published_or_final_version / Physics / Doctoral / Doctor of Philosophy

Chapter 1 briefly introduces the theoretical background of how applications and lists some of resent research progresses in this area. Polymer with various configurations and conformations pass through nanopores; including polymer linear chains, stars polymer, branched polymers, polymer micelles are introduced. Among them, the de Gennes and Brochard-Wyart's predictions of polymer linear and star chains passing through nanopores are emphasized, in which they predicted that qc of linear chain is qc ≃ kBT/(3pieta), where kB, T and eta are the Boltzmann constant, the absolutely temperature, and the viscosity of solvent, respectively, independent of both the chain length and the pore size; and for star chains passing through nanopores, there exist a optimal entering arm numbers, namely, the star chains passing through nanopores. / Chapter 2 details basic theory of static and dynamic laser light scattering (LLS), including its instrumentation and our ultrafiltration setup. / Chapter 3 briefly introduces the sample preparation, including the history and mechanism of anionic living polymerization, as well as how we used a novel home-made set-up to prepare linear polystyrene with different chain lengths and star polystyrene with various arm numbers and lengths. / Chapter 4 summarizes our measured critical flow rates (qc) of linear polymer chains with different lengths for nanopores with different sizes, since the flow rate is directly related to the hydrodynamic force, we have developed a sensitive method (down to tens fN) to directly assess how much the hydrodynamic force (Fh) is required to overcome the weak entropy elasticity and stretch individual coiled chains in solution. Our method is completely different from the using existing optical tweezers or AFM, because they measure the relatively stronger enthalpy elasticity. Our results confirm that qc is indeed independent of the chain length, but decreases as the pore size increases. The value of qc is ∼10--200 times smaller than kBT/(3pieta). Such a discrepancy has been attributed to the rough assumption made by de Gennes and his coworkers; namely, each chain segment "blob" confined inside the pore is not a hard sphere so that the effective length along the flow direction is much longer than the pore diameter. Finally, using the solution temperature, we varied the chain conformation, our result shows that q c has a minimum which is near, but not exactly located at the theta temperature, might leading to a better way to determine the true ideal state of a polymer solution, at which all viral coefficients, not only the second vanish. / Chapter 5 uses polymer solutions made of different mixtures of linear and star chains, we have demonstrated that flushing these solution mixtures through a nanopore with a properly chosen flow rate can effectively and cleanly separate linear and star chains no matter whether linear chains are larger or smaller than star chains. / Chapter 6 further investigates how star-like polystyrene pass through a given nanopore under the flow field. Star polystyrene chains with different arm lengths (LA) and numbers (f) passing through a nanopore (20 nm) under an elongational flow field was investigated in terms of the flow-rate dependent relative retention ((C0 - C)/C0), where C 0 and C are the polymer concentrations before and after the ultrafiltration. Our results reveal that for a given arm length (LA), the critical flow rate (qc,star), below which star chains are blocked, dramatically increases with the total arm numbers (f); but for a given f, is nearly independent on LA, contradictory to the previous prediction made by de Gennes and Brochard-Wyart. We have revised their theory in the region fin < fout and also accounted for the effective length of each blob, where fin and fout are the numbers of arms inside and outside the pore, respectively. In the revision, we show that qc,star is indeed independent of LA but related to f and f in in two different ways, depending on whether fin ≤ f/2 or ≥ f/2. A comparison of our experimental and calculated results reveals that most of star chains pass through the nanopores with fin ∼ f/2. Further study of the temperature dependent (C0 - C)/C 0 of polystyrene in cyclohexane reveals that there exists a minimum of qc,star at ∼38 °C, close to its theta temperature (-34.5 °C). / This Ph. D. thesis presents our study on the ultrafiltration of polymers with different configurations and conformations; namly, theoretically, the passing of polymer chains through a nanopore under an elongational flow filed has been studied for years, but experimental studies are rare because of two following reasons: (1) lacks a precise method to investigate how individual single polymer chain pass through a nanopore; (2) it is difficult, if not impossible, to obtain a set of polymer samples with a narrow molar mass distribution and a uniform structures; except for linear chains. The central question in this study is to find the critical (minimum) flow rate (qc) for each kind of chains, at which the chains can pass through a given nanopore. A comparison of the measured and calculated qc leads to a better understanding how different chains are deformed, stretched and pulled through a nanopore. We have developed a novel method of combinating static and dynamic laser light scattering (LLS) to precisely measure the relative retention concentration ((C0 - C)/C0). / Ge, Hui. / Adviser: Chi Wu. / Source: Dissertation Abstracts International, Volume: 73-01, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

This thesis investigates the geometry of polymers and other miscellaneous results in the Kardar-Parisi-Zhang (KPZ) universality class. Directed polymers have enjoyed a rich history in both probability theory and mathematical physics and have connections to several families of statistical mechanical and random growth models that belong to the KPZ universality class [77]. In this thesis, we focus on 2 integrable polymer models, the (1+1)-dimensional continuum directed random polymer (CDRP) and the half-space log-gamma (HSLG) polymer, and study their path properties. For the CDRP, we show both of its superdiffusivity and localization features. Namely, the annealed law of polymer of length t, upon t²/³ superdiffusive scaling, is tight in the space of C( [0, 1])-valued random variables and the quenched law of any point distance pt from the origin on the path a point-to-point polymer (or the endpoint of a point-to-line polymer) concentrates in a O(1) window around a random favorite point Mp,t. The former marks the first pathwise tightness result for positive temperature models and the latter result confirms the “favorite region conjecture” for the CDRP. Moreover, we provide an explicit random density for the quenched distribution around the favorite point Mp,t. The proofs of both results utilize connections with the KPZ equation and our techniques also allow us to prove properties of the KPZ equation itself, such as ergodicity and limiting Bessel behaviors around the maximum. For the HSLG polymers, we combine our localization techniques from the CDRP and the recently developed HSLG line ensemble results [22, 27] with an innovative combinatorial argument to obtain its limiting quenched endpoint distribution from the diagonal in the boundphase (α < 0). This result proves Kardar’s “pinning” conjecture in the case of HSLG polymers[158]. Finally, this thesis also contains two separate works on the tightness of the Bernoulli Gibbsian line ensemble under mild conditions and the upper-tail large deviation principle (LDP) of the asymmetric simple exclusion process (ASEP) with step initial data. In the first work, we prove that under a mild but uniform control of the one-point marginals of the top curve of the line ensemble, i.e. the shape of the top curve as approximately an inverse parabola and asymptotically covering the entire real line after scaling and recentering, the sequence of line ensembles is tight. With a characterization of [109], our tightness result implies the convergence of the Bernoulli Gibbsian line ensemble to the parabolic Airy line ensemble if the top curve converges to the parabolic Airy2 process in the finite dimensional sense. Compared to a similar work of [93], our result applies to line ensembles with possibly random initial and terminal data, instead of a packed initial condition, and does not rely on exact formulas. In our work on the ASEP, we obtain the exact Lyapunov exponent for the height function of ASEP with step initial data and subsequently its upper-tail LDP, where the rate function matches with that of the TASEP given in a variational form in [156].

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In this thesis we investigate large deviation and path properties of a few models within the Kardar-Parisi-Zhang (KPZ) universality class. The KPZ equation is the central object in the KPZ universality class. It is a stochastic PDE describing various objects in statistical mechanics such as random interface growth, directed polymers, interacting particle systems. In the first project we study one point upper tail large deviations of the KPZ equation 𝜢(t,x) started from narrow wedge initial data. We obtain precise expression of the upper tail LDP in the long time regime for the KPZ equation. We then extend our techniques and methods to obtain upper tail LDP for the asymmetric exclusion process model, which is a prelimit of the KPZ equation. In the next direction, we investigate temporal path properties of the KPZ equation. We show that the upper and lower law of iterated logarithms for the rescaled KPZ temporal process occurs at a scale (log log 𝑡)²/³ and (log log 𝑡)¹/³ respectively. We also compute the exact Hausdorff dimension of the upper level sets of the solution, i.e., the set of times when the rescaled solution exceeds 𝛼(log log 𝑡)²/³. This has relevance from the point of view of fractal geometry of the KPZ equation. We next study superdiffusivity and localization features of the (1+1)-dimensional continuum directed random polymer whose free energy is given by the KPZ equation. We show that for a point-to-point polymer of length 𝑡 and any 𝑝 ⋲ (0,1), the point on the path which is 𝑝𝑡 distance away from the origin stays within a 𝑂(1) stochastic window around a random point 𝙈_𝑝,𝑡 that depends on the environment. This provides an affirmative case of the folklore `favorite region' conjecture. Furthermore, the quenched density of the point when centered around 𝙈_𝑝,𝑡 converges in law to an explicit random density function as 𝑡 → ∞ without any scaling. The limiting random density is proportional to 𝑒^{-𝓡(x)} where 𝓡(x) is a two-sided 3D Bessel process with diffusion coefficient 2. Our proof techniques also allow us to prove properties of the KPZ equation such as ergodicity and limiting Bessel behaviors around the maximum. In a follow up project, we show that the annealed law of polymer of length 𝑡, upon 𝑡²/³ superdiffusive scaling, is tight (as 𝑡 → ∞) in the space of 𝐶([0,1]) valued random variables. On the other hand, as 𝑡 → 0, under diffusive scaling, we show that the annealed law of the polymer converges to Brownian bridge. In the final part of this thesis, we focus on an integrable discrete half-space variant of the CDRP, called half-space log-gamma polymer.We consider the point-to-point log-gamma polymer of length 2𝑁 in a half-space with i.i.d.Gamma⁻¹(2𝛳) distributed bulk weights and i.i.d. Gamma⁻¹(𝛼+𝛳) distributed boundary weights for 𝛳 > 0 and 𝛼 > -𝛳. We establish the KPZ exponents (1/3 fluctuation and 2/3 transversal) for this model when 𝛼 ≥ 0. In particular, in this regime, we show that after appropriate centering, the free energy process with spatial coordinate scaled by 𝑁²/³ and fluctuations scaled by 𝑁¹/³ is tight. The primary technical contribution of our work is to construct the half-space log-gamma Gibbsian line ensemble and develop a toolbox for extracting tightness and absolute continuity results from minimal information about the top curve of such half-space line ensembles. This is the first study of half-space line ensembles. The 𝛼 ≥ 0 regime correspond to a polymer measure which is not pinned at the boundary. In a companion work, we investigate the 𝛼 < 0 setting. We show that in this case, the endpoint of the point-to-line polymer stays within 𝑂(1) window of the diagonal. We also show that the limiting quenched endpoint distribution of the polymer around the diagonal is given by a random probability mass function proportional to the exponential of a random walk with log-gamma type increments.