. Consequently the good fit obtained with our theory is characteristic of an aniso-
12 10
a 6 4 2 0 0
.0005 .OOl .0015 .002
.0025 .003
1/R2(k) Fig. 2. ‘H spin-lattice relaxation rates of liquid methylcyclohexane as a function of pore radius (R-‘) in porous sol-gel silica glasses at 300K. (Continuous lines --t theory.)
3
4
5100O?T(K)7
8
'
Fig. 4. Semilogarithmic plot of the ‘H spin-lattice relaxation rates of liquid methylcyclohexane as a function Of 1000/T(K) in the bulk and in porous silica glasses R = 96 A. (Continuous line -+ theory.)
Nuclear relaxation of liquids l J.-P.
tropic diffusion of a liquid in this confined geometry. The activation energy, Ea = 1.27 Kcal/mole, is found to be much lower than in the bulk.’ EFFECTS OF GEOMETRICAL CONFINEMENT AND SURFACE INTERACTIONS To take into account these effects, we use molecular dynamics (MD) to study the dipolar relaxation of an interacting liquid confined in the slit pore model. We consider a Lennard-Jones (LJ) liquid confined between two flat structureless LJ pore walls separated by a distance d in the z direction.4 We vary the surface density X of the LJ wall sites to cover a large range of liquid-solid interactions, from a non-interacting liquid, when h < 1, up to an interacting liquid when X I 1. Two observables are calculated over the ensemble of the molecular trajectories: the diffusion coefficients Dx,v,Z in the x, y, z directions and the reduced dipolar autocorrelation functions G(“)( 7) (and spectral densities J(m)(w)). Figure 5 shows the J’“)(o) and local relative densities p*( z*)/pg of a liquid confined in a large pore (d* = 10) when increasing h. For X = 0.01, the average molecular density pg is equivalent to p*( z*) and the absence of m-dependence is characteristic of an isotropic dipolar relaxation process. For h = 0.05, p*(z*) f pz. The separation of the different m-contributions of Jcm’( w) proves the anisotropic character of the diffusion. For h = 0.25, one notes an important layering and a logarithmic divergence of J(O)(w) at low frequency at variance with the constant value of Jt2)(w). This form of divergence’and the evidence of a strong anisotropy of the diffusion (9 = 2Q/(Q + Dy = 0.14) prove unambiguously that a two-dimensional process occurs. Finally, the pure geometrical effect (h = 0.01) for a highly confined liquid (d’ = 3) gives also a layering and a large anisotropy of diffusion (7 = 1.4 x 10e2) as well as a logarithmic dependence of J(O)(w) and a constant value for Jc2)(w). CONCLUSION A theory of nuclear relaxation of a non-interacting liquid diffusing in confinements is proposed. This explains why the values of TIP, T2, and T, are much shorter in confinements than in the bulk. At small frequency 1TIPand 1/T, vary logarithmically at the limit of small pores and tend progressively to a constant when increasing the pore radius d. One notes a quadratic pore
181
KORB ET AL.
25 20 15
: w
m=O m=2
10
5 I I
58 40
I
_
$8
30
24 ho
: m=O
20
2
. m=l
___
6
8 10 3
(b)
:
,
”
108
4 z*
\
lo im=2 n
1
I “apl2
109
1o’O
10"
1o12
03 (rad/s) Fig. 5. Semilogarithmic plots of the frequency dependences of dimensionless spectral densities Jcrn) (w)/U and relative densities (in insert), by varying the surface interaction parameter h. U = C A6 ps-*, with C = IO4(a) and lo3 (b).
size dependence of lT, , l/T,, and l/T2 0~l/R2 at variance with the linear (0~ l/R) relation coming from the biphasic fast exchange model. For an interacting liquid, surface-interactions and geometrical confinement lead to the same logarithmic frequency. dependence of spectral densities for small pores. REFERENCES Liu, G.; Li Y.; Jonas, J. Confined geometry effects on reorientational dynamics of molecular liquids in porous silica glasses. J. Chem. Phys. 95:6892-6901; 1991. Korb, J.-P.; Xu, Shu; Jonas, J. Confinement effects on dipolar relaxation by translational dynamics of liquids in porous silicaglasses. J. Chem. Phys. 98:241 l-2422; 1993. Cable, P.C.; Klemperer, W.G.; Simon, C.A. Molecular architecture and its role in silica sol-gel polymerization. In C. J. Brinker (Ed). MRS Symposium Proceedings 180, Better Ceramics Through Chemistry, IV; 1990:~~. 29-37. Korb, J.-P. Delville, A.; Xu, Shu; Demeulenaere, G.; Costa, P.; Jonas, J. Relative role of surface interactions and topological effects in NMR of confined liquids. (submitted to J. Chem. Phys.)
Pergamon
0730-725X(93)EOOOS-9
0 Contributed Paper NUCLEAR RELAXATION OF LIQUIDS IN CONFINEMENTS J.-P. KORB,* A. DELVILLE,? SW Xv, AND J. JONAS~ *Laboratoire de Physique de la Mat&e Conde&e, Ecole Polytechnique, CNRS, 91128 Palaiseau, France, tCentre de Recherche sur la Mat&e Diviske, CNRS, lb rue de la FCrollerie, 45071 OrlCans, France, and SMaterial and Research Laboratory and Department of Chemistry, University of Illinois, Urbana, IL 61801, USA We study theoretically the effects of geometrical confinement on the dipolar relaxation of a non-interacting liquid in porous media. Application to the ‘H relaxation of methylcyclohexane liquid in porous silica glasses is given. The case of an interacting liquid is considered by molecular dynamics simulations. Geometrical confinement and surface interaction lead to similar freauencv behaviour of relaxation rates according to the layering of local density and anisotropy of the molecular mob&. Keywords: Relaxation; Confinement; Silica Glasses; Surface interactions.
porous system has been derived.2 We consider an ensemble of spins I = l/2 belonging to the same species of uniform density diffusing within an infinite layer of finite thickness d, between two flat solid surfaces, in presence of a strong constant magnetic field B,, oriented at the angle /3 from the normal axis n. The possibility of finite size L of the layer has also been considered. Only the intermolecular dipolar relaxation process is considered. We impose a distance of minimal approach 6 <
INTRODUCTION
Nuclear relaxation of a liquid in porous media depends on geometrical confinement and surface interactions. However it is difficult to identify their respective roles, especially when the average pore size approaches molecular dimensions. Experimental attempts have been made, with porous silica glasses, by modifying the surface and varying the polarity of the guest liquids.’ But some results are paradoxical for small pores: First, the spin-lattice relaxation rate T;’ largely depends on the surface modification while the spin-spin relaxation rate TF’ and spin-lattice relaxation rate in the rotating frame Tip’ remain unchanged.’ Second, there is a logarithmic frequency behaviour of T;’ and T,’ for quadrupolar and dipolar nuclei either for interacting or non interacting liquids. Is2All these seemingly paradoxical results indicate that geometrical confinement dominates over surface modification, at low frequency and for small pores. In the absence of a theoretical treatment it is difficult however, to take both effects into account. The aim of this work is precisely to propose such a treatment by considering both the case of non-interacting and interacting liquids. We apply such a theory to the ‘H relaxation of methylcyclohexane liquid in porous silica glasses.2
It-
(JL(u)) = -
-
u
10 d2iY2 ‘l
x[3ln[l+&]+2(1+$) x
1 - i 07~ + 07~ Arctan( wT1)
II (1)
EFFECT OF GEOMETRICAL CONFINEMENT ON DIPOLAR RELAXATION
The frequency dependence of Eq. (1) is displayed in Fig. 1. For small values of d/6, one has a logarithmic dependence when WT~< 1 as observed experimentally. 1,2This logarithmic dependence decreases when
A theory of nuclear dipolar relaxation by translational diffusion of a non-interacting liquid in a model 179
Magnetic Resonance Imaging 0 Volume 12, Number 2, 1994
I. Calculated
variation
o
of ( JL (a)) vs ml for different
(rad s-1)
d/6. Fig. 3. Semilogarithmic plot of the ‘H spin-lattice tion rates of liquid methylcyclohexane as a function
relaxaof w in
porous silica glasses at 300K. (Continuous lines + theory.) d/6 increases and at the limit d/6 130, (JL) tends to a size independent value. These variations are then intermediate between the usual two-dimensional and threedimensional results. An immediate consequence of Eq. (1) is the quadratic pore size dependence (0~l/d’) for T;‘, TF1 and T1,‘,* which is at variance with the linear dependence (0: l/d) observed for T;’ for interacting liquids according to the two-fraction fastexchange model.’ APPLICATION TO WEAK INTERACTING LIQUIDS IN POROUS SILICA GLASSES We apply this theory to the ‘H relaxation of methylcyclohexane liquid in sol-gel porous silica glasses with narrow pore-size distribution.* The observed pore size dependences of the relaxation rates l/T,, l/T2 and l/T,, (Fig. 2) confirm the theoretical predictions (0: 1/R*) for very weak interacting solvents in systems of pore sizes in the range of 18.4-87.2 A and in the
bulk. At the limit of small pores, the logarithmic frequency dependences of 1/TIP and 1/T, observed over several decades of frequency are interpreted in terms of a model of unbounded two-dimensional diffusion in a layered geometry (Fig. 3). The leveling off of the 1T,, at low frequency is interpreted as a bounded twodimensional diffusion due to the finite length L of the pores.2 An estimate of a finite pore size of L = 100 A is in excellent agreement with the experimental results of the transmission electron microscopy study of platinum-carbon replicated xerogels.3 Finally the temperature dependence of T, is shown in Fig. 4, both in the bulk and in a porous silica glass of radius R = 96 A. One notes clearly the phase transition on the bulk diagram for T = 147 K. This temperature is shifted to 124 K in the porous system. Consequently the good fit obtained with our theory is characteristic of an aniso-
12 10
a 6 4 2 0 0
.0005 .OOl .0015 .002
.0025 .003
1/R2(k) Fig. 2. ‘H spin-lattice relaxation rates of liquid methylcyclohexane as a function of pore radius (R-‘) in porous sol-gel silica glasses at 300K. (Continuous lines --t theory.)
3
4
5100O?T(K)7
8
'
Fig. 4. Semilogarithmic plot of the ‘H spin-lattice relaxation rates of liquid methylcyclohexane as a function Of 1000/T(K) in the bulk and in porous silica glasses R = 96 A. (Continuous line -+ theory.)
Nuclear relaxation of liquids l J.-P.
tropic diffusion of a liquid in this confined geometry. The activation energy, Ea = 1.27 Kcal/mole, is found to be much lower than in the bulk.’ EFFECTS OF GEOMETRICAL CONFINEMENT AND SURFACE INTERACTIONS To take into account these effects, we use molecular dynamics (MD) to study the dipolar relaxation of an interacting liquid confined in the slit pore model. We consider a Lennard-Jones (LJ) liquid confined between two flat structureless LJ pore walls separated by a distance d in the z direction.4 We vary the surface density X of the LJ wall sites to cover a large range of liquid-solid interactions, from a non-interacting liquid, when h < 1, up to an interacting liquid when X I 1. Two observables are calculated over the ensemble of the molecular trajectories: the diffusion coefficients Dx,v,Z in the x, y, z directions and the reduced dipolar autocorrelation functions G(“)( 7) (and spectral densities J(m)(w)). Figure 5 shows the J’“)(o) and local relative densities p*( z*)/pg of a liquid confined in a large pore (d* = 10) when increasing h. For X = 0.01, the average molecular density pg is equivalent to p*( z*) and the absence of m-dependence is characteristic of an isotropic dipolar relaxation process. For h = 0.05, p*(z*) f pz. The separation of the different m-contributions of Jcm’( w) proves the anisotropic character of the diffusion. For h = 0.25, one notes an important layering and a logarithmic divergence of J(O)(w) at low frequency at variance with the constant value of Jt2)(w). This form of divergence’and the evidence of a strong anisotropy of the diffusion (9 = 2Q/(Q + Dy = 0.14) prove unambiguously that a two-dimensional process occurs. Finally, the pure geometrical effect (h = 0.01) for a highly confined liquid (d’ = 3) gives also a layering and a large anisotropy of diffusion (7 = 1.4 x 10e2) as well as a logarithmic dependence of J(O)(w) and a constant value for Jc2)(w). CONCLUSION A theory of nuclear relaxation of a non-interacting liquid diffusing in confinements is proposed. This explains why the values of TIP, T2, and T, are much shorter in confinements than in the bulk. At small frequency 1TIPand 1/T, vary logarithmically at the limit of small pores and tend progressively to a constant when increasing the pore radius d. One notes a quadratic pore
181
KORB ET AL.
25 20 15
: w
m=O m=2
10
5 I I
58 40
I
_
$8
30
24 ho
: m=O
20
2
. m=l
___
6
8 10 3
(b)
:
,
”
108
4 z*
\
lo im=2 n
1
I “apl2
109
1o’O
10"
1o12
03 (rad/s) Fig. 5. Semilogarithmic plots of the frequency dependences of dimensionless spectral densities Jcrn) (w)/U and relative densities (in insert), by varying the surface interaction parameter h. U = C A6 ps-*, with C = IO4(a) and lo3 (b).
size dependence of lT, , l/T,, and l/T2 0~l/R2 at variance with the linear (0~ l/R) relation coming from the biphasic fast exchange model. For an interacting liquid, surface-interactions and geometrical confinement lead to the same logarithmic frequency. dependence of spectral densities for small pores. REFERENCES Liu, G.; Li Y.; Jonas, J. Confined geometry effects on reorientational dynamics of molecular liquids in porous silica glasses. J. Chem. Phys. 95:6892-6901; 1991. Korb, J.-P.; Xu, Shu; Jonas, J. Confinement effects on dipolar relaxation by translational dynamics of liquids in porous silicaglasses. J. Chem. Phys. 98:241 l-2422; 1993. Cable, P.C.; Klemperer, W.G.; Simon, C.A. Molecular architecture and its role in silica sol-gel polymerization. In C. J. Brinker (Ed). MRS Symposium Proceedings 180, Better Ceramics Through Chemistry, IV; 1990:~~. 29-37. Korb, J.-P. Delville, A.; Xu, Shu; Demeulenaere, G.; Costa, P.; Jonas, J. Relative role of surface interactions and topological effects in NMR of confined liquids. (submitted to J. Chem. Phys.)