Effects of Disorder in Polymer Morphology on Spin Diffusion - The

J. Phys. Chem. B 1999, 103, 9423-9431

9423

Effects of Disorder in Polymer Morphology on Spin Diffusion T. T. P. Cheung Phillips Petroleum Company, Phillips Research Center, BartlesVille, Oklahoma 74004 ReceiVed: February 24, 1999; In Final Form: June 5, 1999

A diffusion equation appropriate to the nuclear magnetic resonance spin diffusion experiments is analyzed using a periodic lattice model. In this model, the initial magnetization resides only in domains of one of the phases in a multiphase system. These domains are arranged according to a lattice with well-defined lattice spacing. Disorders are incorporated by introducing variations in the domain sizes, in orientations of the domains with respect to primitive basis vectors of the lattice, and in the occupancy of the lattice site. By means of a spatial Fourier analysis, it is shown that because of spin diffusion, the initial magnetization decays within a time scale of (B/2π)2/D when there is no disorder. B is the lattice spacing, and D is the spin diffusion coefficient. When there are disorders due to vacancies at the lattice sites or variations in the domain sizes, the magnetization decays much slower and follows a power law at long time. The slower decay is due to the introduction of the long wavelength modes by the disorders. The orientation disorder, however, does not lead to a slower decay, and the zero wavevector mode is forbidden.

Introduction A key piece of information about a polymer solid that can be inferred from proton nuclear magnetic resonance (NMR) spin diffusion experiments,1-12 is the spatial distribution of the domains of different phases.3,13 In a solid polymer, whether it is a homopolymer, block copolymer, or polymer blend, there are usually two or more phases present. The spatial extent of each phase consists of many separate domains distributed among the domains of other phases. In spin diffusion experiments, NMR pulses generate a spatial nonequilibrium in the total magnetization, in which the magnetization resides only in domains of one phase. Then spin diffusion is allowed to redistribute the magnetization in a time scale shorter than the longitudinal spin relaxation time. By measuring the evolution of the magnetization over time and matching it to the solutions of the spin diffusion equation obtained from different trial initial conditions, one can estimate the spatial distribution of the initial magnetization, which, as mentioned earlier, describes the spatial distribution of domains of the phases. The spatial information extracted includes domain sizes and distances between domains. It is generally difficult to obtain analytical solutions to the spin diffusion equation when the spin diffusion coefficient and spin density of each phase are different because of the complicated boundary conditions,14 namely, matching over time the spin flux across each interface between any two domains belonging to different phases. Although the brute-force numerical solution15 to the spin diffusion equation can be obtained because of the computing power available, it remains valuable to seek the analytical solutions. In addition to providing insights on how the solutions vary with the initial conditions, the analytical approach is most useful when the system is inherently disordered in nature, which is the case for most polymer systems. In this paper, we shall show how the randomness in the distribution of the domains affects the analytical solutions of the spin diffusion equation. We start with an idealistic uniform distribution such as that shown in Figure 1a and introduce disorders in terms of vacancies in one phase (Figure 1b), variations in the domain sizes (Figure 1c), and variations of

Figure 1. (a) The initial condition in a two-dimensional PPL model. Shaded areas represent domains of phase I, in which the initial magnetization resides. The unshaded area belongs to phase II and does not contain any magnetization initially. The lattice sites are represented by the circles at the centers of the domains of phase I. The lattice extends to infinity in both x- and y-directions with lattice constants X and Y, respectively. bx and by are, respectively, the domain sizes along these directions, and they are the same for all the domains of phase I. (b) A two-dimensional DPL model in which vacancies created by the absence of the domains of phase I at the lattice sites are introduced to an otherwise perfect periodic lattice. The probability of a vacancy at a lattice site is completely independent of the probabilities on other sites. (c) A two-dimensional DPL model in which the domain width bx along the x-direction can take on the values of b - , b, and b + with the probabilities of (1 - f )/2, f, and (1 - f )/2 respectively. The domain width along the y-direction remains constant. The variation of bx in a lattice site is completely independent of those in other sites. (d) A twodimensional disordered periodic lattice model in which the orientation of the domain of phase I can be either parallel or perpendicular to the x-direction with equal probability. The orientation at each site is independent of those at other sites. The domain length a and width b remain the same for all lattice sites.

the domain orientations (Figure 1d). To focus our attention on the spatial distribution of the domains, namely, the initial

10.1021/jp9906684 CCC: $18.00 © 1999 American Chemical Society Published on Web 10/15/1999

9424 J. Phys. Chem. B, Vol. 103, No. 44, 1999

Cheung

conditions for solving the spin diffusion equation, we shall circumvent the boundary condition problems by assuming that the spin diffusion coefficient of each phase is the same. Because the spin density and spin diffusion coefficient are dictated by the proton-proton separations in the solid, the assumption of a single diffusion coefficient also implies a single spin density for all of the phases. With a single spin diffusion coefficient, we can apply Fourier analysis to solve the diffusion equation. We shall show that except for the fluctuation in the domain orientation, other disorders lead to a slower decay of the total magnetization. The slower decay is due to the introduction of long wavelength modes by the disorders. These modes follow a power-law decay. Solution for Single Spin Diffusion Coefficient For simplicity, we consider the solid consisting of just two phases, which we label as phase I and II. The inclusion of a third phase will be discussed later. Let V be the volume of the polymer solid, and N be the number of domains in phase I. The volume V and the number N are infinitely large but N/V remains finite. We introduce the spatial Fourier transform,

mq ) V-1/2

∫Vm(r, t ) 0) exp(-iq‚r)dr

Substituting eqs 3 and 5 into eq 6, one obtains

M(t))

∑q mqm-q exp(-Dq2t)

) V-1

exp[iq‚(Ri - Rj)] × ∑q ∑ i,j C(Ωi,q)C(Ωj,-q)exp(-Dq2t) (7)

where C(Ωi,q) is defined as

C(Ωi,q) )

(1)

(2)

M(t ) 0)) V-1 )

m(r, t) ) V

∑q mq exp(iq‚r - Dq t) 2

(3)

The initial condition is given by

m(r, t ) 0)) 1 for r ∈ phase I ) 0 for r ∈ phase II

(4)

where the magnetization resides only in one of the phases which was chosen to be phase I. This leads to

mq) V-1/2 ) V-1/2

∑i ∫

∑q ∑i C(Ωi,q)C(Ωi,-q)

∑i Ωi

(9)

and

M(t ) ∞) ) V-1(

with D being the spin diffusion coefficient, can be expressed in terms of the Fourier transform mq, -1/2

(8)

i

We shall refer C(Ωi,q) as the domain form factor in view of the close similarity in the mathematical form of C(Ωi,q) and the atomic form factor in X-ray diffraction. Equation 7 is the general solution for a two-phase system with a single diffusion coefficient. The sum in eq 7 with i ) j represents the autodiffusion, whereas the sum with i * j describes the correlated diffusion. If phase I occupies only a small fraction of the total volume V, the autodiffusion suffices to describe M(t).3 Otherwise, correlated diffusion must be included. Note that

where m(r, t) is the local magnetization density at location r and time t. The solution to the spin diffusion equation

∂ m(r, t) ) D32m(r, t) ∂t

∫Ω dF exp(iq‚F)

∑i Ωi)2

(10)

We first consider the case of a perfect periodic lattice (PPL) in which the centers of the domains of phase I form a threedimensional orthorhombic periodic lattice with primitive basis vectors X, Y, and Z along the x, y, and z directions, and in which the domain form factor C(Ωi,q) is translation invariant with respect to the lattice. The latter implies that both Ωi and the orientation of the domain are the same at each lattice site. A two-dimensional example is depicted in Figure 1a. When the properties of a periodic lattice are used, ∞

∞

∞

exp[iq‚(Ri - Rj)] ) N ∑ ∑ ∑ exp[i(lqxX + ∑ i,j l)-∞m)-∞n)-∞ mqyY + nqzZ)]

dr exp(-iq‚r) Ω i

∑i exp(-iq‚Ri)∫Ω dF exp(-iq‚F)

(2π)3

)N (5)

M(t) )

∫V m(r, t)dr

(6)

I

The integration in eq 6 is confined to the volume of phase I.

∞

∞

2πl/X)δ(qy - 2πm/Y)δ(qz - 2πn/Z) (11)

i

where Ωi denotes the volume of the ith domain of phase I. Ri is the position of the center of Ωi, and F is measured from the center of Ωi, i.e., r ) Ri + F. The integration limit in eq 5 indicates that the integration is over the volume Ωi. The observable in the spin diffusion experiment is the total magnetization M(t) associated with phase I

∞

∑ ∑ ∑ δ(qx XYZ l)-∞ m)-∞ n)-∞

and the site index i is dropped, thus M(t) is reduced to

M(t) )

N

∞

∞

∑ ∑

∑| ( ∞

XYZl)-∞ m)-∞ n)-∞

C Ω,q )

)|

2πl 2πm 2πn , , X Y Z

2

{ [( ) ( ) ( ) ] }

exp -D

2πl X

2

+

2πm Y

2

+

2πn Z

×

2

t

(12)

M(t ) ∞) is given by the l ) m ) n ) 0 term, which indeed satisfies eq 10. As in X-ray diffraction,16 the periodicity of the lattice limits the wavevector q to the reciprocal lattice vectors

Disorder in Polymer Morphology

J. Phys. Chem. B, Vol. 103, No. 44, 1999 9425

in Fourier space. The key feature of the PPL model is that the minimum decay rate of [M(t) - M(t ) ∞)] is set by the smallest nonzero reciprocal lattice vector, namely, the vector with a magnitude 2π/max{X, Y, Z}. Therefore [M(t) - M(t ) ∞)] must go to zero in time of (max{X, Y, Z}/2π)2/D. Here we denote the maximum of a set of numbers by max{...}. Next, we incorporate randomness in the PPL model by introducing disorders at each lattice site. In this disordered periodic lattice (DPL) model, the disorders are manifested as fluctuations in the domain form factor. Examples of the fluctuations include variations in the volume Ωi and in the orientations of the domain with respect to the primitive basis vectors. We assume that the fluctuations at one site are independent of the fluctuations at other sites. An average over the fluctuations is denoted by 〈...〉. Because there is no correlation in the fluctuations between sites, we can write

exp[iq‚(Ri - Rj)C(Ωi,q)C(Ωj,-q) ) ∑ i,j N〈C(Ω,q)C(Ω,-q)〉 + 〈C(Ω,q)〉〈C(Ω,-q)〉 × exp[iq‚(Ri - Rj) (13)

∑ i*j

Substitution of eqs 11 and 13 into eq 7 yields

M(t) ) Ξ1(t) + Ξ2(t) Ξ1(t) )

(14)

∫-∞dq[〈C(Ω,q)C(Ω,-q)〉 ∞

N (2π)3

gamma function. The parameter γ equals 1, 2π, and 4π, respectively, when d equals 1, 2, and 3. Thus Ξ1(t) follows a power-law decay instead of the exponential form in Ξ2(t) or eq 12. At long t, M(t) is dominated by Ξ1(t). Therefore, disorders which satisfy condition 17 will always lead to a slower decay in [M(t) - M(t ) ∞)] in comparison to that of the PPL model in eq 12. Mathematically, the disorders introduce the q ∼ 0 modes to M(t). These long wavelength modes, which are absent in the PPL model, decay much slower than the slowest modes in that model. The dimensionality of the spin diffusion should not be confused with the dimension of the lattice. The morphology and ordering of the domains of phase I dictate the dimensionality of the spin diffusion.3 Spin diffusion from domains arranged in parallel layers is essentially perpendicular to the flat surfaces of the layers and is therefore one-dimensional (d ) 1), whereas spin diffusion from parallel rodlike domains and cubelike domains is considered, respectively, as two-dimensional (d ) 2) and three-dimensional (d ) 3). Equation 18 suggests that when Ξ1(t) contributes significantly to M(t) and condition 17 is satisfied, the domain morphology can be discerned by following the long t decay of M(t). Furthermore, one notes that the integration of [M(t) - M(t ) ∞)] over time diverges for low-d diffusion. Namely,

∫0t[M(t′) - M(∞)]dt′∝ t1/2

for d ) 1

ln(t) for d ) 2

(19)

|〈C(Ω,q)〉|2]exp(-Dq2t) (15) and

Ξ2(t) )

N

〈(

)〉| { [( ) ( ) ( ) ] }

∑ ∑ ∑| ∞

∞

∞

XYZl)-∞ m)-∞ n)-∞ exp -D

C Ω,q )

2πl X

2

+


2πm Y

2

+

2πn Z

The effect of the disorders on the slowing of the decay of M(t) can be expressed in terms of the disorder index ξ defined by the ratio

2

×

ξ)

2

t

qf0

〈Ω〉 -

(17)

Ξ1(t) decays in a much slower rate than [Ξ2(t) - Ξ2(t ) ∞)]. This slow decay can be understood as follows. When condition 17 is satisfied, the long time behavior of Ξ1(t) is governed by the q ∼ 0 modes. To a first approximation at long t, we can write

∫0∞dq[〈C(Ω,q)C(Ω,-q)〉 - |〈C(Ω,q)〉|2]exp(-Dq2t) ∫0∞dq exp(-Dq2t)

≈ [〈C(Ω,q ) 0)2〉 - 〈C(Ω,q ) 0)〉2]

Γ(d/2) γ ) [〈C(Ω,q ) 0)2〉 - 〈C(Ω,q ) 0)〉2] 2(Dt)d/2

M(t ) 0)

(16)

Notes the similarity between eq 12 and Ξ2(t), and that they both decay in a rate faster than D(2π/max{X, Y, Z})2 because only q in the reciprocal lattice can contribute. Note that M (t ) ∞) is given by the l ) m ) n ) 0 term in Ξ2(t), i.e., M(t ) ∞) ) Ξ2(t ) ∞). Therefore Ξ1(t f ∞) f 0. In Ξ1(t), the wavevector can take on continuous values, including those around the origin q ) 0. When

lim〈C(Ω,q)C(Ω,-q)〉 - |〈C(Ω,q)〉|2 * 0

Ξ1(t ) 0)

(18)

where d is the dimensionality of the spin diffusion and Γ is the

)

∫-∞∞dq|〈C(Ω,q)〉|2

1 (2π)3

〈Ω〉

Effects of Disorders on Spin Diffusion Many types of disorders are possible for the domain at each lattice site. We shall focus on three general types: (1) vacancy disorder, in which a lattice site may be devoid of the domain of phase I, (2) volume disorder, in which the volume of the domain of phase I takes on a range of values, and (3) orientation disorder, in which the orientation of the domain varies with respect to the primitive basis vectors. Examples of these types of disorders are shown, respectively, in Figures 1b, c, and d for a two-dimensional lattice. To determine M(t), one needs to evaluate the averages in eqs 15 and 16. Let p be the probability that a lattice site is occupied by a domain of phase I, ϑ be the orientation of the directional vector of the domain with respect to the primitive basis vectors, and P(Ω,ϑ) be the conditional probability that if a lattice site is occupied by a domain of phase I, the domain will have a volume Ω and a direction ϑ. Then the average of a function F(Ω,ϑ) is given by

∫

〈F(Ω,ϑ)〉) p dΩdϑP(Ω,ϑ)F(Ω,ϑ) ≡ p〈〈F(Ω,ϑ)〉〉

(20)


Cheung

The last identity defines 〈〈F(Ω,ϑ)〉〉. Using eq 20, we find

Ξ1(t) )

∫-∞∞dq[p〈〈C(Ω,q)C(Ω,-q)〉〉 -

N (2π)3

p2|〈〈C(Ω,q)〉〉|2]exp(-Dq2t) (21) and Ξ2(t) )

N

∞

∞

)〉〉| { [( ) ( ) ( ) ] } ∞

∑ ∑ ∑

XYZl)-∞ m)-∞ n)-∞

p2

|〈〈 (

C Ω,q )

2πl

exp -D

2


2πm

+

X

2

+

2πn

Y

2

×

2

Z

t

(22)

In the following, we treat each type of disorder separately and show how the observable Φ(t) defined by

Φ(t) ) [M(t) - M(t ) ∞)]/[M(t ) 0) - M(t ) ∞)]

(23)

is affected by the disorders. 1. Vacancy Disorder. We consider the case that the only disorder present is the vacancy disorder. Then

〈〈C(Ω,q)C(Ω,-q)〉〉) |〈〈C(Ω,q)〉〉| )

∏

R)x,y,z

[

2

]

sin(qRbR/2)

2

qR

2

(24)

For the two-dimensional lattice in Figure 1b, we need only the x and y terms in the product in the last equation. It is straightforward to show that the disorder index ξ is given by ξ ) (1 - p). Therefore, the effect of the vacancy disorder on the decay of Φ(t) is directly proportional to the probability of a vacancy at the lattice site. To evaluate Ξ1(t), we need to calculate integrals involving products of two sine functions. Because these products can be rewritten as cosine functions, the following integral

K(t,s)≡

q2

( )

s2 π|s| erf ) - xπDt exp 4Dt 2

(x ) s2 4Dt

disorder is much slower than that depicted for the twodimensional case. When there is a vacancy between two domains, the centerto-center separation between these two domains increases by one unit of the lattice constant. The probability of this occurrence is p2(1 - p). When there are two vacancies in a row, the separation increases by two units. The probability is p2(1 p)2. Note that the vacancy disorder is equivalent to a binomial distribution in the domain separations. This is a generalization of the one-dimensional treatment given in ref 13. 2. Volume Disorder. For clarity, we consider the case depicted in Figure 1c where the domain volume fluctuates only in the x-direction with the domain width bx taking on the values of b - , b, b + with the probabilities of (1 - f)/2, f, (1 f)/2 respectively. The domain width by along the y-direction remains constant. The averages of C(Ω,q) become

[

sin2[qx(b - )/2] 〈〈C(Ω,q)C(Ω,-q)〉〉 ) 2(1 - f) + q2x

exp(-Dq2t)

∫0∞dq cos(qs)

Figure 2. A comparison of Φ(t) of a two-dimensional PPL given by eq A3 with those according to eq A4 of a two-dimensional DPL due to vacancy disorders (see Figure 1b). The values of 0.5 and 0.75 are chosen for the probability p of a lattice site being occupied. The ratios Y/X ) 5, bx/X ) 0.5, and by/X ) 4.5 are used. X and Y are the lattice constants in the x- and y-direction, respectively. bx and by are the width and the length of the domain.

sin [qx(b + )/2] 2

(25)

suffices for the complete determination of Ξ1(t). Substituting eq 24 into eqs 21 and 22 and making use of eq 25 lead to the analytical solution of Φ(t). The complete expression for Φ(t) is given in eq A4 in the Appendix. (The analytical solution for the one-dimensional case can be found in equation (A7′) in the Appendix with ) 0 and b ) b.) In Figure 2, we show Φ(t) for the two-dimensional lattice in Figure 1b when p is equal to 0.75 and 0.5, and compare them to the case without vacancy, i.e., p ) 1 (see eq A3). Because

〈C(Ω,q ) 0)C(Ω,-q ) 0)〉 - |〈C(Ω,q ) 0)〉| ) p(1 - p)Ω2 2

the vacancy disorder satisfies condition 17. It is expected that the decay of Φ(t) is slower than that of the PPL model as confirmed by Figure 2. Because of the t-1/2 dependence, the decay of the Φ(t) in the one-dimensional lattice with vacancy

2(1 - f)

q2x

[

]

sin2[qxb/2]

+ 4f

[

q2x

×

]

(26)

]

(27)

sin(qyby/2) 2 qy

sin[qx(b - )/2] + qx sin[qx(b + )/2] sin[qxb/2] 2 (1 - f) + 2f × qx qx sin(qyby/2) 2 qy

|〈〈C(Ω,q)〉〉|2 ) (1 - f)

[

]

2

2

Using the above two equations, we obtain Φ(t) from eqs 21 and 22. The analytical form of Φ(t) is given by eq A7 in the Appendix. (The analytical solution for the one-dimensional case can be found in equation (A7′) in the Appendix by setting p ) 1.) Figure 3 shows Φ(t) when equals 0.5b, and f equals 1/5, 1/3, and 1/2. As in the case of the vacancy disorder, the volume



Figure 3. A comparison of Φ(t) of a two-dimensional PPL given by eq A3 with those according to eq A7 of a two-dimensional DPL due to volume disorders (see Figure 1c). The domain width bx along the x-direction can have the values b - , b, and b + with the probabilities of (1 - f )/2, f, and (1 - f )/2, respectively. The results for f being 1/5, 1/3, and 1/2 are shown. The ratios Y/X ) 5, b/X ) 0.5, by/X ) 4.5, and /X ) 0.25 are used. X and Y are the lattice constants in the x- and y-direction, respectively. by is the domain width in the y-direction and is the same for all domains of phase I. b is the average domain width at the x-direction.

disorder satisfies condition 17 because

〈C(Ω,q ) 0)C(Ω,-q ) 0)〉 - 〈C(Ω,q ) 0)〉2 ) 〈Ω2〉 - 〈Ω〉2 This leads to a slower decay of Φ(t). Because of the power law, the solution of the one-dimensional lattice decays much more gradually than the two-dimensional case shown in Figure 3. The disorder index given by

Figure 4. A comparison of Φ(t) of a two-dimensional PPL given by eq A3 with that according to eq A8 of a two-dimensional DPL due to orientation disorders (see Figure 1d). The ratios Y/X ) 1, a/X ) 0.9, and b/X ) 0.2 are used. The length a and width b of the domain remain the same at all lattice sites.

Figure 4, we compare eq A8 with that of the PPL model. In contrast to the vacancy and volume disorders, the orientation disorder does not lead to a slower decay in Φ(t). The orientation disorder does not satisfy condition 17 and therefore Φ(t) does not contain the slow decay component given by eq 18. This conclusion is not limited to the special case in Figure 1d, but is also valid in general for any orientation disorder. This follows because the equalities

〈〈C(Ω,0)C(Ω,0)〉〉 ) 〈〈C(Ω,0)〉〉2 ) Ω2

(31)

hold for any type of orientation disorder. Generalization

ξ ) (/b)(1 - f )/2 2

(28)

indicates the linear increase of the contribution of Ξ1(t) to M(t) with . As a function of f, the contribution is largest when f ) 0. This represents a special case13 in which the domain width takes on only two values with equal probability. 3. Orientation Disorder. The calculation of M(t) when the orientation of the domain is completely random is quite complicated. However, much can be learned from the simple case depicted in Figure 1d, in which the orientation of the domain can be in only two perpendicular directions with equal probability. This simplification yields

〈〈C(Ω,q)C(Ω,-q)〉〉 ) sin2(qxa/2) sin2(qyb/2) sin2(qxb/2) sin2(qya/2) + (29) 8 q2x q2y q2x q2y

(

)

From the results in the previous section, we can ignore the orientation disorder in the general treatment in the DPL model because it has negligible effects on the time evolution of M(t). One can see from eqs A5-A7 that the solution for the volume disorder is complicated. It becomes more complex when one includes more than three possible values for the domain width fluctuations. According to eqs 15 and 16, the solutions for Φ(t) are hinged on the calculations of 〈C(Ω,q)C(Ω,-q)〉 and 〈C(Ω,q)〉. In the following, we shall show that a general compact form of Φ(t), which takes into account the effects of vacancy and volume disorder, may be possible by using a Gaussian approximation for C(Ω,q) and expressing the volume disorder in terms of the standard deviation σ defined as

σ2) 〈〈C(Ω,q ) 0)2〉〉 - 〈〈C(Ω,q ) 0)〉〉2 )

and

|〈〈C(Ω,q)〉〉|2 ) sin(qxa/2) sin(qyb/2) sin(qxb/2) sin(qya/2) + 4 qx qy qx qy

(

)

2

(30)

The parameters a and b are, respectively, the length and width of the domains of phase I. The solution for Φ(t) is calculated from eqs 21 and 22 and is shown in eq A8 in the Appendix. In

∫Ω2P(Ω)dΩ - (∫ΩP(Ω)dΩ)2

(32)

The usage of σ avoids the need to determine the distribution of the disorder a priori. Let us first consider a one-dimensional periodic lattice with domains of phase I having a width b. We then have

C(b,q) ) 2

sin(qb/2) q

(33)


Cheung

To a good approximation,

sin2(qb/2) C(b,q)C(b,-q)) 4 q2 1 = b2 exp - (b/2)2q2 π

(

)

(34)

The Gaussian approximation in eq 34 satisfies the initial condition in eq 9. Figure 5 compares the exact solution of the one-dimensional PPL model (with a lattice constant B)

( ) ( ( )) ( ) ( ) ∑ ( )

sin2

∞

∑

2πn b

B 2

2πn

n)1

exp -D

2

2πn B

2

t

B

Φ(t) )

∞

sin

2

B 2

2πn

n)1

(35)

2πn b

2

B

with that obtained with the Gaussian approximation,

( ( )( )) ( ( ) ) ∑ ( ( )( ))

∞

Φ(t) )

1 b

∑exp - π 2

n)1

2πn

2

2

B

∞

exp -

n)1

1 b

exp -D

2

2πn

π2

2πn B

Figure 5. Φ(t) calculated with a Gaussian approximation for a onedimensional PPL (eq 36) is compared with the exact solution eq 35. B and b are the lattice constant and domain width, respectively.

2

t

(36)

2

(

Ξ1(t) 1 sxs2 + 4πDt 4πDt ) × pN 2 2 2

B

Equation 36 appears to be a reasonably good approximation for eq 35 at long time t. At short time, it deviates from the exact solution. This can be understood because the Gaussian approximation in eq 34 is most appropriate when q is small but is less adequate at large q where there are oscillations in the sine function. Because at short time, all q modes, both large and small, contribute to Φ(t), the deviation in eq 36 is expected. On the other hand, because only the modes with small q can contribute to Φ(t) at long time, we expect the Gaussian approximation to be adequate. The comparison between eq 35 and 36 can be considered as the worst-case scenario because the smallest mode allowed is q ) (2π/B). In Ξ1(t) in eq 15, the contributing modes extend to q ∼ 0 when condition 17 is satisfied. Therefore, for disordered systems where Ξ1(t) is the main component in Φ(t), the Gaussian approximation is appropriate. If the domain width b has a continuous range of values from b - to b + with equal probability of being at any one of these values, then

(

|〈C(b,q)〉|2) p2b2

)(

sin(qb/2) qb/2

2

)

sin(q/2) q/2

2

) (

ln(s + xs2 + 4πDt)

)

(37)

)|

s)b+

-

s)b-

pb2 2π

x [( )

π 1 b2 2 + + Dt π 2 2 (38)

( )]

and

Ξ2(t) pN

)

pb2

∞

( ( ( ) ( ) )( ) ) 1 b

∑ exp - π 2 B n)-∞

2

+

1

π2

2

+ Dt

2πn B

2

(39)

It is interesting to compare Φ(t) calculated from eqs 38 and 39, in which the volume fluctuation can have a continuous range of values, with that in equation (A7′) in the Appendix for the discrete volume fluctuation in a one-dimensional periodic lattice. To make the comparison meaningful, one must choose a common measure for the fluctuations. The range of the fluctuation is insufficient because it does not convey any information regarding the difference in P(Ω). The simplest parameter that contains information about and P(Ω) is the standard deviation σ defined in eq 32. For a given σ, for the continuous case is determined from

σ 2)

1 1 = p2b2exp - (b/2)2q2 exp - (/2)2q2 π π

(

Substituting eqs 37 and 34 into eqs 15 and 16, one obtains

1 2

b+ db(b - b)2 ∫b-

) 2/3


J. Phys. Chem. B, Vol. 103, No. 44, 1999 9429 ) 0) when measured from the center of a domain in phase I can have a Gaussian form

[ (bx) ]

m(x,t ) 0) ) exp -π

2

(40)

where b is roughly the distance from the lattice site within which there is a nonzero magnetization density. The partition of the initial magnetization is dictated by other NMR signatures such as differences in the transverse spin relaxation times, or chemical shifts. As long as there is no overlap in m(x,t ) 0) between adjacent lattice sites, the formalism developed above remains valid provided that m(x,t ) 0) in eq 4 is replaced by the distribution like that in eq 40, and C(Ωi,q) in eq 8 is replaced by

C(Ωi,q) )

∫VdF exp(iq‚F)mi(F,t ) 0)

(41)

where mi(F,t ) 0) is the initial magnetization density measured from the center of ith lattice site. Note that substituting eq 40 into eq 41 leads to

2 C(b,q)C(b,-q) ) b2 exp - (b/2)2q2 π

(

Figure 6. Φ(t) of a one-dimensional DPL is calculated from eqs 38 and 39 where the domain width b of phase I can have a continuous range of values and the Gaussian approximation is used. It is compared with Φ(t) given by eq A7′ where the domain width b can take on three discrete values. A common standard deviation σ is used for both the continuous and discrete domain size fluctuation. The variation is given by σx3 for the former and by σx(1 - f ) for the latter. A ratio b/B ) 0.5 and a probability p of 0.9 are used. (a) σ/B ) 0.245, f ) 1/3. (b) σ/B ) 0.3, f ) 0.

Therefore, ) σx3. Similarly, for the discrete case, ) σx(1 - f). In Figure 6, we depict Φ(t) of the continuous and discrete cases for σ ) 0.245 and 0.3. The values of 0 and 1/3 are chosen for the probability f in the calculations of the discrete fluctuations. We also introduce a small vacancy disorder by setting p ) 0.9. Note that, at long t, Φ(t) with a continuous volume fluctuation is almost identical to that with a discrete fluctuation. At small t, there are differences between the two. However, the differences are sufficiently small that the continuous volume fluctuation model can be used as a phenomenological model for describing the volume fluctuations in general, as long as one uses the standard deviation σ as the measure of the fluctuation. So far we have considered systems consisting of just two phases. It is not difficult to include an interphase region between phase I and II. As long as a single spin diffusion coefficient suffices to describe the transfer of the magnetization, the only complication that arises from the interphase is the partition of the initial magnetization. If there is no magnetization in the interphase initially, we can treat this phase as if it were a part of phase II. If the initial magnetization density in the interphase is the same as that in phase I, then we treat it as if it were a part of phase I. More often, there is a gradient in the magnetization density across the interphase. One must decide where the interphase begins and where it ends. For instance in a one-dimensional lattice, the initial magnetization density m(x,t

)

which amounts to using the Gaussian approximation described earlier. Then solutions such as eqs 38 and 39 or eq 36 in Figure 5 obtained by the Gaussian approximation become the exact solutions. Note that a gradient in the initial magnetization, whether because of the presence of an interphase or as the result of experimental conditions, always leads to a smaller initial slope in the decay of Φ(t) as a function of t1/2 when compared to that obtained from eq 4 where there is a sharp boundary in the initial magnetization. This is clearly illustrated in Figure 5. The fact that a gradient in the initial magnetization leads to a smaller initial slope in Φ(t) is not limited to the Gaussian form of the gradient, as in eq 40, but is true in general. This can be seen by integrating eq 2 over the volume of phase I followed by converting the volume integral to a surface integral using the Green’s theorem. It is important to note that the condition 17 provides a sufficient condition for a slow decay in Φ(t). That is, if condition 17 is satisfied, the slow decay is guaranteed. However, condition 17 is not necessary; one can still have a slow decay even if condition 17 fails. For instance, consider the case that the standard deviation of the domain form factor [〈C(Ω,q)C(Ω,q)〉 - 〈C(Ω,q)〉〈C(Ω,-q)〉] vanishes at q ) 0 but has a maximum at qm, with qm much smaller than 2π/B. Although Φ(t) no longer follows the power-law decay given by eq 18, it still has a slow component decaying at a rate of the order of Dqm2, which is much smaller than D(2π/B)2, the rate in the absence of disorders. An alternate way of understanding the effects of disorders on the decay of the magnetization is by examining the distribution of the initial magnetization. We first imagine dividing the polymer solid into an ensemble of cubes of equal size and compare the total magnetization in each of the cubes. If they all are the same, we redivide the solid further into cubes of a smaller size until they become different. Let L be the length of the side of the cube when differences appear. Because of spin diffusion, these differences will disappear in time of the order of L2/D, and Φ(t) must also decay to zero in the same time scale. When L is much larger than the lattice constant B, the decay of Φ(t) will be slower than that of the PPL model.

9430 J. Phys. Chem. B, Vol. 103, No. 44, 1999 For the orientation disorder, it is clear that L is about the same as B. Therefore Φ(t) is expected to decay in the same rate as that of PPL model. On the other hand, because of the statistical fluctuation in the local magnetization in the vacancy and volume disorders, there is a high probability of finding L much larger than B and a slow decay of Φ(t) follows. This alternate view of the slow decay of Φ(t) also holds when the spin diffusion coefficients and the spin densities are different in different phases. The difference in the initial magnetization will disappear in time of the order of L2/Deff, where Deff is an effective diffusion coefficient. We expect that Df > Deff > Ds. Df and Ds are, respectively, the largest and smallest of the spin diffusion coefficients. As long as L is larger than the lattice constant B, the decay of Φ(t) will be slower. If Ds is much smaller than Df, for instance by an order of magnitude, the rate determining step for the transportation of the magnetization is dictated by the slow diffusion in the phase with the smaller Ds. An asymptotic power law in Φ(t) due to disorders is expected. However, in the regime between Ds ∼ Df and Ds , Df, it is unclear if the simple power-law decay remains valid. According to eqs 38 and 39, five independent parameters are needed to determine Φ(t). They are p, b, σ, D, and B. The spin diffusion coefficient D can be estimated from the second moment of the wideline NMR spectrum.3 The ratio b/B can be obtained by other measurements, such as the multiple component analysis of the NMR free induction decay or density analysis, if one knows the dimensionality of the spin diffusion. One can assume the dimensionality or estimate it using eq 18 or 19 when the long t behavior of Φ(t) is available experimentally. Therefore when one applies the DPL model to spin diffusion experiments, one obtains p which describes the heterogeneity in the distribution of phase I among phase II; σ which represents the fluctuation in the size of the domains of phase I; b which gives the average size of the domains of phase I; and, indirectly from the ratio b/B, the separation B. In block copolymers and polymer blends, microscopy reveals phase morphology ranging from lamellar to spherical.17 The existence of lamellar structures in semicrystalline homopolymers is also well documented. In principle, the phase morphology should be reflected by the dimensionality of the spin diffusion. However, the dimensionality d has never been established from spin diffusion experiments alone. The main reason is that both the domain width b and the dimensionality d are to be determined by the same experiment. If one uses the short time decay of Φ(t), only the product of b and d can be determined. If there are disorders in the polymers, the long time behavior of Φ(t) may reveal the dimensionality d independently, but the long time behavior of Φ(t) can be distorted by the longitudinal spin relaxation. This distortion might be suppressed by subjecting the sample to zero field condition during the spin diffusion stage of the experiment, or by subjecting the sample to very low temperatures to lengthen the T1 relaxation. However, no such experiments have been performed so far. The annealing of a polymer reduces the extent of disorders in the polymer. Annealed polymers can reveal whether Φ(t) indeed decays faster after the annealing process. However, the annealing process also tends to promote domain growth. This complicates the analysis because domain growth in phase II would lead to a faster decay of Φ(t), whereas the growth in phase I has the opposite effect. Few spin diffusion experiments on annealed polymers18,19 have been reported. A faster decay in Φ(t) was found in a composite latex18 after annealing when a weak dipolar filter was used. Because the faster decay can be explained in terms of the reduction of disorders or domain

Cheung growth, it will require a detailed analysis of the experimental Φ(t) to determine which mechanisms would best fit the data. In using the spatial Fourier analysis, we find a close parallelism between spin diffusion and X-ray diffraction analysis.16 There are the similarities between C(Ω,q) and the atomic form factor, between eq 11 and Laue’s condition for diffraction, and in the usage of wavevectors in the reciprocal lattice space. In fact, we can generalize our treatment of the spin diffusion using tools and methodologies established in X-ray diffraction analysis. What makes the spin diffusion analysis different from that of X-ray diffraction is that the former is essentially an integral method. This is highlighted by the spatial integration in eq 6. The spatial integration translates to the summation or integration over the reciprocal lattice in the Fourier analysis. The shortcoming of the integration is that only spatially averaged information can be obtained. Conclusion In spin diffusion experiments, the spatial transfer of magnetization leads to the decay of the magnetization initially associated with domains of one of the phases. We showed that the existence of a distribution in the domain size and the irregular placing of domains of this phase within the other phase will lead to a slower decay of the magnetization. In terms of a spatial Fourier transform, we find that when there are no fluctuations in the domain size and in the placing of the domains, the magnetization decays at least in a time of (B/2π)2/D, where B is the center-to-center separation between adjacent domains of one phase, because the wavevector of the slowest mode allowed is (2π/B). However, the presence of the fluctuations allows the long wavelength modes to contribute to the decay. Because these modes follow a power-law decay, they decrease much slower in the long time regime. On the other hand, we show that the variation in the orientation of the domain does not lead to a slower decay in the magnetization and the q ∼ 0 modes are forbidden. Appendix The two-dimensional solutions of the spin diffusion equation are summarized here according to the type of disorders. Let us define the following functions

2 G1(t,s) ≡ [K(t,s ) 0) - K(t,s)] π

(A1)

and

W1(t,s,S) ≡

1

[ ( )] ∑ [ ( )] ( ) ∞

Sn)-∞

2 sin

2πn s

S 2

2πn

2

2

exp -D

2πn S

2

t

(A2)

S

K(t, s) is defined in eq 25. Recall that in the periodic lattice, the lattice constants (i.e., the center-to-center separations of the domains) in the X, Y, and Z directions are, respectively, X, Y, and Z, and a domain of phase I has a width bx in the x-direction, by in the y-direction, and bz in the z-direction. For an orthorhombic lattice, the x-, y- and z-directions coincide with X, Y, and Z. We first give the result of the two-dimensional PPL model in which all of the domains of phase I have the same bx and by



bx2 by2 X Y Φ(t) ) (A3) bx2 by2 W1(t ) 0,bx,X)W1(t ) 0,by,Y) X Y W1(t,bx,X)W1(t,by,Y) -

1. Vacancy Disorder. The probability of a vacancy at a lattice site is (1 - p), and all of the domains of phase I have the same bx and by

{

[

Φ(t) ) (1 - p)G1(t,bx)G1(t,by) + p W1(t,bx,X)W1(t,by,Y) bx2 by2 X Y

[

]}/{

[

G2(t,b,) - pG ˜ 2(t,b,) + p W2(t,b,X,) -

bx2 by2 X Y

]}

{

{

}

X (A7′)

G3(t, a, b) + W3(t, a, b, X, Y) - (ab)2/XY G3(t ) 0, a, b) + W3(t ) 0, a, b, X, Y) - (ab)2/XY

where G3(t, a, b) and W3(t, a, b, X, Y) are defined as

G3(t,a,b) ≡

2 {[K(t,0) - K(t,a)][K(t,0) - K(t,b)] π2 [K(t,b/2 - a/2) - K(t,b/2 + a/2)]2} (A9)

and

(A5)

( )

{

]

b2

(A8)

1-f2 2 2 f [K(t,0) - K(t,b)] + [2K(t,0) π 2 2 K(t,b - ) - K(t,b + )] + {f(1 - f)[K(t,/2) + π K(t,-/2) - K(t,b - /2) - K(t,b + /2)]} + 2 2 (1 - f) [K(t,) - K(t,b)] (A5′) π 2

}

X

3. Orientation Disorder. For domains with a length a and a width b, we have

(A4)

(1 - f) 2 f[K(t,0) - K(t,b)] + [2K(t,0) π 2 K(t,b - ) - K(t,b + )]

[

]

b2

G2(t ) 0,b,) - pG ˜ 2(t ) 0,b,) + p W2(t ) 0,b,X,) -

Φ(t) )

2. Volume Disorder. p equals 1 and bx can be b - , b, and b + , with the probabilities (1 - f)/2, f, and (1 - f)/2, respectively. by is the same for all domains. We first define

G ˜ 2(t,b,) ≡

Φ(t) )

(1 - p)G1(t ) 0,bx)G1(t ) 0,by) +

p W1(t ) 0,bx,X)W1(t ) 0,by,Y) -

G2(t,b,) ≡

vacancy and volume disorders, the solution is

}

[

W3(t,a,b,X,Y) ≡

4

∞

∞

∑ ∑

XYn)-∞m)-∞ 2πn a 2πm b 2πn b 2πm a sin sin + sin sin X 2 Y 2 X 2 Y 2

]

( ) ( ) ( ) ( ) ( )( ) { [( ) ( ) ] } 2πn 2πm X

Y

exp -D

2πn X

2

+

2πm Y

2

t

2

×

(A10)

and References and Notes

W2(t,b,X,) ≡ 4

∞

∑ Xn)-∞

{[ ( ) f sin

sin

2πn b

+

X 2 2πn (b + )

(

X

2

(

)

2πn (b - ) 1-f 1-f sin × + 2 X 2 2 2πn 2 2πn 2 × exp -D t (A6) X X

)]/[ ]} [ ( ) ]

Then

{

˜ 2(t,b,)]G1(t,by) + Φ(t) ) [G2(t,b,) - G W2(t,b,X,)W1(t,by,Y) -

b2 by2 X Y

}{

/ [G2(t ) 0,b,) -

G ˜ 2(t ) 0,b,)]G1(t ) 0,by) + W2(t ) 0,b,X,) × W1(t ) 0,by,Y) -

b2 by2 X Y

}

(A7)

For a one-dimensional periodic lattice which includes both

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Effects of Disorder in Polymer Morphology on Spin Diffusion - The

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