Investigation of porous materials with large surface heterogeneity using the generalized Porod's scattering law method

Wei-Shan Chiang, Jin-Hong Chen, and Yun Liu

Phys. Rev. E 99, 042801 – Published 9 April 2019

Abstract

Surface heterogeneity is ubiquitous in both natural and man-made materials, and can significantly influences material properties. However, it is very challenging to noninvasively probe the variation of surface properties in porous materials. Recently, we have proposed a method, i.e., the generalized Porod's scattering law method (GPSLM), to obtain the surface heterogeneity information in bulk porous materials by extending the classic Porod's scattering method. However, it was not clear if the GPSLM can be applied to other more complex materials, such as porous materials with dead pores, i.e., pores that guest fluid molecules cannot access or porous materials whose solid matrix can adsorb small guest molecules. In this paper, we theoretically extend the GPSLM to study those more complex situations. For all five cases with different levels of complexity discussed in this work, the scattering intensity at the Porod's law region always follows a parabolic function of scattering length density (SLD) of the guest fluid. Moreover, the minimum value of the scattering intensity is all related to the surface heterogeneity of the porous materials. The SLD of the guest fluid at which the minimum intensity is reached is always related to the surface-averaged SLD of materials. We also discuss the potential limitations and possible future applications of the GPSLM. As the GPSLM is based on the contrast variation method commonly used for a wide range of materials, such as geological materials, biomaterials, and colloidal suspensions, the theoretical development here is potentially useful for researchers who would like to apply the GPSLM to more complicated materials besides porous materials.

Received 1 May 2018

DOI:https://doi.org/10.1103/PhysRevE.99.042801

Physics Subject Headings (PhySH)

Adsorption Composition Domains Microstructure Scattering theory

Heterostructures

Neutron scattering Surface scattering

Condensed Matter, Materials & Applied PhysicsPhysics of Living SystemsGeneral PhysicsPolymers & Soft Matter

Authors & Affiliations

Wei-Shan Chiang^1,2, Jin-Hong Chen³, and Yun Liu^1,2,4,*

¹Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
²Department of Chemical and Biomolecular Engineering, University of Delaware, Newark, Delaware 19716, USA
³Aramco Services Company: Aramco Research Center-Houston, Houston, Texas 77084, USA
⁴Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA

^*Corresponding author: yunliu@nist.gov; yunliu@udel.edu

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Vol. 99, Iss. 4 — April 2019

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Images

Figure 1
(a) A homogeneous two-phase system. (b)–(f) Five different heterogeneous systems discussed in this work. The detectable structure is comparable to length scale of Porod's $Q$ range $\approx \frac{2 π}{Q}$ . (b) Case 1: solid matrix is not changed by loading fluid and all the pores with length scale $\approx L$ are accessible. (c) Case 2: solid matrix is not changed by loading fluid and there are inaccessible pores or compact solid pockets with length scale $\approx L$ . (d) Case 3: solid domains have the same dependence of average domain SLD on guest fluid SLD and all the pores with length scale $\approx L$ are accessible. (e) Case 4: solid domains have different dependence of average domain SLD on guest fluid SLD and all the pores with length scale $\approx L$ are accessible. (f) Case 5: solid domains have different dependence of average domain SLD on guest fluid SLD and there are inaccessible pores or compact solid pockets with length scale $\approx L$ . The solid lines outline the interfaces between the matrix and the pores or solid pockets inaccessible to the guest fluid; the dashed lines outline the interfaces between the matrix and the accessible open pores; the dotted lines [in (f)] outline the interfaces between inaccessible solid pocket (with nonchanging SLD) and open pores.
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Figure 2
(a) Illustration of the simulated heterogeneous model system. Different colors represent different SLDs. Solid and dashed curves outline the inaccessible and accessible interfaces, respectively, to the guest fluid (the continuous medium with green color). Particles have shells with different SLDs and hollow cores inaccessible to the fluid. The structure parameters of the four different kinds of particles are listed in Table 1. (b) The simulated scattering intensity $I (Q)$ vs $Q$ for the model system loaded with fluid of different SLDs, $ρ_{f}$ . (c) $R (ρ_{f}) = \frac{C_{P} (ρ_{f})}{C_{P} (ρ_{f} = 0)}$ vs $ρ_{f}$ for the model system with $C_{P} (ρ_{f})$ as the Porod's constant obtained at guest fluid $SLD = ρ_{f}$ using Eq. (2).
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Figure 3
(a) The simulated scattering intensity $I (Q)$ vs $Q$ for the model system (sample 1) loaded with fluid of different SLDs, $ρ_{f}$ . The structure parameters of the four different kinds of particles are listed in Table 3. (b) The simulated scattering intensity $I (Q)$ vs $Q$ for the model system (sample 2) loaded with fluid of different SLDs, $ρ_{f}$ . The structure parameters of the four different kinds of particles are listed in Table 4. (c) $R (ρ_{f}) = \frac{C_{P} (ρ_{f})}{C_{P} (ρ_{f} = 0)}$ vs $ρ_{f}$ for the model system with $C_{P} (ρ_{f})$ as the Porod's constant obtained at guest fluid $SLD = ρ_{f}$ using Eq. (2) for sample 1. (d) $R (ρ_{f}) = \frac{C_{P} (ρ_{f})}{C_{P} (ρ_{f} = 0)}$ vs $ρ_{f}$ for the model system with $C_{P} (ρ_{f})$ as the Porod's constant obtained at guest fluid $SLD = ρ_{f}$ using Eq. (2) for sample 2.
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