About Nucleation and Growth

About Nucleation and Growth 150 150 Lyne Ismail

As a woman of science, trained to think logically and rationalizing through empirical data, immersing myself in the process of art making creates the in-between space that thrives in mystery and ambiguity. In my line of work, I see materials at their most basic form. The domain of material science deals with its physical form at its most fundamental level where unseen forces of repulsion and attraction play the main role. Things that can’t be seen by the naked eyes; things that are not there but always exist beyond physicality. I am used to looking at the character of a material and how form affect its bulk properties. Of how the basic properties can be engineered or re-engineered and further developed for enhancement in order to meet the ever-rising demand in applications.  The scientist in me sees materials like palette of colours similar to an artist. Just as the artist in me can create different paintings using different colours, the material scientist in me can create and improve upon different materials using different elements of the periodic tables with different synthesis and processes to create objects.

It then begs the question, how does one view materials and objects. Does the object comes first or the materials. If materials are like palette of colours waiting to be applied in different processes to make objects then what is material made of? In the materials science paradigm, structure–properties-processing relationships are connected in a triangulated shape of an interrelated form of action and consequences. Ultimately, the testing of these inter-dependent entities resulted in the performance of the materials [1]. These relationships can also depict the cohesive existence of each entity in oneself based on some assumptions. Lets say that the fundamental structure of a solid which is the periodic arrangement of atoms stacked in its lattice is the core value of a human being; the bulk properties is the inherent character or the inner strength of a person and processing of a material is akin to how life experiences shapes oneself. The result of this synergistic connection is portrayed by the performance of the materials or how one passed the life test.

In the world where materials possession defines our social relationships and economic quality of life, this triangle connection defines us. In a holistic living albeit idealistic, one achieves a balance in the triangle where the movement from one pointed end to the other flows through an inner circle that seamlessly connect all three forms. But in reality, as we move towards achieving each form in the triangle of structure-properties-processing, we tend to place different importance of one over the other. As we go further along, our structure shifted with the presence of kinks and defects in the arrangement of our core values. These imperfections in life made us who we are from the experiences and life challenges. But how do these experiences form a cohesion existence that is diffracted into fragments of different energy level? How do we arrange these events into a cohesive solid solution? Do these events substitute one of our values or do they occupy the interstitial spaces between these values?

In mapping out my journey in life with all its challenges and experiences, I see this as the nucleation and growth of inner strength that transformed my core values in a process I called as solid-state phase transformation. This transformational of phase in one’s life occurred whenever a major life changing experiences visited us. One can say that you can never be the same after it happened; these are the experiences that indelibly left a mark in our life. How it occurred and how we respond to it, may well be what shapes us today.

If I can relate these observations and experiences empirically and measure them systematically, I can come up with a specific depiction for my life journey. I may well be adopting John Locke’s empiricism epistemology, a branch in the philosophy of science, which emphasizes evidence, especially as discovered in experiments [2,3], where the fundamental part of making hypothesis and theories is that it must be tested against observations of the natural world rather than resting solely on a priori reasoning, intuition, or revelation. I can also think along the way of Aristotle’s theory of tabula rasa that evolved into an elaborate theory by Avicenna (Ibn Sina) who argued that”…human intellect at birth resembled a tabula rasa, a pure potentiality that is actualized through education and comes to know” and that knowledge is attained through “…empirical familiarity with objects in this world from which one abstracts universal concepts” which develops through a syllogistic method of reasoning; observations lead to propositional statements, which when compounded lead to further abstract concepts[4]

The modern treatment on the concept of tabula rasa by John Locke in his writings “An Essay Concerning Human Understanding; Book 1”, has outlined that our mind is like a clean slate and it can be educated using three distinct methods: the development of a healthy body; the formation of a virtuous character; and the choice of an appropriate academic curriculum [5].  Immanuel Kant in his quote: “Although all our knowledge begins with experience, it does not follow that it arises from experience” argued that even though all our knowledge comes from observations and experiences, these occurrences could also be constrained by the inherent structure of thoughts itself [6]. In my view, we are what we want to believe; human mind’s is wired to make only certain kind of observations, this “pick and choose method” is limited by logic rationality stemming from the part in our own defense mechanism protecting us from the unsavoury experiences.

Adopting this epistemology of thinking, I can therefore view my life journey as solid-state phase transformation. Without emphasizing on specific incidents but only major altering life events, the overall kinetics of transformation process depends on both nucleation[1] and growth.

If nucleation is assumed as my inherent character or internal force then growth is how this internal force developed in response to life challenges or application of heat/temperature (T)

If more nuclei are present at a particular temperature, growth occurs from a larger number of sites and the phase transformation is completed in a shorter period of time (t). At higher temperature, the diffusion coefficient[1] is higher, growth rates are more rapid, and again we expect the transformation in a shorter time, assuming an equal number of nuclei. Expressing the rate of transformation given by Avrami equation, where the fraction of transformed, f related to time, t by

f = 1 – exp (-ctⁿ)

where, c and n are constant for a particular temperature

[1] Nucleation – As in solidification, nucleation occurs most easily on surfaces already present in the structure, thereby minimizing the surface energy term. Thus, the precipitates heterogeneously nucleate most easily at grain boundaries and other defects.
[2] Diffusion Coefficient – The rate at which atoms, ions, particles or other species diffuse in a material can be measured by the flux, J. The flux J is defined as the number of atoms passing through a plane of unit area per unit  time. Fick’s first law explains the net flux of atoms:
J= -D dc/dx

Avrami relationship produces a sigmoidal, or S-shaped curve describing the solid state transformation process or how my life transformed when the external force/challenges first came.

An incubation time, t0 during where no observable transformation occurs, is the time required for nucleation. Initially, the transformation occurs slowly as nuclei formed. During this time, as the first initial setbacks came, I would be disconcerted and stand down to gather my thoughts, this incubation time gave me the space to feel my way, to think and set my strategies on how best to cope and manage it.

This is the time when I sourced my inner character deep in the recesses of myself while nuclei started to form in different sites. Once I have gathered all these little nuclei/inner force forming, incubation is followed by rapid growth as the atoms diffuse to the growing precipitate. This is the time where I see myself going into the eye of the storm and fought my way through as my inner character rapidly grew with time. Near the end of the transformation, the rate again slows down as the source of atoms available to diffuse to the growing precipitate is depleted.

Almost always, life throws you curve balls or challenges, which in this case, if I consider these curve balls as the effect of temperature as in many phase transformation. Nucleation and growth are temperature-dependent; the rate of phase transformation depends on the temperature gradient or the undercooling (ΔT). If the thermodynamic driving force is low, the undercooling is small, then nucleation rate slows down. This means, if the challenges that happened are not that significant, the growth rate of my inherent character will not be that fast. However, if it is one of a life altering experiences such as death or lost of someone I loved, the undercooling is very big and the thermodynamic driving force is also high, the system. Thus, the temperature dependent growth rate follows the Arrhenius relationship as shown in the equation below:

where,  Q is the activation energy or the barrier of grief that I need to overcome in order for phase transformation to occur, T is the temperature or the unwanted event that happened, R is the gas temperature or the environment that I am in and A is a constant entity in my life. All the factors in this equation are inter dependent on each other and one cannot exist in isolation. In this context, in order for me to increase the growth rate of my inherent character/strength when such incidences occurred, I need to be able to overcome the barrier of grief, Q in relation to my environment and the constant entities around me.

Solid-state transformations, which have profound effect on the structure and properties of materials, are greatly influenced by heat. The nucleation of inherent character and the rate of its growth are influenced by the initial response during incubation time.  During this period of time, incubation gives me the in-between space needed to reflect and slows down to seek my inner character in order to gather myself to face any outcomes and possibilities which can be depicted as sigmoidal curve using Avrami relationship. However, when there is an incident where temperature is not constant anymore, the rate of phase transformation depends on the total rate of nucleation and rate of growth, where the maximum transformation is at a critical temperature (Fig 4a). In my view, the critical temperature is when the challenges are at its peak and the entropy or chaos in one’s self is at its maximum. Consequently, if one is able to overcome this critical temperature or the so-called barrier of grief, phase transformation will reach its optimum value. However, if I want to measure how long will it take for me to overcome this barrier or to quantitatively describe the rate of a phase transformation, I can defined is as reciprocal of time for transformation to proceed halfway to completion:

Plotting this transformation of time vs. temperature results in a characteristic C-shaped curve as shown in Fig 4b.

In general, this treatment of measuring transformation starting with nucleation, then growth and to determine how nucleation and growth are affected by temperature can be applied to any other transformation. Such as during a creative process of art making, it started out as an idea or nuclei in different parts of our mind. Then these nuclei would rearranged and combined in a process called nucleation. How growth of ideas takes on a physical form and how fast do these growth needed to occur for phase transformation to reach its maximum and how long does it take. In addition, what if during the ideation process, certain challenges came in, how would these challenges affect the ideation itself?  In going back to the materials science paradigm of structure-properties-processing, we can see that all these entities are inter dependent and cannot exists in isolation. Ultimately, they are connected in a triangulated form that cohesively existed.

References:
[1] Askeland, Donald R., Fulay, Pradeep P. and  Wright, Wendelin J., The Science and Engineering of Materials, SI 6th Edition, Cengage Learning, 2011
[2] Psillos, Stathis; Curd, Martin (2010). The Routledge companion to philosophy of science (1. publ. in paperback ed.). London: Routledge. pp. 129–138. ISBN 978-0415546133
[3] The American International Encyclopedia, 9, New York: JJ Little Co, 1954.
[4] Sajjad H. Rizvi (2006), Avicenna/Ibn Sina (CA. 980–1037), Internet Encyclopedia of Philosophy
[5] Jolley, Nicholas. Locke: His Philosophical Thought. Oxford: Oxford University Press, 1999.
[6] Gardner, Sebastian:  Routledge Philosophy GuideBook to Kant and the Critique of Pure Reason: Routledge Philosophy GuideBooks, 1999
[7] Ibid [1], pg. 453
[8] Ibid [1], pg. 454